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%!PS-Adobe-2.0 %%Creator: dvipsk 5.86 p1.5d Copyright 1996-2001 ASCII Corp.(www-ptex@ascii.co.jp) %%based on dvipsk 5.86 Copyright 1999 Radical Eye Software (www.radicaleye.com) %%Title: book.dvi %%Pages: 34 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Helvetica %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -t letter -o book.ps book %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.09.30:1309 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: psfrag.pro %% %% This is file `psfrag.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% psfrag.dtx (with options: `filepro') %% %% Copyright (c) 1996 Craig Barratt, Michael C. 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b(of) i(sources) e(of) h(information,) g (including) h(amino) f(acid) g(sequence,) 714 2832 y(h) n(ydropath) n (y) e(pro\014les,) g(gene) h(expression) f(data,) h(kno) n(wn) g (protein-protein) f(in) n(teractions) h(and) 714 2940 y(kno) n(wn) e(protein) h(complexes.) f(W) -7 b(e) 29 b(sho) n(w) e(that) i(a) f(supp) r(ort) g(v) n(ector) f(mac) n(hine) h (\(SVM\)) h(trained) 714 3047 y(from) 22 b(all) g(of) g(these) g(data,) g(using) g(the) h(com) n(bined) f(k) n(ernel,) g(p) r(erforms) f (signi\014can) n(tly) h(b) r(etter) g(than) 714 3155 y(the) 28 b(same) f(algorithm) g(trained) h(on) f(an) n(y) h(single) f (t) n(yp) r(e) h(of) g(data,) g(and) f(b) r(etter) i(than) f (previously) 714 3263 y(describ) r(ed) f(approac) n(hes.) p Fv -34 3612 a(1.1) 101 b(In) m(tro) s(duction) p -34 3479 3736 3 v Fx 714 3827 a(Muc) n(h) 27 b(researc) n(h) e(in) j (computational) e(biology) g(in) n(v) n(olv) n(es) f(dra) n(wing) h (statistically) h(sound) g(infer-) 714 3935 y(ences) c(from) g (collections) f(of) h(data.) g(F) -7 b(or) 23 b(example,) g(the) h (function) g(of) f(an) g(unannotated) g(protein) 714 4043 y(sequence) 34 b(can) h(b) r(e) g(predicted) h(based) e(on) h(an) g (observ) n(ed) e(similarit) n(y) h(b) r(et) n(w) n(een) i(that) f (protein) 714 4151 y(sequence) 29 b(and) g(the) h(sequence) f(of) g(a) g (protein) g(of) g(kno) n(wn) g(function.) h(Related) g(metho) r (dologies) 714 4259 y(in) n(v) n(olv) n(e) 18 b(inferring) i(related) g (functions) g(of) g(t) n(w) n(o) g(proteins) g(if) g(they) h(o) r(ccur) e(in) i(fused) g(form) f(in) g(some) 714 4367 y(other) 25 b(organism,) g(if) h(they) g(co-o) r(ccur) f(in) h(m) n(ultiple) h(sp) r (ecies,) f(if) h(their) f(corresp) r(onding) e(mRNAs) 714 4475 y(share) i(similar) h(expression) f(patterns,) h(or) g(if) h(the) g (proteins) f(in) n(teract) g(with) h(one) f(another.) 797 4583 y(It) 46 b(seems) g(natural) f(that,) i(while) f(all) g(suc) n(h) g (data) f(sets) h(con) n(tain) g(imp) r(ortan) n(t) g(pieces) g(of) 714 4691 y(information) 32 b(ab) r(out) g(eac) n(h) g(gene) g(or) g (protein,) g(the) h(comparison) e(and) h(fusion) h(of) f(these) h(data) 714 4799 y(should) 25 b(pro) r(duce) f(a) h(m) n(uc) n(h) g(more) f (sophisticated) h(picture) g(of) g(the) h(relations) e(among) g (proteins,) 714 4907 y(and) c(a) g(more) g(detailed) h(represen) n (tation) e(of) h(eac) n(h) g(protein.) g(Esp) r(ecially) g(the) h (recen) n(t) f(a) n(v) -5 b(ailabilit) n(y) 714 5015 y(of) 26 b(m) n(ultiple) h(t) n(yp) r(es) g(of) f(genome-wide) f(data) h (that) h(pro) n(vide) f(biologists) f(with) i(complemen) n(tary) p 90 rotate dyy eop %%Page: 3 3 3 2 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.1) 78 b(Intr) l(o) l(duction) 3109 b(3) p Fx 945 349 a(views) 28 b(of) i(a) e(single) h(genome) f(highligh) n(ts) h (the) g(need) g(for) g(mac) n(hine) g(learning) f(algorithms) f(that) 945 457 y(unify) 20 b(these) g(views) f(and) g(exploit) h(this) g (fused) g(represen) n(tation.) e(Com) n(bining) h(information) g(from) 945 565 y(di\013eren) n(t) 24 b(sources) f(con) n(tributes) h(to) g (forming) g(a) g(complete) g(picture) g(of) h(the) g(relations) e(b) r (et) n(w) n(een) 945 673 y(the) 31 b(di\013eren) n(t) g(comp) r(onen) n (ts) g(of) g(a) g(genome,) f(enhancing) h(the) g(total) g(information) f (ab) r(out) h(the) 945 781 y(problem) c(at) g(hand.) 1028 889 y(In) g(y) n(east,) g(for) f(example,) h(for) g(a) g(giv) n(en) g (gene) f(w) n(e) h(t) n(ypically) g(kno) n(w) g(the) g(protein) g(it) h (enco) r(des,) 945 997 y(that) 35 b(protein's) f(similarit) n(y) h(to) f (other) h(proteins,) f(its) h(h) n(ydrophobicit) n(y) f(pro\014le,) g (the) i(mRNA) 945 1105 y(expression) e(lev) n(els) g(asso) r(ciated) g (with) i(the) g(giv) n(en) e(gene) h(under) g(h) n(undreds) g(of) g (exp) r(erimen) n(tal) 945 1213 y(conditions,) 29 b(the) i(o) r (ccurrences) d(of) i(kno) n(wn) f(or) g(inferred) h(transcription) f (factor) g(binding) h(sites) 945 1321 y(in) 36 b(the) g(upstream) g (region) f(of) h(that) g(gene,) f(the) i(iden) n(tities) f(of) g(man) n (y) g(of) g(the) g(proteins) f(that) 945 1429 y(in) n(teract) 29 b(with) h(the) g(giv) n(en) f(gene's) g(protein) g(pro) r(duct) h(or) f (form) g(a) g(complex) h(with) g(it.) g(Eac) n(h) f(of) 945 1536 y(these) e(distinct) h(data) f(t) n(yp) r(es) h(pro) n(vides) e (one) h(view) g(of) g(the) h(molecular) e(mac) n(hinery) g(of) i(the) g (cell.) 945 1644 y(In) g(the) h(near) e(future,) i(researc) n(h) d(in) j (bioinformatics) e(will) h(fo) r(cus) h(more) e(and) h(more) g(hea) n (vily) f(on) 945 1752 y(metho) r(ds) h(of) f(data) g(fusion.) 1028 1860 y(One) h(problem) g(with) i(this) f(approac) n(h,) d(ho) n(w) n (ev) n(er,) h(is) i(that) g(genomic) f(data) g(come) g(in) h(a) g(wide) 945 1968 y(v) -5 b(ariet) n(y) 36 b(of) h(data) f(formats:) g (expression) g(data) g(are) g(expressed) g(as) g(v) n(ectors) f(or) h (time) i(series;) 945 2076 y(protein) e(sequence) f(data) h(as) g (strings) f(from) h(a) g(20-sym) n(b) r(ol) e(alphab) r(et;) i(gene) g (sequences) g(are) 945 2184 y(strings) 30 b(from) h(a) f(di\013eren) n (t) h(\(4-sym) n(b) r(ol\)) g(alphab) r(et;) g(protein-protein) e(in) n (teractions) h(are) g(b) r(est) 945 2292 y(expressed) c(as) h(graphs,) f (and) i(so) f(on.) 1028 2400 y(This) h(c) n(hapter) g(presen) n(ts) f (a) h(computational) g(and) h(statistical) f(framew) n(ork) e(for) i (in) n(tegrating) 945 2508 y(heterogeneous) 40 b(descriptions) h(of) h (the) h(same) e(set) h(of) g(genes,) f(proteins) h(or) f(other) g(en) n (tities.) 945 2616 y(The) 36 b(approac) n(h) e(relies) h(on) g(the) i (use) e(of) h(k) n(ernel-based) e(statistical) h(learning) g(metho) r (ds) h(that) 945 2724 y(ha) n(v) n(e) 41 b(already) f(pro) n(v) n(en) h (to) h(b) r(e) g(v) n(ery) f(useful) i(to) r(ols) e(in) i (bioinformatics) e(\(Jaakk) n(ola) e(et) k(al.,) 945 2832 y(1999;) 36 b(Bro) n(wn) i(et) h(al.,) f(2000;) f(F) -7 b(urey) 38 b(et) h(al.,) f(2000;) f(Zien) h(et) h(al.,) f(2000\).) f (These) h(metho) r(ds) 945 2940 y(represen) n(t) 22 b(the) i(data) e(b) n(y) h(means) g(of) g(a) g(k) n(ernel) g(function,) h(whic) n(h) f (de\014nes) g(similarities) g(b) r(et) n(w) n(een) 945 3047 y(pairs) 44 b(of) h(genes,) f(proteins,) h(etc.) g(Suc) n(h) g (similarities) g(can) f(b) r(e) i(quite) f(complex) g(relations,) 945 3155 y(implicitly) 40 b(capturing) f(asp) r(ects) h(of) g(the) g (underlying) f(biological) g(mac) n(hinery) -7 b(.) 39 b(One) g(reason) 945 3263 y(for) d(the) i(success) e(of) i(k) n(ernel) e (metho) r(ds) h(is) g(that) h(the) f(k) n(ernel) g(function) h(tak) n (es) e(relationships) 945 3371 y(that) 27 b(are) g(implicit) h(in) f (the) h(data) f(and) g(mak) n(es) f(them) i(explicit,) g(so) e(that) i (it) g(is) f(easier) f(to) h(detect) 945 3479 y(patterns.) j(Eac) n(h) f (k) n(ernel) h(function) h(th) n(us) g(extracts) e(a) h(sp) r(eci\014c) h(t) n(yp) r(e) g(of) f(information) g(from) g(a) 945 3587 y(giv) n(en) 35 b(data) h(set,) g(thereb) n(y) g(pro) n(viding) f (a) h(partial) f(description) h(or) g(view) g(of) g(the) h(data.) f (The) 945 3695 y(goal) 31 b(of) h(this) h(c) n(hapter) e(is) h(to) g (\014nd) h(a) f(k) n(ernel) f(that) i(b) r(est) g(represen) n(ts) d (all) i(of) h(the) f(information) 945 3803 y(a) n(v) -5 b(ailable) 33 b(for) h(a) g(giv) n(en) g(statistical) g(learning) g (task.) g(Giv) n(en) g(man) n(y) g(partial) g(descriptions) g(of) 945 3911 y(the) h(data,) f(w) n(e) g(solv) n(e) g(the) h(mathematical) f (problem) g(of) g(com) n(bining) g(them) i(using) e(a) g(con) n(v) n (ex) 945 4019 y(optimization) 21 b(metho) r(d) h(kno) n(wn) e(as) h (semide\014nite) h(programming) d(\(SDP\)) j(\(Bo) n(yd) f(et) g(al.,) g (1994;) 945 4127 y(Nestero) n(v) 27 b(and) i(Nemiro) n(vsky,) f(1994;) f (V) -7 b(anden) n(b) r(erghe) 28 b(and) h(Bo) n(yd,) f(1996\).) f(This) i(SDP-based) 945 4235 y(approac) n(h) 22 b(\(Lanc) n(kriet) h(et) h (al.,) g(2002\)) e(yields) i(a) g(general) e(metho) r(dology) h(for) h (com) n(bining) f(man) n(y) 945 4343 y(partial) 31 b(descriptions) g (of) h(data) f(that) h(is) g(statistically) f(sound,) h(as) f(w) n(ell) g(as) g(computationally) 945 4451 y(e\016cien) n(t) c(and) h(robust.) 1028 4558 y(In) i(order) f(to) g(demonstrate) h(the) g(feasibilit) n(y) g(of) g(these) g(metho) r(ds,) g(w) n(e) g(describ) r(e) f(t) n(w) n(o) h(prob-) 945 4666 y(lems:) g(iden) n(tifying) h(mem) n(brane) e (proteins) h(in) g(y) n(east) g(and) g(predicting) g(the) g(function) h (of) g(y) n(east) 945 4774 y(proteins.) c(Both) h(problems) f(are) g (statistical) g(learning) g(problems) g(in) i(whic) n(h) f(a) f(single) h(t) n(yp) r(e) g(of) 945 4882 y(feature) 35 b(deriv) n(ed) g(from) h (the) g(protein) g(sequence) f(pro) n(vides) f(only) i(partial) f (information.) g(W) -7 b(e) 945 4990 y(demonstrate) 25 b(that) i(incorp) r(orating) e(kno) n(wledge) g(deriv) n(ed) h(from) g (the) h(amino) f(acid) g(sequences,) p 90 rotate dyy eop %%Page: 4 4 4 3 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(4) 709 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Fx 714 349 a(protein) 19 b(complex) h(data,) f(h) n(ydropath) n(y) g (pro\014les,) g(gene) g(expression) g(data) g(and) h(kno) n(wn) f (protein-) 714 457 y(protein) g(in) n(teractions) g(signi\014can) n (tly) h(impro) n(v) n(es) e(classi\014cation) h(p) r(erformance) g (relativ) n(e) g(to) h(pre-) 714 565 y(viously) 30 b(describ) r(ed) i (metho) r(ds) g(and) f(relativ) n(e) g(to) g(our) g(metho) r(d) h (trained) f(on) g(an) n(y) g(single) g(t) n(yp) r(e) 714 673 y(of) c(data.) 797 781 y(W) -7 b(e) 22 b(b) r(egin) f(b) n(y) g (describing) g(related) g(w) n(ork.) f(Afterw) n(ards,) h(the) h(main) f (ideas) g(of) h(the) g(k) n(ernel) e(ap-) 714 889 y(proac) n(h) 25 b(to) h(pattern) g(analysis) f(are) h(explained) g(and) g (semide\014nite) h(programming) e(tec) n(hniques) 714 997 y(are) 33 b(in) n(tro) r(duced) h(as) g(an) g(adv) -5 b(anced) 34 b(instance) h(of) f(con) n(v) n(ex) f(optimization.) i (After) f(presen) n(ting) 714 1105 y(the) 28 b(necessary) f (mathematical) h(bac) n(kground,) e(w) n(e) i(describ) r(e) g(ho) n(w) g (di\013eren) n(t) g(k) n(ernels) f(de\014ned) 714 1213 y(on) j(di\013eren) n(t) h(data) f(can) g(b) r(e) h(in) n(tegrated) f (using) g(semide\014nite) h(programming) d(tec) n(hniques) j(to) 714 1321 y(pro) n(vide) 23 b(a) h(uni\014ed) h(description.) f(Finally) -7 b(,) 24 b(w) n(e) g(describ) r(e) g(the) h(t) n(w) n(o) e(biological) g (applications) h(of) 714 1429 y(mem) n(brane) i(protein) i(recognition) e(and) h(protein) g(function) i(prediction) e(in) h(y) n(east.) p Fv -34 1777 a(1.2) 101 b(Related) 34 b(W) -9 b(ork) p -34 1644 3736 3 v Fx 714 1993 a(Considerable) 32 b(w) n(ork) g(has) h (b) r(een) g(dev) n(oted) g(to) g(the) h(problem) f(of) h (automatically) e(in) n(tegrating) 714 2101 y(genomic) 40 b(datasets,) f(lev) n(eraging) g(the) i(in) n(teractions) e(and) i (correlations) d(b) r(et) n(w) n(een) i(them) i(to) 714 2209 y(obtain) 37 b(more) g(re\014ned) g(and) g(higher-lev) n(el) f (information.) h(Previous) f(researc) n(h) g(in) h(this) h(\014eld) 714 2316 y(can) 27 b(b) r(e) h(divided) g(in) n(to) f(three) g(classes) g (of) g(metho) r(ds.) 797 2424 y(The) 18 b(\014rst) h(class) e(treats) h (eac) n(h) g(data) g(t) n(yp) r(e) h(indep) r(enden) n(tly) -7 b(.) 20 b(Inferences) e(are) g(made) g(separately) 714 2532 y(from) 28 b(eac) n(h) g(data) g(t) n(yp) r(e,) h(and) g(an) f (inference) h(is) g(deemed) g(correct) e(if) i(the) g(v) -5 b(arious) 28 b(data) g(agree.) 714 2640 y(This) i(t) n(yp) r(e) h(of) f (analysis) f(has) h(b) r(een) h(used) g(to) f(v) -5 b(alidate,) 31 b(for) e(example,) i(gene) f(expression) f(and) 714 2748 y(protein-protein) j(in) n(teraction) g(data) h(\(Ge) h(et) g(al.,) f (2001;) f(Grigoriev,) f(2001;) h(Mro) n(wk) -5 b(a) 32 b(et) i(al.,) 714 2856 y(2003\),) 19 b(to) h(v) -5 b(alidate) 20 b(protein-protein) f(in) n(teractions) g(predicted) i(using) f(\014v) n (e) g(di\013eren) n(t) g(metho) r(ds) 714 2964 y(\(v) n(on) 29 b(Mering) g(et) h(al.,) f(2002\),) f(and) i(to) f(infer) h(protein) f (function) i(\(Marcotte) e(et) h(al.,) f(1999\).) f(A) 714 3072 y(sligh) n(tly) 21 b(more) g(complex) h(approac) n(h) e(com) n (bines) h(m) n(ultiple) h(data) g(sets) f(using) h(in) n(tersections) f (and) 714 3180 y(unions) 27 b(of) h(the) g(o) n(v) n(erlapping) d(sets) i(of) h(predictions) f(\(Jansen) g(et) h(al.,) f(2002\).) 797 3288 y(The) 47 b(second) f(formalism) h(to) f(represen) n(t) g (heterogeneous) f(data) i(is) g(to) g(extract) f(binary) 714 3396 y(relations) 29 b(b) r(et) n(w) n(een) i(genes) f(from) h(eac) n (h) f(data) g(source,) g(and) h(represen) n(t) e(them) j(as) e(graphs.) f(As) 714 3504 y(an) g(example,) f(sequence) h(similarit) n(y) -7 b(,) 29 b(protein-protein) f(in) n(teraction,) g(gene) h(co-expression) d(or) 714 3612 y(closeness) 32 b(in) i(a) g(metab) r(olic) g(path) n(w) n(a) n(y) e(can) i(b) r(e) g(used) g(to) g(de\014ne) g(binary) f (relations) g(b) r(et) n(w) n(een) 714 3720 y(genes.) c(Sev) n(eral) g (groups) g(ha) n(v) n(e) h(attempted) h(to) f(compare) f(the) i (resulting) f(gene) g(graphs) f(using) 714 3827 y(graph) c(algorithms) h (\(Nak) -5 b(a) n(y) n(a) 25 b(et) j(al.,) e(2001;) g(T) -7 b(ana) n(y) 25 b(et) i(al.,) g(2002\),) f(in) h(particular) e(to) i (extract) 714 3935 y(clusters) g(of) g(genes) g(that) h(share) e (similarities) h(with) h(resp) r(ect) g(to) f(di\013eren) n(t) h(sorts) e(of) i(data.) 797 4043 y(The) g(third) g(class) g(of) g(tec) n (hniques) g(uses) g(statistical) g(metho) r(ds) g(to) h(com) n(bine) e (heterogeneous) 714 4151 y(data.) c(F) -7 b(or) 23 b(example,) h (Holmes) g(and) g(Bruno) f(use) g(a) h(join) n(t) g(lik) n(eliho) r(o) r (d) g(mo) r(del) g(to) f(com) n(bine) h(gene) 714 4259 y(expression) c(and) i(upstream) f(sequence) g(data) h(for) f (\014nding) h(signi\014can) n(t) f(gene) g(clusters) g(\(Holmes) 714 4367 y(and) i(Bruno,) f(2000\).) g(Similarly) -7 b(,) 23 b(Deng) g(et) h(al.) f(\(2003b\)) f(use) h(a) g(maxim) n(um) g(lik) n (eliho) r(o) r(d) g(metho) r(d) 714 4475 y(to) 39 b(predict) g (protein-protein) f(in) n(teractions) g(and) g(protein) h(function) h (from) f(three) f(t) n(yp) r(es) i(of) 714 4583 y(data.) 30 b(Alternativ) n(ely) -7 b(,) 31 b(protein) f(lo) r(calization) g(can) h (b) r(e) g(predicted) g(b) n(y) g(con) n(v) n(erting) e(eac) n(h) h (data) 714 4691 y(source) 25 b(in) n(to) i(a) f(conditional) g (probabilistic) g(mo) r(del) h(and) g(in) n(tegrating) e(via) h(Ba) n (y) n(esian) f(calculus) 714 4799 y(\(Dra) n(wid) 36 b(and) g(Gerstein,) g(2000\).) f(The) h(general) f(formalism) h(of) g (graphical) f(mo) r(dels,) i(whic) n(h) 714 4907 y(includes) 29 b(Ba) n(y) n(esian) e(net) n(w) n(orks) h(and) h(Mark) n(o) n(v) e (random) i(\014elds) g(as) g(sp) r(ecial) g(cases,) f(pro) n(vides) g (a) 714 5015 y(systematic) 34 b(metho) r(dology) g(for) g(building) h (suc) n(h) g(in) n(tegrated) f(probabilistic) g(mo) r(dels.) h(As) g (an) p 90 rotate dyy eop %%Page: 5 5 5 4 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.3) 78 b(Kernel) 28 b(Metho) l(ds) 2991 b(5) p Fx 945 349 a(instance) 24 b(of) g(this) g(metho) r(dology) -7 b(,) 23 b(Deng) h(et) h(al.) f(\(2003a\)) e(dev) n(elop) r(ed) i(a) f (Mark) n(o) n(v) f(random) h(\014eld) 945 457 y(mo) r(del) e(to) g (predict) g(y) n(east) g(protein) f(function.) i(They) f(found) h(that) f(the) h(use) f(of) g(di\013eren) n(t) g(sources) 945 565 y(of) 29 b(information) f(indeed) i(impro) n(v) n(ed) d(prediction) i(accuracy) e(when) j(compared) d(to) i(using) g(only) 945 673 y(one) e(t) n(yp) r(e) h(of) f(data.) 1028 781 y(This) 36 b(c) n(hapter) f(describ) r(es) g(a) h(fourth) g(t) n(yp) r(e) g(of) g (data) g(fusion) g(tec) n(hnique,) g(also) f(statistical,) 945 889 y(but) 41 b(of) f(a) g(more) f(nonparametric) g(and) h (discriminativ) n(e) f(\015a) n(v) n(or.) g(The) h(metho) r(d,) h (describ) r(ed) 945 997 y(in) j(detail) g(b) r(elo) n(w,) f(consists) h (of) f(represen) n(ting) g(eac) n(h) g(t) n(yp) r(e) h(of) g(data) f (indep) r(enden) n(tly) i(as) e(a) 945 1105 y(matrix) d(of) h(k) n (ernel) g(similarit) n(y) f(v) -5 b(alues.) 41 b(These) g(k) n(ernel) f (matrices) g(are) g(then) i(com) n(bined) f(to) 945 1213 y(mak) n(e) 29 b(o) n(v) n(erall) e(predictions.) j(An) g(early) f (example) g(of) h(this) g(approac) n(h,) e(based) h(on) h(\014xed) g (sums) 945 1321 y(of) k(k) n(ernel) f(matrices,) h(sho) n(w) n(ed) f (that) h(com) n(binations) g(of) g(k) n(ernels) f(can) h(yield) g (impro) n(v) n(ed) f(gene) 945 1429 y(classi\014cation) 26 b(p) r(erformance) g(in) h(y) n(east,) g(relativ) n(e) f(to) h (learning) f(from) h(a) g(single) g(k) n(ernel) f(matrix) 945 1536 y(\(P) n(a) n(vlidis) f(et) i(al.,) f(2001\).) f(The) h(curren) n (t) g(w) n(ork) f(tak) n(es) h(this) g(metho) r(dology) g(further|w) n (e) g(use) g(a) p Fy 945 1644 a(weighte) l(d) p Fx 33 w(linear) c(com) n(bination) f(of) i(k) n(ernels,) f(and) g (demonstrate) g(ho) n(w) g(to) h(estimate) f(the) i(k) n(ernel) 945 1752 y(w) n(eigh) n(ts) 32 b(from) g(the) h(data.) f(This) h(yields) g (not) f(only) h(predictions) f(that) h(re\015ect) f(con) n(tributions) 945 1860 y(from) 25 b(m) n(ultiple) h(data) e(sources,) g(but) i(also) e (yields) h(an) g(indication) g(of) g(the) h(relativ) n(e) e(imp) r (ortance) 945 1968 y(of) j(these) h(sources.) 1028 2076 y(The) 44 b(graphical) e(mo) r(del) i(formalism,) g(as) f (exempli\014ed) h(b) n(y) g(the) h(Mark) n(o) n(v) c(random) i(\014eld) 945 2184 y(mo) r(del) 33 b(of) g(Deng) g(et) g(al.) g(\(2003a\),) e (has) i(sev) n(eral) e(adv) -5 b(an) n(tages) 31 b(in) j(the) f (biological) e(setting.) i(In) 945 2292 y(particular,) 39 b(prior) h(kno) n(wledge) f(can) h(b) r(e) i(readily) d(incorp) r (orated) h(in) n(to) g(suc) n(h) g(mo) r(dels,) h(with) 945 2400 y(standard) 28 b(Ba) n(y) n(esian) g(inference) h(algorithms) f(a) n(v) -5 b(ailable) 28 b(to) i(com) n(bine) f(suc) n(h) g(kno) n(wledge) f(with) 945 2508 y(data.) 36 b(Moreo) n(v) n(er,) f(the) i(mo) r(dels) g (are) f(\015exible,) i(accommo) r(dating) d(a) i(v) -5 b(ariet) n(y) 36 b(of) h(data) g(t) n(yp) r(es) 945 2616 y(and) c(pro) n(viding) g(a) g(mo) r(dular) g(approac) n(h) f(to) i (com) n(bining) f(m) n(ultiple) h(data) g(sources.) e(Classical) 945 2724 y(discriminativ) n(e) 46 b(statistical) h(approac) n(hes,) e(on) i (the) g(other) g(hand,) g(can) g(pro) n(vide) f(sup) r(erior) 945 2832 y(p) r(erformance) 30 b(in) h(simple) g(situations,) f(b) n(y) g (fo) r(cusing) h(explicitly) g(on) f(the) i(b) r(oundary) e(b) r(et) n (w) n(een) 945 2940 y(classes,) 46 b(but) j(tend) f(to) g(b) r(e) g (signi\014can) n(tly) f(less) g(\015exible) h(and) g(less) f(able) h (to) f(incorp) r(orate) 945 3047 y(prior) 34 b(kno) n(wledge.) g(As) i (w) n(e) f(discuss) g(in) g(this) h(c) n(hapter,) f(ho) n(w) n(ev) n (er,) e(recen) n(t) i(dev) n(elopmen) n(ts) f(in) 945 3155 y(k) n(ernel) 26 b(metho) r(ds) h(ha) n(v) n(e) f(yielded) h(a) f (general) f(class) h(of) h(discriminativ) n(e) f(metho) r(ds) h(that) h (readily) 945 3263 y(accommo) r(date) 33 b(non-standard) f(data) i(t) n (yp) r(es) g(\(suc) n(h) g(as) f(strings,) g(trees) h(and) g(graphs\),) f(allo) n(w) 945 3371 y(prior) 18 b(kno) n(wledge) g(to) i(b) r(e) g (brough) n(t) e(to) i(b) r(ear,) f(and) g(pro) n(vide) g(general) f (mac) n(hinery) g(for) h(com) n(bining) 945 3479 y(m) n(ultiple) 28 b(data) f(sources.) p Fv 198 3827 a(1.3) 100 b(Kernel) 35 b(Metho) s(ds) p 198 3695 3736 3 v Fx 945 4043 a(Kernel) 24 b(metho) r(ds) h(w) n(ork) f(b) n(y) h(em) n(b) r(edding) g(data) f (items) i(\(genes,) e(proteins,) h(etc.\)) g(in) n(to) g(a) g(v) n (ector) 945 4151 y(space) p Fu 30 w(F) p Fx 8 w(,) 31 b(called) f(a) p Fy 30 w(fe) l(atur) l(e) j(sp) l(ac) l(e) p Fx(,) e(and) g(searc) n(hing) e(for) h(linear) g(relations) g(in) h (suc) n(h) f(a) g(space.) 945 4259 y(This) 24 b(em) n(b) r(edding) h (is) f(de\014ned) g(implicitly) -7 b(,) 26 b(b) n(y) e(sp) r(ecifying) g (an) g(inner) g(pro) r(duct) g(for) g(the) h(feature) 945 4367 y(space) d(via) h(a) f(p) r(ositiv) n(e) h(semide\014nite) p Fy 24 w(kernel) j(function) p Fx 6 w(:) p Ft 23 w(k) p Fx 3 w(\() p Fs(x) p Fr 2835 4379 a(1) p Ft 2872 4367 a(;) p Fs 14 w(x) p Fr 2959 4379 a(2) p Fx 2997 4367 a(\)) d(=) p Fu 23 w(h) p Fx(\010\() p Fs(x) p Fr 3314 4379 a(1) p Fx 3352 4367 a(\),) g(\010\() p Fs(x) p Fr 3572 4379 a(2) p Fx 3610 4367 a(\)) p Fu(i) p Fx(,) h(where) 945 4475 y(\010\() p Fs(x) p Fr 1087 4487 a(1) p Fx 1125 4475 a(\)) 38 b(and) f(\010\() p Fs(x) p Fr 1508 4487 a(2) p Fx 1546 4475 a(\)) h(are) e(the) i(em) n(b) r(eddings) f(of) h (data) f(items) p Fs 38 w(x) p Fr 2965 4487 a(1) p Fx 3040 4475 a(and) p Fs 37 w(x) p Fr 3261 4487 a(2) p Fx 3299 4475 a(.) h(Note) f(that) h(if) g(all) 945 4583 y(w) n(e) c(require) g(in) h(order) f(to) h(\014nd) g(those) g(linear) f (relations) g(are) g(inner) g(pro) r(ducts,) h(then) g(w) n(e) g(do) 945 4691 y(not) k(need) h(to) f(ha) n(v) n(e) g(an) g(explicit) h(represen) n(tation) e(of) h(the) h(mapping) f(\010,) h(nor) f(do) g(w) n(e) g(ev) n(en) 945 4799 y(need) 28 b(to) g(kno) n(w) f(the) h(nature) f(of) h (the) h(feature) e(space.) g(It) i(su\016ces) e(to) h(b) r(e) g(able) g (to) g(ev) -5 b(aluate) 27 b(the) 945 4907 y(k) n(ernel) h(function,) h (whic) n(h) g(is) f(often) h(m) n(uc) n(h) g(easier) e(than) i (computing) g(the) g(co) r(ordinates) f(of) g(the) 945 5015 y(p) r(oin) n(ts) 21 b(explicitly) -7 b(.) 21 b(Ev) -5 b(aluating) 20 b(the) i(k) n(ernel) e(on) h(all) f(pairs) g(of) h(data) g(items) g(yields) g(a) g(symmetric,) p 90 rotate dyy eop %%Page: 6 6 6 5 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(6) 709 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Fx 714 349 a(p) r(ositiv) n(e) e(semide\014nite) h(matrix) p Ft 27 w(K) p Fx 32 w(kno) n(wn) f(as) g(the) p Fy 28 w(kernel) j(matrix) p Fx(,) e(whic) n(h) g(can) f(b) r(e) h(regarded) 714 457 y(as) g(a) g(matrix) g(of) g(generalized) g(similarit) n(y) f (measures) h(among) f(the) i(data) f(p) r(oin) n(ts.) 797 565 y(The) 40 b(k) n(ernel-based) e(binary) h(classi\014cation) g (algorithm) f(that) j(w) n(e) e(will) h(describ) r(e) g(in) g(this) 714 673 y(c) n(hapter,) i(the) p Fy 45 w(1-norm) i(soft) h(mar) l(gin) g (supp) l(ort) g(ve) l(ctor) f(machine) p Fx 52 w(\(SVM\)) g(\(Boser) f (et) h(al.,) 714 781 y(1992;) 24 b(Sc) n(h\177) -42 b(olk) n(opf) 26 b(and) g(Smola,) g(2002\),) e(forms) i(a) g(linear) g(discriminan) n(t) g(b) r(oundary) f(in) i(feature) 714 889 y(space) p Fu 33 w(F) p Fx 8 w(,) p Ft 33 w(f) p Fx 9 w(\() p Fs(x) p Fx(\)) 34 b(=) p Fs 32 w(w) p Fq 1431 859 a(T) p Fx 1484 889 a(\010\() p Fs(x) p Fx(\)) 23 b(+) p Ft 22 w(b) p Fx(,) 34 b(where) p Fs 33 w(w) p Fu 34 w(2) g(F) p Fx 41 w(and) p Ft 34 w(b) p Fu 32 w(2) p Fp 34 w(R) p Fx(.) 40 b(Giv) n(en) 33 b(a) g(lab) r(eled) h(sample) p Ft 714 997 a(S) p Fq 765 1009 a(n) p Fx 854 997 a(=) p Fu 44 w(f) p Fx(\() p Fs(x) p Fr 1087 1009 a(1) p Ft 1125 997 a(;) 14 b(y) p Fr 1203 1009 a(1) p Fx 1239 997 a(\)) p Ft(;) g(:) g(:) g(:) g(;) p Fx 14 w(\() p Fs(x) p Fq 1538 1009 a(n) p Ft 1584 997 a(;) g(y) p Fq 1662 1009 a(n) p Fx 1707 997 a(\)) p Fu(g) p Fx(,) p Fs 40 w(w) p Fx 42 w(and) p Ft 40 w(b) p Fx 40 w(are) 40 b(optimized) h(to) f (maximize) g(the) h(distance) 714 1105 y(\(\\margin"\)) j(b) r(et) n(w) n(een) h(the) h(p) r(ositiv) n(e) e(and) i(negativ) n(e) e(class,) g (allo) n(wing) g(misclassi\014cations) 714 1213 y(\(therefore) 27 b(\\soft) g(margin"\):) 923 1429 y(min) p Fo 908 1500 a(w) p Fq 1 w(;b;) p Fn(\030) p Fs 1242 1429 a(w) p Fq 1312 1394 a(T) p Fs 1365 1429 a(w) p Fx 20 w(+) p Ft 18 w(C) p Fq 1655 1325 a(n) p Fm 1616 1350 a(X) p Fq 1622 1527 a(i) p Fr(=1) p Ft 1749 1429 a(\030) p Fq 1785 1441 a(i) p Fx 3532 1429 a(\(1.1\)) 714 1634 y(sub) 5 b(ject) 27 b(to) p Ft 166 w(y) p Fq 1283 1646 a(i) p Fx 1311 1634 a(\() p Fs(w) p Fq 1413 1600 a(T) p Fx 1465 1634 a(\010\() p Fs(x) p Fq 1607 1646 a(i) p Fx 1636 1634 a(\)) 18 b(+) p Ft 18 w(b) p Fx(\)) p Fu 23 w(\025) p Fx 23 w(1) p Fu 18 w(\000) p Ft 18 w(\030) p Fq 2127 1646 a(i) p Ft 2155 1634 a(;) 69 b(i) p Fx 23 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(n) 1242 1767 y(\030) p Fq 1278 1779 a(i) p Fu 1329 1767 a(\025) p Fx 23 w(0) p Ft(;) 69 b(i) p Fx 22 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(n) p Fx 714 1933 a(where) p Ft 24 w(C) p Fx 32 w(is) 24 b(a) h (regularization) e(parameter,) g(trading) i(o\013) g(error) e(against) g (margin.) i(By) f(consid-) 714 2041 y(ering) 30 b(the) h(corresp) r (onding) e(dual) i(problem) g(of) g(\(1.1\),) g(one) g(can) f(pro) n(v) n(e) g(\(see,) h(e.g.,) g(Sc) n(h\177) -42 b(olk) n(opf) 714 2149 y(and) 26 b(Smola,) g(2002\)) e(that) j(the) g(w) n(eigh) n(t) e (v) n(ector) g(can) h(b) r(e) h(expressed) e(as) p Fs 26 w(w) p Fx 24 w(=) p Fm 3115 2086 a(P) p Fq 3202 2107 a(n) 3202 2173 y(i) p Fr(=1) p Ft 3328 2149 a(\013) p Fq 3381 2161 a(i) p Ft 3408 2149 a(y) p Fq 3449 2161 a(i) p Fx 3477 2149 a(\010\() p Fs(x) p Fq 3619 2161 a(i) p Fx 3647 2149 a(\),) 714 2256 y(where) k(the) i(supp) r(ort) f(v) -5 b(alues) p Ft 30 w(\013) p Fq 1715 2268 a(i) p Fx 1773 2256 a(are) 30 b(solutions) f(of) h(the) h(follo) n(wing) e(dual) p Fy 31 w(quadr) l(atic) k(pr) l(o) l(gr) l(am) p Fx 714 2364 a(\(QP\):) 714 2530 y(max) p Fn 759 2581 a(\013) p Fx 1034 2530 a(2) p Fn(\013) p Fq 1139 2496 a(T) p Fs 1191 2530 a(e) p Fu 18 w(\000) p Fn 18 w(\013) p Fq 1400 2496 a(T) p Fx 1452 2530 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 52 w(:) p Ft 51 w(C) p Fu 29 w(\025) p Fn 23 w(\013) p Fu 23 w(\025) p Fx 23 w(0) p Ft(;) p Fn 59 w(\013) p Fq 2790 2496 a(T) p Fs 2843 2530 a(y) p Fx 25 w(=) 22 b(0) p Ft(:) p Fx 462 w(\(1.2\)) 797 2705 y(The) 34 b(\014rst) f(stage) g(of) h(pro) r(cessing) e(in) i(a) f(k) n (ernel) g(metho) r(d) i(is) e(th) n(us) h(to) g(reduce) f(the) h(data) g (b) n(y) 714 2813 y(computing) f(the) i(k) n(ernel) e(matrix.) g(Giv) n (en) h(this) g(matrix,) f(and) h(giv) n(en) f(the) h(lab) r(els) p Ft 34 w(y) p Fq 3365 2825 a(i) p Fx 3392 2813 a(,) g(w) n(e) g(can) 714 2921 y(thro) n(w) 23 b(a) n(w) n(a) n(y) f(the) j(original) d(data;) i (the) g(problem) g(of) g(\014tting) h(the) f(SVM) h(to) f(data) f (reduces) h(to) g(an) 714 3029 y(optimization) k(pro) r(cedure) h(that) g(is) g(based) g(en) n(tirely) f(on) h(the) g(k) n(ernel) g(matrix) f (and) h(the) h(lab) r(els.) 714 3137 y(Di\013eren) n(t) i(k) n(ernels) f (corresp) r(ond) f(to) i(di\013eren) n(t) g(em) n(b) r(eddings) g(of) g (the) h(data) e(and) h(th) n(us) g(can) g(b) r(e) 714 3245 y(view) n(ed) 26 b(as) h(capturing) f(di\013eren) n(t) h(notions) g (of) g(similarit) n(y) -7 b(.) 26 b(F) -7 b(or) 26 b(example,) h(in) g (a) g(space) f(deriv) n(ed) 714 3353 y(from) 20 b(amino) h(acid) f (sequences,) h(t) n(w) n(o) f(genes) g(that) h(are) f(close) g(to) h (one) g(another) f(will) h(ha) n(v) n(e) f(protein) 714 3461 y(pro) r(ducts) 28 b(with) h(v) n(ery) f(similar) g(amino) g(acid) g(sequences.) g(This) h(amino) f(acid) g(space) g(w) n(ould) g(b) r(e) 714 3569 y(quite) 21 b(di\013eren) n(t) g(from) f(a) h(space) f(deriv) n (ed) g(from) h(microarra) n(y) d(gene) i(expression) g(measuremen) n (ts,) 714 3677 y(in) 28 b(whic) n(h) f(closeness) g(w) n(ould) g (indicate) h(similarit) n(y) f(of) h(the) g(expression) e(pro\014les) h (of) h(the) g(genes.) 714 3785 y(Finally) -7 b(,) 38 b(an) h(unlab) r(eled) g(data) f(item) p Fs 39 w(x) p Fq 1987 3797 a(new) p Fx 2152 3785 a(can) g(b) r(e) i(classi\014ed) d (b) n(y) i(computing) f(the) h(linear) 714 3892 y(function) p Ft 1249 4104 a(f) p Fx 9 w(\() p Fs(x) p Fq 1381 4116 a(new) p Fx 1508 4104 a(\)) 23 b(=) p Fs 22 w(w) p Fq 1720 4070 a(T) p Fx 1773 4104 a(\010\() p Fs(x) p Fq 1915 4116 a(new) p Fx 2042 4104 a(\)) c(+) p Ft 18 w(b) p Fx 22 w(=) p Fq 2362 4001 a(n) p Fm 2322 4025 a(X) p Fq 2328 4202 a(i) p Fr(=1) p Ft 2456 4104 a(\013) p Fq 2509 4116 a(i) p Ft 2537 4104 a(y) p Fq 2578 4116 a(i) p Ft 2605 4104 a(k) p Fx 3 w(\() p Fs(x) p Fq 2733 4116 a(i) p Ft 2761 4104 a(;) p Fs 14 w(x) p Fq 2848 4116 a(new) p Fx 2975 4104 a(\)) f(+) p Ft 18 w(b:) p Fx 714 4348 a(If) p Ft 24 w(f) p Fx 9 w(\() p Fs(x) p Fq 925 4360 a(new) p Fx 1052 4348 a(\)) 24 b(is) h(p) r(ositiv) n(e,) e(then) i (w) n(e) f(classify) p Fs 24 w(x) p Fq 2154 4360 a(new) p Fx 2305 4348 a(as) f(b) r(elonging) h(to) g(class) f(+1;) g(otherwise,) h(w) n(e) 714 4455 y(classify) p Fs 27 w(x) p Fq 1051 4467 a(new) p Fx 1205 4455 a(as) j(b) r(elonging) g(to) g(class) p Fu 27 w(\000) p Fx(1.) p Fv -34 4804 a(1.4) 101 b(Semide\014nite) 33 b(Programming) i(\(SDP\)) p -34 4671 3736 3 v 90 rotate dyy eop %%Page: 7 7 7 6 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.4) 78 b(Semide\014nite) 28 b(Pr) l(o) l(gr) l(amming) h (\(SDP\)) 2390 b(7) p Fx 945 349 a(In) 31 b(this) f(section) h(w) n(e) f (review) g(the) h(basic) f(de\014nition) h(of) f(semide\014nite) h (programming) e(as) h(w) n(ell) 945 457 y(as) k(some) g(imp) r(ortan) n (t) g(concepts) g(and) h(k) n(ey) f(results.) g(Details) h(and) f(pro) r (ofs) g(can) g(b) r(e) h(found) g(in) 945 565 y(Bo) n(yd) 26 b(and) i(V) -7 b(anden) n(b) r(erghe) 27 b(\(2001\).) 1028 673 y(Semide\014nite) k(programming) c(\(Nestero) n(v) i(and) h(Nemiro) n(vsky,) e(1994;) h(V) -7 b(anden) n(b) r(erghe) 29 b(and) 945 781 y(Bo) n(yd,) e(1996;) f(Bo) n(yd) h(and) h(V) -7 b(anden) n(b) r(erghe,) 27 b(2001\)) f(deals) i(with) g(the) h (optimization) e(of) h(con) n(v) n(ex) 945 889 y(functions) g(o) n(v) n (er) d(the) j(con) n(v) n(ex) f(cone) p Fl 2058 861 a(1) p Fx 2123 889 a(of) h(symmetric,) f(p) r(ositiv) n(e) g(semide\014nite) h (matrices) p Fu 1776 1072 a(P) p Fx 29 w(=) p Fm 1951 1004 a(\010) p Ft 2000 1072 a(X) p Fu 29 w(2) p Fp 23 w(R) p Fq 2230 1037 a(p) p Fk(\002) p Fq(p) p Fu 2388 1072 a(j) p Ft 28 w(X) p Fx 30 w(=) p Ft 22 w(X) p Fq 2701 1037 a(T) p Ft 2753 1072 a(;) 14 b(X) p Fu 29 w(\027) p Fx 23 w(0) p Fm 3018 1004 a(\011) p Ft 3079 1072 a(;) p Fx 945 1254 a(or) 44 b(a\016ne) g(subsets) g(of) h(this) g(cone.) f(As) h(explained) f(b) r(efore,) h(ev) n(ery) e(p) r(ositiv) n(e) h (semide\014nite) 945 1362 y(and) 35 b(symmetric) h(matrix) f(is) h(a) f (k) n(ernel) g(matrix,) g(and) h(con) n(v) n(ersely) -7 b(,) 33 b(ev) n(ery) i(k) n(ernel) g(matrix) g(is) 945 1470 y(symmetric) 28 b(and) h(p) r(ositiv) n(e) f(semide\014nite.) h (Therefore) p Fu 27 w(P) p Fx 36 w(can) f(b) r(e) h(view) n(ed) f(as) g (a) g(searc) n(h) f(space) 945 1578 y(for) d(p) r(ossible) h(k) n (ernel) g(matrices.) f(This) h(consideration) f(leads) h(to) g(the) h (k) n(ey) e(problem) h(addressed) 945 1686 y(in) 33 b(this) g(c) n (hapter|w) n(e) f(wish) h(to) g(sp) r(ecify) g(a) g(con) n(v) n(ex) e (cost) i(function) h(that) f(will) g(enable) g(us) g(to) 945 1794 y(learn) 27 b(the) h(optimal) f(k) n(ernel) g(matrix) g(within) p Fu 28 w(P) p Fx 34 w(using) g(semide\014nite) i(programming.) p Fs 945 2010 a(1.4.1) 98 b(De\014nition) 30 b(of) i(Semide\014nite) d (Programming) p Fx 945 2226 a(A) p Fy 28 w(line) l(ar) h(matrix) g(ine) l(quality,) p Fx 29 w(abbreviated) c(LMI,) i(is) f(a) h(constrain) n(t) e(of) i(the) g(form) p Ft 1773 2408 a(F) p Fx 12 w(\() p Fs(u) p Fx(\)) 23 b(:=) p Ft 23 w(F) p Fr 2142 2420 a(0) p Fx 2198 2408 a(+) p Ft 18 w(u) p Fr 2329 2420 a(1) p Ft 2366 2408 a(F) p Fr 2419 2420 a(1) p Fx 2475 2408 a(+) p Ft 18 w(:) 14 b(:) g(:) p Fx 18 w(+) p Ft 18 w(u) p Fq 2804 2420 a(q) p Ft 2840 2408 a(F) p Fq 2893 2420 a(q) p Fu 2954 2408 a(\026) p Fx 22 w(0) p Ft(:) p Fx 945 2591 a(Here,) p Fs 23 w(u) p Fx 25 w(is) 23 b(the) i(v) n(ector) d (of) i(decision) g(v) -5 b(ariables,) 22 b(and) p Ft 24 w(F) p Fr 2678 2603 a(0) p Ft 2716 2591 a(;) 14 b(:) g(:) g(:) f(;) h (F) p Fq 2953 2603 a(q) p Fx 3014 2591 a(are) 23 b(giv) n(en) h (symmetric) p Ft 23 w(p) p Fu 11 w(\002) p Ft 11 w(p) p Fx 945 2699 a(matrices.) 29 b(The) i(notation) p Ft 30 w(F) p Fx 12 w(\() p Fs(u) p Fx(\)) p Fu 28 w(\026) p Fx 28 w(0) f(means) g(that) g(the) h(symmetric) f(matrix) p Ft 30 w(F) p Fx 42 w(is) h(negativ) n(e) 945 2807 y(semide\014nite.) h (Note) g(that) g(suc) n(h) f(a) g(constrain) n(t) g(is) g(in) h (general) e(a) p Fy 31 w(nonline) l(ar) p Fx 33 w(constrain) n(t;) g (the) 945 2915 y(term) i(\\linear") e(in) i(the) h(name) e(LMI) h (merely) g(emphasizes) f(that) p Ft 32 w(F) p Fx 44 w(is) h(a\016ne) g (in) p Fs 32 w(u) p Fx(.) g(P) n(erhaps) 945 3023 y(the) 27 b(most) g(imp) r(ortan) n(t) g(feature) g(of) g(an) g(LMI) g(constrain) n(t) f(is) h(its) h(con) n(v) n(exit) n(y:) d(the) j(set) f(of) p Fs 27 w(u) p Fx 27 w(that) 945 3131 y(satisfy) g(the) h(LMI) g(is) f(a) g(con) n(v) n(ex) f(set.) 1028 3238 y(An) 37 b(LMI) g(constrain) n(t) f (can) h(b) r(e) h(seen) e(as) h(an) p Fy 37 w(in\014nite) p Fx 37 w(set) g(of) g(scalar,) e(a\016ne) i(constrain) n(ts.) 945 3346 y(Indeed,) g(for) f(a) h(giv) n(en) p Fs 35 w(u) p Fx(,) p Ft 37 w(F) p Fx 12 w(\() p Fs(u) p Fx(\)) p Fu 39 w(\026) p Fx 38 w(0) g(if) g(and) g(only) f(if) p Fs 37 w(z) p Fq 2777 3316 a(T) p Ft 2830 3346 a(F) p Fx 12 w(\() p Fs(u) p Fx(\)) p Fs(z) p Fu 39 w(\024) p Fx 38 w(0) h(for) f(ev) n(ery) p Fs 36 w(z) p Fx(;) h(ev) n(ery) 945 3454 y(constrain) n(t) 28 b(indexed) i(b) n(y) p Fs 29 w(z) p Fx 31 w(is) f(an) g(a\016ne) h(inequalit) n(y) -7 b(,) 29 b(in) h(the) g(ordinary) e(sense,) h(i.e.,) h(the) g(left-) 945 3562 y(hand) 21 b(side) g(of) g(the) h(inequalit) n(y) e(is) h(a) g (scalar,) f(comp) r(osed) g(of) i(a) e(linear) h(term) g(in) p Fs 21 w(u) p Fx 22 w(and) g(a) f(constan) n(t) 945 3670 y(term.) 29 b(Alternativ) n(ely) -7 b(,) 29 b(using) g(a) g(standard) g (result) g(from) g(linear) g(algebra,) e(w) n(e) i(ma) n(y) g(state) g (the) 945 3778 y(constrain) n(t) p Ft 26 w(F) p Fx 12 w(\() p Fs(u) p Fx(\)) p Fu 24 w(\026) p Fx 22 w(0) f(as) p Fu 1907 3961 a(8) p Ft(Z) p Fu 28 w(2) 23 b(P) p Fx 57 w(:) 51 b(trace) o(\() p Ft(F) p Fx 12 w(\() p Fs(u) p Fx(\)) p Ft(Z) p Fx 6 w(\)) p Fu 24 w(\024) p Fx 23 w(0) p Ft(:) p Fx 791 w(\(1.3\)) 945 4143 y(This) 31 b(can) g(b) r(e) h(seen) f (b) n(y) g(writing) g(do) n(wn) g(the) g(sp) r(ectral) g(decomp) r (osition) g(of) p Ft 31 w(Z) p Fx 37 w(and) g(using) g(the) 945 4251 y(fact) d(that) p Fs 27 w(z) p Fq 1330 4221 a(T) p Ft 1383 4251 a(F) p Fx 12 w(\() p Fs(u) p Fx(\)) p Fs(z) p Fu 24 w(\024) p Fx 23 w(0) f(for) g(ev) n(ery) p Fs 26 w(z) p Fx(.) 1028 4359 y(A) 21 b(semide\014nite) g(program) e(\(SDP\)) i (is) g(an) f(optimization) h(problem) f(with) h(a) f(linear) g(ob) 5 b(jectiv) n(e,) 945 4467 y(and) 27 b(linear) g(matrix) g(inequalit) n (y) g(and) h(a\016ne) f(equalit) n(y) g(constrain) n(ts.) p 945 4627 798 3 v Fl 945 4739 a(1.) p Fj 39 w(S) p Fi 25 w(\022) p Fh 21 w(R) p Fg 1247 4708 a(d) p Fl 1312 4739 a(is) f(a) g(con) n(v) n(ex) f(cone) h(i\013) p Fi 26 w(8) p Ff(x) p Fj(;) p Ff 12 w(y) p Fi 22 w(2) p Fj 22 w(S;) p Fi 13 w(8) p Fj(\025;) 13 b(\026) p Fi 21 w(\025) p Fl 21 w(0) 48 b(:) p Fj 21 w(\025) p Ff(x) p Fl 17 w(+) p Fj 17 w(\026) p Ff(y) p Fi 22 w(2) p Fj 22 w(S) p Fl 4 w(.) p 90 rotate dyy eop %%Page: 8 8 8 7 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(8) 709 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Fz 714 349 a(De\014nition) 33 b(1.1) p Fx 714 457 a(A) 28 b(semide\014nite) g(program) d(is) j(a) f(problem) g(of) h(the) g(form) 938 623 y(min) p Fo 986 673 a(u) p Fs 1242 623 a(c) p Fq 1284 589 a(T) p Fs 1337 623 a(u) p Fx 2142 w(\(1.4\)) 714 785 y(sub) 5 b(ject) 27 b(to) p Ft 166 w(F) p Fq 1307 750 a(j) p Fx 1342 785 a(\() p Fs(u) p Fx(\)) d(=) p Ft 22 w(F) p Fq 1635 745 a(j) p Fr 1623 807 a(0) p Fx 1689 785 a(+) p Ft 18 w(u) p Fr 1820 797 a(1) p Ft 1857 785 a(F) p Fq 1922 745 a(j) p Fr 1910 807 a(1) p Fx 1975 785 a(+) p Ft 18 w(:) 14 b(:) g(:) p Fx 18 w(+) p Ft 18 w(u) p Fq 2304 797 a(q) p Ft 2340 785 a(F) p Fq 2405 750 a(j) 2393 805 y(q) p Fu 2463 785 a(\026) p Fx 23 w(0) p Ft(;) 96 b(j) p Fx 28 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(L) 1242 917 y(A) p Fs(u) p Fx 24 w(=) p Fs 22 w(b) p Ft(;) p Fx 714 1083 a(where) p Fs 24 w(u) p Fu 23 w(2) p Fp 23 w(R) p Fq 1159 1053 a(q) p Fx 1227 1083 a(is) 25 b(the) g(v) n(ector) f(of) g(decision) h(v) -5 b(ariables,) p Fs 23 w(c) p Fu 24 w(2) p Fp 23 w(R) p Fq 2660 1053 a(q) p Fx 2728 1083 a(is) 25 b(the) g(ob) 5 b(jectiv) n(e) 24 b(v) n(ector,) g(and) 714 1191 y(matrices) p Ft 26 w(F) p Fq 1111 1152 a(j) 1099 1215 y(i) p Fx 1169 1191 a(=) f(\() p Ft(F) p Fq 1354 1152 a(j) 1342 1215 y(i) p Fx 1389 1191 a(\)) p Fq 1421 1161 a(T) p Fu 1497 1191 a(2) p Fp 23 w(R) p Fq 1629 1161 a(p) p Fk(\002) p Fq(p) p Fx 1787 1191 a(are) k(giv) n(en.) 714 1357 y(By) 20 b(con) n(v) n(exit) n(y) f(of) h(their) g(LMI) h(constrain) n(ts,) e (SDPs) h(are) g(con) n(v) n(ex) f(optimization) h(problems.) f(The) 714 1465 y(usefulness) 24 b(of) h(the) g(SDP) g(formalism) f(stems) h(from) f(t) n(w) n(o) g(imp) r(ortan) n(t) h(facts.) f(First,) h(despite) g (the) 714 1573 y(seemingly) 19 b(v) n(ery) g(sp) r(ecialized) h(form) f (of) h(SDPs,) g(they) g(arise) f(in) i(a) e(host) h(of) g (applications;) f(second,) 714 1681 y(there) 28 b(exist) g(\\in) n (terior-p) r(oin) n(t") e(algorithms) g(to) i(globally) f(solv) n(e) g (SDPs) i(that) f(ha) n(v) n(e) f(extremely) 714 1789 y(go) r(o) r(d) i(theoretical) h(and) g(practical) f(computational) g (e\016ciency) i(\(V) -7 b(anden) n(b) r(erghe) 29 b(and) h(Bo) n(yd,) 714 1897 y(1996\).) 797 2005 y(One) 42 b(v) n(ery) f(useful) i(to) r (ol) f(to) g(reduce) g(a) g(problem) g(to) g(an) h(SDP) f(is) g(the) h (so-called) e(Sc) n(h) n(ur) 714 2113 y(complemen) n(t) 27 b(lemma,) h(whic) n(h) f(will) h(b) r(e) g(in) n(v) n(ok) n(ed) e (later) h(in) h(this) g(c) n(hapter.) p Fz 714 2279 a(L) -5 b(emma) 35 b(1.2) p Fs 714 2387 a(Sc) m(h) m(ur) e(complemen) m(t) c (lemma) p Fx 24 w(Consider) e(the) h(partitioned) f(symmetric) g (matrix) p Ft 1735 2628 a(X) p Fx 30 w(=) p Ft 22 w(X) p Fq 1997 2594 a(T) p Fx 2072 2628 a(=) p Fm 2160 2486 a( ) p Ft 2296 2568 a(A) 111 b(B) 2267 2698 y(B) p Fq 2334 2668 a(T) p Ft 2470 2698 a(C) p Fm 2578 2486 a(!) p Ft 2658 2628 a(;) p Fx 714 2874 a(where) p Ft 38 w(A) p Fx 39 w(and) p Ft 38 w(C) p Fx 45 w(are) 38 b(square) f(and) h (symmetric.) h(If) g(det\() p Ft(A) p Fx(\)) p Fu 42 w(6) p Fx(=) i(0,) d(w) n(e) g(de\014ne) h(the) g(Sc) n(h) n(ur) 714 2982 y(complemen) n(t) 31 b(of) p Ft 31 w(A) p Fx 31 w(in) p Ft 31 w(X) p Fx 38 w(b) n(y) f(the) i(matrix) p Ft 30 w(S) p Fx 34 w(=) p Ft 28 w(C) p Fu 27 w(\000) p Ft 20 w(B) p Fq 2533 2952 a(T) p Ft 2586 2982 a(A) p Fk 2648 2952 a(\000) p Fr(1) p Ft 2737 2982 a(B) p Fx 4 w(.) f(The) g(Sc) n(h) n(ur) g(complemen) n(t) 714 3090 y(lemma) c(states) g(that) h(if) p Ft 28 w(A) p Fu 24 w(\037) p Fx 22 w(0,) f(then) p Ft 29 w(X) p Fu 29 w(\027) p Fx 23 w(0) g(if) h(and) f(only) h(if) p Ft 28 w(S) p Fu 27 w(\027) p Fx 23 w(0.) 797 3256 y(T) -7 b(o) 22 b(illustrate) h(ho) n(w) f(this) i(lemma) e(can) h(b) r(e) g (used) g(to) g(cast) g(a) f(nonlinear) g(con) n(v) n(ex) g (optimization) 714 3364 y(problem) 27 b(as) g(an) g(SDP) -7 b(,) 28 b(consider) e(the) i(follo) n(wing) f(result:) p Fz 714 3530 a(L) -5 b(emma) 35 b(1.3) p Fx 714 3638 a(The) 27 b(quadratically) f(constrained) h(quadratic) f(program) g(\(QCQP\)) 938 3804 y(min) p Fo 986 3853 a(u) p Ft 1270 3804 a(f) p Fr 1311 3816 a(0) p Fx 1348 3804 a(\() p Fs(u) p Fx(\)) 2067 b(\(1.5\)) 714 3949 y(sub) 5 b(ject) 27 b(to) p Ft 194 w(f) p Fq 1311 3961 a(i) p Fx 1338 3949 a(\() p Fs(u) p Fx(\)) p Fu 24 w(\024) p Fx 22 w(0) p Ft(;) 69 b(i) p Fx 23 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(M) t(;) p Fx 714 4115 a(with) p Ft 31 w(f) p Fq 947 4127 a(i) p Fx 974 4115 a(\() p Fs(u) p Fx(\)) p Fe 29 w(,) p Fx 27 w(\() p Ft(A) p Fq 1307 4127 a(i) p Fs 1336 4115 a(u) p Fx 20 w(+) p Fs 20 w(b) p Fq 1547 4127 a(i) p Fx 1575 4115 a(\)) p Fq 1607 4085 a(T) p Fx 1660 4115 a(\() p Ft(A) p Fq 1754 4127 a(i) p Fs 1782 4115 a(u) p Fx 20 w(+) p Fs 21 w(b) p Fq 1994 4127 a(i) p Fx 2021 4115 a(\)) p Fu 21 w(\000) p Fs 20 w(c) p Fq 2201 4085 a(T) 2201 4137 y(i) p Fs 2254 4115 a(u) p Fu 21 w(\000) p Ft 20 w(d) p Fq 2456 4127 a(i) p Fx 2484 4115 a(,) 31 b(is) g(equiv) -5 b(alen) n(t) 30 b(to) h(the) g(semide\014nite) 714 4223 y(programming) 25 b(problem:) 938 4389 y(min) p Fo 964 4439 a(u) p Fq(;t) p Ft 1270 4389 a(t) p Fx 2232 w(\(1.6\)) 714 4634 y(sub) 5 b(ject) 27 b(to) p Fm 1270 4492 a( ) p Ft 1547 4574 a(I) 378 b(A) p Fr 2023 4586 a(0) p Fs 2061 4574 a(u) p Fx 18 w(+) p Fs 18 w(b) p Fo 2268 4586 a(0) p Fx 1336 4703 a(\() p Ft(A) p Fr 1430 4715 a(0) p Fs 1467 4703 a(u) p Fx 19 w(+) p Fs 18 w(b) p Fo 1675 4715 a(0) p Fx 1717 4703 a(\)) p Fq 1749 4673 a(T) p Fs 1884 4703 a(c) p Fo 1926 4715 a(0) p Fq 1969 4673 a(T) p Fs 2021 4703 a(u) p Fx 18 w(+) p Ft 18 w(d) p Fr 2218 4715 a(0) p Fx 2274 4703 a(+) p Ft 18 w(t) p Fm 2387 4492 a(!) p Fu 2476 4634 a(\027) p Fx 23 w(0) p Ft(;) p Fm 1270 4776 a( ) p Ft 1535 4858 a(I) 290 b(A) p Fq 1923 4870 a(i) p Fs 1951 4858 a(u) p Fx 18 w(+) p Fs 18 w(b) p Fq 2158 4870 a(i) p Fx 1336 4987 a(\() p Ft(A) p Fq 1430 4999 a(i) p Fs 1458 4987 a(u) p Fx 18 w(+) p Fs 18 w(b) p Fq 1665 4999 a(i) p Fx 1693 4987 a(\)) p Fq 1725 4957 a(T) p Fs 1863 4987 a(c) p Fq 1905 4957 a(T) 1905 5009 y(i) p Fs 1958 4987 a(u) p Fx 18 w(+) p Ft 18 w(d) p Fq 2155 4999 a(i) p Fm 2186 4776 a(!) p Fu 2274 4917 a(\027) p Fx 23 w(0) p Ft(;) 97 b(i) p Fx 22 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(M) t(:) p 90 rotate dyy eop %%Page: 9 9 9 8 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.4) 78 b(Semide\014nite) 28 b(Pr) l(o) l(gr) l(amming) h (\(SDP\)) 2390 b(9) p Fx 945 349 a(This) 27 b(can) h(b) r(e) g(seen) f (b) n(y) g(rewriting) g(the) h(QCQP) e(\(1.5\)) h(as:) 1169 515 y(min) p Fo 1195 565 a(u) p Fq(;t) p Ft 1501 515 a(t) p Fx 945 672 a(sub) 5 b(ject) 28 b(to) p Ft 193 w(t) p Fu 18 w(\000) p Ft 18 w(f) p Fr 1673 684 a(0) p Fx 1710 672 a(\() p Fs(u) p Fx(\)) p Fu 24 w(\025) p Fx 23 w(0) p Ft(;) p Fu 1501 805 a(\000) p Ft(f) p Fq 1607 817 a(i) p Fx 1634 805 a(\() p Fs(u) p Fx(\)) p Fu 24 w(\025) p Fx 22 w(0) p Ft(;) 69 b(i) p Fx 23 w(=) 22 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(M) t(:) p Fx 945 971 a(Note) 27 b(that) g(for) g(a) f(\014xed) h(and) g(feasible) p Fs 27 w(u) p Fx(,) p Ft 27 w(t) p Fx 23 w(=) p Ft 23 w(f) p Fr 2458 983 a(0) p Fx 2509 971 a(\() p Fs(u) p Fx(\)) g(is) g(the) h(optimal) e(solution.) h(The) g(con) n(v) n(ex) 945 1079 y(quadratic) e(inequalit) n(y) p Ft 26 w(t) p Fu 15 w(\000) p Ft 15 w(f) p Fr 1865 1091 a(0) p Fx 1902 1079 a(\() p Fs(u) p Fx(\)) f(=) e(\() p Ft(t) p Fx 16 w(+) p Fs 15 w(c) p Fo 2330 1091 a(0) p Fq 2372 1049 a(T) p Fs 2425 1079 a(u) p Fx 15 w(+) p Ft 15 w(d) p Fr 2616 1091 a(0) p Fx 2654 1079 a(\)) p Fu 16 w(\000) p Fx 15 w(\() p Ft(A) p Fr 2876 1091 a(0) p Fs 2914 1079 a(u) p Fx 15 w(+) p Fs 15 w(b) p Fo 3115 1091 a(0) p Fx 3157 1079 a(\)) p Fq 3189 1049 a(T) p Ft 3242 1079 a(I) p Fk 3285 1049 a(\000) p Fr(1) p Fx 3374 1079 a(\() p Ft(A) p Fr 3468 1091 a(0) p Fs 3506 1079 a(u) p Fx 15 w(+) p Fs 15 w(b) p Fo 3707 1091 a(0) p Fx 3749 1079 a(\)) p Fu 23 w(\025) p Fx 23 w(0) 945 1187 y(is) 27 b(no) n(w) g(equiv) -5 b(alen) n(t) 28 b(to) f(the) h(follo) n(wing) f (LMI,) g(using) h(the) f(Sc) n(h) n(ur) g(complemen) n(t) h(Lemma) f (1.2:) p Fm 1760 1286 a( ) p Ft 2037 1368 a(I) 378 b(A) p Fr 2513 1380 a(0) p Fs 2551 1368 a(u) p Fx 18 w(+) p Fs 19 w(b) p Fo 2759 1380 a(0) p Fx 1826 1498 a(\() p Ft(A) p Fr 1920 1510 a(0) p Fs 1958 1498 a(u) p Fx 18 w(+) p Fs 18 w(b) p Fo 2165 1510 a(0) p Fx 2207 1498 a(\)) p Fq 2239 1468 a(T) p Fs 2374 1498 a(c) p Fo 2416 1510 a(0) p Fq 2459 1468 a(T) p Fs 2511 1498 a(u) p Fx 18 w(+) p Ft 18 w(d) p Fr 2708 1510 a(0) p Fx 2764 1498 a(+) p Ft 18 w(t) p Fm 2877 1286 a(!) p Fu 2966 1428 a(\027) p Fx 23 w(0) p Ft(:) p Fx 945 1674 a(Similar) 33 b(steps) g(for) h(the) g(other) f(quadratic) f(inequalit) n(y) h (constrain) n(ts) f(\014nally) i(yield) g(\(1.6\),) f(an) 945 1781 y(SDP) d(in) f(standard) g(form) g(\(1.4\),) h(equiv) -5 b(alen) n(t) 29 b(to) h(\(1.5\).) f(This) h(sho) n(ws) e(that) i(a) g (QCQP) e(can) h(b) r(e) 945 1889 y(cast) c(as) f(an) h(SDP) -7 b(.) 26 b(Of) f(course,) f(in) i(practice) e(a) h(QCQP) f(should) h (not) h(b) r(e) f(solv) n(ed) f(using) h(general-) 945 1997 y(purp) r(ose) i(SDP) g(solv) n(ers,) f(since) h(the) h (particular) e(structure) h(of) h(the) f(problem) g(at) h(hand) f(can) g (b) r(e) 945 2105 y(e\016cien) n(tly) 33 b(exploited.) g(The) g(ab) r (o) n(v) n(e) e(do) r(es) i(sho) n(w) f(that) h(QCQPs,) f(and) g(in) i (particular,) d(linear) 945 2213 y(programming) 25 b(problems,) i(b) r (elong) g(to) h(the) g(SDP) f(family) -7 b(.) p Fs 945 2429 a(1.4.2) 98 b(Dualit) m(y) p Fx 945 2645 a(An) 19 b(imp) r(ortan) n(t) f(principle) g(in) h(optimization|p) r(erhaps) e (ev) n(en) h(the) h(most) f(imp) r(ortan) n(t) g(principle|) 945 2753 y(is) 31 b(that) h(of) p Fy 31 w(duality) p Fx(.) h(T) -7 b(o) 31 b(illustrate) g(dualit) n(y) h(in) f(the) h(case) f(of) g(an) h (SDP) -7 b(,) 31 b(w) n(e) g(will) h(\014rst) f(review) 945 2861 y(basic) 42 b(concepts) g(in) h(dualit) n(y) g(theory) f(and) g (then) h(sho) n(w) f(ho) n(w) g(they) h(can) g(b) r(e) g(extended) g (to) 945 2969 y(semide\014nite) 31 b(programming.) f(In) h(particular,) f(dualit) n(y) h(will) g(giv) n(e) f(insigh) n(ts) h(in) n(to) g (optimalit) n(y) 945 3077 y(conditions) c(for) g(the) h(semide\014nite) g(program.) 1028 3185 y(Consider) e(an) i(optimization) f(problem) g (with) p Ft 28 w(n) p Fx 28 w(v) -5 b(ariables) 26 b(and) p Ft 27 w(m) p Fx 28 w(scalar) g(constrain) n(ts) 1169 3351 y(min) p Fo 1217 3400 a(u) p Ft 1473 3351 a(f) p Fr 1514 3363 a(0) p Fx 1551 3351 a(\() p Fs(u) p Fx(\)) 2095 b(\(1.7\)) 945 3496 y(sub) 5 b(ject) 28 b(to) p Ft 165 w(f) p Fq 1514 3508 a(i) p Fx 1542 3496 a(\() p Fs(u) p Fx(\)) p Fu 23 w(\024) p Fx 23 w(0) p Ft(;) 96 b(i) p Fx 23 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(m;) p Fx 945 3662 a(where) p Fs 23 w(u) p Fu 23 w(2) p Fp 24 w(R) p Fq 1390 3632 a(n) p Fx 1441 3662 a(.) 24 b(In) g(the) g(con) n (text) g(of) g(dualit) n(y) -7 b(,) 24 b(problem) f(\(1.7\)) h(is) g (called) f(the) p Fy 25 w(primal) k(pr) l(oblem) p Fx(;) 945 3770 y(w) n(e) g(denote) h(its) f(optimal) h(v) -5 b(alue) p Ft 27 w(p) p Fk 2011 3740 a(\003) p Fx 2049 3770 a(.) 28 b(F) -7 b(or) 27 b(no) n(w,) g(w) n(e) g(do) g(not) h(assume) f(con) n (v) n(exit) n(y) -7 b(.) p Fz 945 3936 a(De\014nition) 33 b(1.4) p Fs 945 4044 a(Lagrangian) p Fx 32 w(The) p Fy 31 w(L) l(agr) l(angian) p Fu 39 w(L) p Fx 60 w(:) p Fp 29 w(R) p Fq 2276 4014 a(n) p Fr(+) p Fq(m) p Fu 2467 4044 a(!) p Fp 29 w(R) p Fx 37 w(corresp) r(onding) c(to) i(the) g (minimization) 945 4152 y(problem) c(\(1.7\)) g(is) h(de\014ned) g(as) p Fu 1633 4334 a(L) p Fx(\() p Fs(u) p Ft(;) p Fn 14 w(\025) p Fx(\)) c(=) p Ft 22 w(f) p Fr 2052 4346 a(0) p Fx 2089 4334 a(\() p Fs(u) p Fx(\)) 19 b(+) p Ft 18 w(\025) p Fr 2356 4346 a(1) p Ft 2394 4334 a(f) p Fr 2435 4346 a(1) p Fx 2472 4334 a(\() p Fs(u) p Fx(\)) g(+) p Ft 18 w(:) 14 b(:) g(:) p Fx 18 w(+) p Ft 18 w(\025) p Fq 2937 4346 a(m) p Ft 3001 4334 a(f) p Fq 3042 4346 a(m) p Fx 3104 4334 a(\() p Fs(u) p Fx(\)) p Ft(:) p Fx 945 4517 a(The) p Ft 27 w(\025) p Fq 1163 4529 a(i) p Fu 1215 4517 a(2) p Fp 23 w(R) p Ft(;) 47 b(i) p Fx 23 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(m) p Fx 27 w(are) 27 b(called) p Fy 27 w(L) l(agr) l(ange) k(multipliers) p Fx 35 w(or) p Fy 27 w(dual) f(variables) p Fx(.) 1028 4683 y(One) d(can) g(no) n(w) g(notice) h(that) p Ft 1350 4908 a(h) p Fx(\() p Fs(u) p Fx(\)) c(=) e(max) p Fn 1633 4980 a(\025) p Fk(\025) p Fr(0) p Fu 1795 4908 a(L) p Fx(\() p Fs(u) p Ft(;) p Fn 14 w(\025) p Fx(\)) h(=) p Fm 2173 4766 a(\() p Ft 2281 4848 a(f) p Fr 2322 4860 a(0) p Fx 2359 4848 a(\() p Fs(u) p Fx(\)) 84 b(if) p Ft 28 w(f) p Fq 2677 4860 a(i) p Fx 2704 4848 a(\() p Fs(u) p Fx(\)) p Fu 24 w(\024) p Fx 22 w(0) p Ft(;) 42 b(i) p Fx 22 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(m) p Fx 2281 4978 a(+) p Fu(1) p Fx 131 w(otherwise) p Ft -1 w(:) p Fx 3763 4908 a(\(1.8\)) p 90 rotate dyy eop %%Page: 10 10 10 9 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(10) 670 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Fx 714 349 a(So,) 43 b(the) i(function) p Ft 44 w(h) p Fx(\() p Fs(u) p Fx(\)) g(coincides) e(with) h(the) h(ob) 5 b(jectiv) n(e) p Ft 43 w(f) p Fr 2721 361 a(0) p Fx 2758 349 a(\() p Fs(u) p Fx(\)) 44 b(in) g(regions) f(where) g(the) 714 457 y(constrain) n(ts) p Ft 37 w(f) p Fq 1189 469 a(i) p Fx 1216 457 a(\() p Fs(u) p Fx(\)) p Fu 42 w(\024) p Fx 40 w(0) p Ft(;) 52 b(i) p Fx 41 w(=) 41 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(m) p Fx 38 w(are) 37 b(satis\014ed) h(and) p Ft 39 w(h) p Fx(\() p Fs(u) p Fx(\)) j(=) g(+) p Fu(1) p Fx 38 w(in) e(infeasible) 714 565 y(regions.) g(In) h(other) g(w) n (ords,) p Ft 40 w(h) p Fx 40 w(acts) g(as) g(a) g(\\barrier") e(of) j (the) g(feasible) f(set) g(of) h(the) g(primal) 714 673 y(problem.) 21 b(Th) n(us) g(w) n(e) g(can) g(as) f(w) n(ell) h(use) p Ft 22 w(h) p Fx(\() p Fs(u) p Fx(\)) g(as) g(ob) 5 b(jectiv) n(e) 21 b(function) h(and) f(rewrite) f(the) i(original) 714 781 y(primal) 27 b(problem) g(\(1.7\)) g(as) g(an) p Fy 27 w(unc) l(onstr) l(aine) l(d) p Fx 28 w(optimization) g(problem:) p Ft 1793 964 a(p) p Fk 1835 929 a(\003) p Fx 1896 964 a(=) c(min) p Fo 2032 1013 a(u) p Fx 2164 964 a(max) p Fn 2171 1035 a(\025) p Fk -1 w(\025) p Fr(0) p Fu 2332 964 a(L) p Fx(\() p Fs(u) p Ft(;) p Fn 14 w(\025) p Fx(\)) p Ft(:) p Fx 910 w(\(1.9\)) 714 1187 y(The) g(notion) h(of) f(w) n(eak) g (dualit) n(y) g(amoun) n(ts) g(to) h(exc) n(hanging) e(the) i(\\min") f (and) g(\\max") f(op) r(erators) 714 1295 y(in) 33 b(the) g(ab) r(o) n (v) n(e) f(form) n(ulation,) g(resulting) g(in) h(a) g(lo) n(w) n(er) e (b) r(ound) j(on) e(the) i(optimal) e(v) -5 b(alue) 33 b(of) g(the) 714 1402 y(primal) 26 b(problem.) h(Strong) f(dualit) n(y) h(refers) f(to) h(the) g(case) f(where) h(this) g(exc) n(hange) f(can) g (b) r(e) i(done) 714 1510 y(without) 33 b(altering) f(the) h(v) -5 b(alue) 33 b(of) g(the) g(result:) g(the) g(lo) n(w) n(er) f(b) r(ound) h(is) g(actually) f(equal) g(to) h(the) 714 1618 y(optimal) 25 b(v) -5 b(alue) p Ft 25 w(p) p Fk 1270 1588 a(\003) p Fx 1308 1618 a(.) 25 b(While) h(w) n(eak) e(dualit) n(y) h(alw) n(a) n (ys) f(hold,) h(ev) n(en) g(if) h(the) f(primal) g(problem) g(\(1.9\)) 714 1726 y(is) 34 b(not) g(con) n(v) n(ex,) f(strong) f(dualit) n(y) i (ma) n(y) g(not) g(hold.) g(Ho) n(w) n(ev) n(er,) e(for) i(a) g(large) e (class) h(of) i(generic) 714 1834 y(con) n(v) n(ex) 26 b(problems,) h(strong) f(dualit) n(y) h(holds.) p Fz 714 2000 a(L) -5 b(emma) 35 b(1.5) p Fs 714 2108 a(W) -8 b(eak) 33 b(dualit) m(y) p Fx 28 w(F) -7 b(or) 28 b(all) g(functions) p Ft 28 w(f) p Fr 1961 2120 a(0) p Ft 1998 2108 a(;) 14 b(f) p Fr 2076 2120 a(1) p Ft 2113 2108 a(;) g(:) g(:) g(:) g(;) g(f) p Fq 2339 2120 a(m) p Fx 2430 2108 a(in) 28 b(\(1.7\),) g(not) g (necessarily) f(con) n(v) n(ex,) g(w) n(e) 714 2216 y(can) g(exc) n (hange) f(the) i(max) f(and) h(the) g(min) g(and) f(get) g(a) h(lo) n (w) n(er) e(b) r(ound) i(on) p Ft 27 w(p) p Fk 3016 2186 a(\003) p Fx 3054 2216 a(:) p Ft 1334 2399 a(d) p Fk 1377 2364 a(\003) p Fx 1439 2399 a(=) 22 b(max) p Fn 1533 2470 a(\025) p Fk(\025) p Fr(0) p Fx 1722 2399 a(min) p Fo 1771 2449 a(u) p Fu 1874 2399 a(L) p Fx(\() p Fs(u) p Ft(;) p Fn 14 w(\025) p Fx 1 w(\)) p Fu 23 w(\024) p Fx 23 w(min) p Fo 2301 2449 a(u) p Fx 2432 2399 a(max) p Fn 2439 2470 a(\025) p Fk(\025) p Fr(0) p Fu 2601 2399 a(L) p Fx(\() p Fs(u) p Ft(;) p Fn 14 w(\025) p Fx(\)) h(=) p Ft 23 w(p) p Fk 3021 2364 a(\003) p Ft 3059 2399 a(:) p Fx 714 2626 a(The) 30 b(ob) 5 b(jectiv) n(e) 29 b(function) i(of) f (the) h(maximization) e(problem) h(is) g(no) n(w) f(called) h(the) g (\(Lagrange\)) 714 2734 y(dual) d(function.) p Fz 714 2900 a(De\014nition) 33 b(1.6) p Fs 714 3008 a(\(Lagrange\)) h(dual) g (function) p Fx 29 w(The) p Fy 29 w(\(L) l(agr) l(ange\)) e(dual) f (function) p Ft 36 w(g) p Fx 28 w(:) p Fp 25 w(R) p Fq 3067 2978 a(m) p Fu 3162 3008 a(!) p Fp 25 w(R) p Fx 35 w(is) e(de\014ned) 714 3116 y(as) p Ft 714 3282 a(g) p Fx 3 w(\() p Fn(\025) p Fx -1 w(\)) 24 b(=) e(min) p Fo 1035 3332 a(u) p Fu 1167 3282 a(L) p Fx(\() p Fs(u) p Ft(;) p Fn 14 w(\025) p Fx(\)) 900 3427 y(=) g(min) p Fo 1035 3477 a(u) p Ft 1167 3427 a(f) p Fr 1208 3439 a(0) p Fx 1245 3427 a(\() p Fs(u) p Fx(\)) d(+) p Ft 18 w(\025) p Fr 1512 3439 a(1) p Ft 1550 3427 a(f) p Fr 1591 3439 a(1) p Fx 1628 3427 a(\() p Fs(u) p Fx(\)) g(+) p Ft 18 w(:) 14 b(:) g(:) p Fx 18 w(+) p Ft 18 w(\025) p Fq 2093 3439 a(m) p Ft 2157 3427 a(f) p Fq 2198 3439 a(m) p Fx 2260 3427 a(\() p Fs(u) p Fx(\)) p Ft(:) p Fx 1090 w(\(1.10\)) 714 3606 y(F) -7 b(urthermore) p Ft 26 w(g) p Fx 3 w(\() p Fn(\025) p Fx(\)) 28 b(is) f(conca) n(v) n (e,) f(ev) n(en) h(if) h(the) p Ft 28 w(f) p Fq 2248 3618 a(i) p Fx 2275 3606 a(\() p Fs(u) p Fx(\)) h(are) d(not) i(con) n (v) n(ex.) 714 3772 y(The) g(conca) n(vit) n(y) e(can) i(easily) f(b) r (e) h(seen) g(b) n(y) g(considering) f(\014rst) g(that) i(for) e(a) h (giv) n(en) p Fs 27 w(u) p Fx(,) p Fu 28 w(L) p Fx(\() p Fs(u) p Ft(;) p Fn 14 w(\025) p Fx(\)) h(is) 714 3880 y(an) c(a\016ne) h(function) h(of) p Fn 26 w(\025) p Fx 25 w(and) f(hence) g(is) g(a) g(conca) n(v) n(e) e(function.) j (Since) p Ft 26 w(g) p Fx 3 w(\() p Fn(\025) p Fx -1 w(\)) g(is) f(the) g(p) r(oin) n(t) n(wise) 714 3988 y(minim) n(um) i(of) f(suc) n(h) h(conca) n(v) n(e) d(functions,) j(it) g(is) g(conca) n(v) n(e.) p Fz 714 4154 a(De\014nition) 33 b(1.7) p Fs 714 4262 a(Lagrange) f(dual) g(problem) p Fx 26 w(The) p Fy 27 w(L) l(agr) l(ange) f(dual) f(pr) l(oblem) p Fx 35 w(is) e(de\014ned) g(as) p Ft 1935 4444 a(d) p Fk 1978 4410 a(\003) p Fx 2039 4444 a(=) 23 b(max) p Fn 2134 4516 a(\025) p Fk(\025) p Fr(0) p Ft 2295 4444 a(g) p Fx 3 w(\() p Fn(\025) p Fx(\)) p Ft(:) p Fx 714 4672 a(Since) p Ft 32 w(g) p Fx 3 w(\() p Fn(\025) p Fx(\)) 32 b(is) h(conca) n(v) n(e,) d(this) j(will) f(alw) n(a) n(ys) f (b) r(e) h(a) g(con) n(v) n(ex) f(optimization) h(problem,) g(ev) n(en) g(if) 714 4780 y(the) h(primal) f(is) g(not.) h(By) p Fy 32 w(we) l(ak) i(duality) p Fx(,) e(w) n(e) g(alw) n(a) n(ys) d(ha) n (v) n(e) p Ft 31 w(d) p Fk 2669 4750 a(\003) p Fu 2739 4780 a(\024) p Ft 31 w(p) p Fk 2877 4750 a(\003) p Fx 2915 4780 a(,) j(ev) n(en) f(for) g(non-con) n(v) n(ex) 714 4888 y(problems.) g(The) i(v) -5 b(alue) p Ft 33 w(p) p Fk 1537 4858 a(\003) p Fu 1598 4888 a(\000) p Ft 22 w(d) p Fk 1728 4858 a(\003) p Fx 1800 4888 a(is) 33 b(called) g(the) h(dualit) n(y) f(gap.) g(F) -7 b(or) p Fs 33 w(con) m(v) m(ex) p Fx 35 w(problems,) 33 b(w) n(e) p 90 rotate dyy eop %%Page: 11 11 11 10 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.4) 78 b(Semide\014nite) 28 b(Pr) l(o) l(gr) l(amming) h (\(SDP\)) 2351 b(11) p Fx 945 349 a(usually) 27 b(\(although) g(not) h (alw) n(a) n(ys\)) e(ha) n(v) n(e) p Fy 26 w(str) l(ong) j(duality) p Fx 29 w(at) e(the) h(optim) n(um,) g(i.e.,) p Ft 2292 532 a(d) p Fk 2335 498 a(\003) p Fx 2396 532 a(=) p Ft 23 w(p) p Fk 2526 498 a(\003) p Ft 2564 532 a(;) p Fx 945 715 a(whic) n(h) 35 b(is) g(also) g(referred) f(to) h(as) g(a) p Fy 35 w(zer) l(o) i(duality) h(gap) p Fx(.) f(F) -7 b(or) 35 b(con) n(v) n(ex) f(problems,) h(a) g(su\016cien) n(t) 945 822 y(condition) 27 b(for) g(zero) g(dualit) n(y) g(gap) g(is) g(pro) n (vided) g(b) n(y) p Fy 27 w(Slater's) j(c) l(ondition) p Fx 6 w(:) p Fz 945 989 a(L) -5 b(emma) 35 b(1.8) p Fs 945 1096 a(Slater's) 27 b(condition) p Fx 24 w(If) e(the) g(primal) f (problem) g(\(1.7\)) g(is) h(con) n(v) n(ex) e(and) h(is) p Fs 24 w(strictly) 29 b(feasible) p Fx(,) 945 1204 y(i.e.,) p Fu 28 w(9) p Fs 27 w(u) p Fr 1228 1216 a(0) p Fx 1316 1204 a(:) p Ft 51 w(f) p Fq 1431 1216 a(i) p Fx 1458 1204 a(\() p Fs(u) p Fr 1543 1216 a(0) p Fx 1581 1204 a(\)) p Ft 23 w(<) p Fx 23 w(0) p Ft(;) 41 b(i) p Fx 23 w(=) 22 b(1) p Ft(;) 14 b(:) g(:) g(:) g(;) g(m) p Fx(,) 27 b(then) p Ft 2292 1387 a(p) p Fk 2334 1353 a(\003) p Fx 2395 1387 a(=) p Ft 22 w(d) p Fk 2525 1353 a(\003) p Ft 2564 1387 a(:) p Fs 945 1603 a(1.4.3) 98 b(SDP) 32 b(Dualit) m(y) g(and) h(Optimalit) m(y) d(Conditions) p Fx 945 1819 a(Consider) i(for) g(simplicit) n(y) i(the) f(case) f(of) h (an) g(SDP) g(with) h(a) e(single) h(LMI) g(constrain) n(t,) f(and) h (no) 945 1927 y(a\016ne) 27 b(equalities:) p Ft 1258 2109 a(p) p Fk 1300 2075 a(\003) p Fx 1361 2109 a(=) c(min) p Fo 1497 2159 a(u) p Fs 1620 2109 a(c) p Fq 1662 2075 a(T) p Fs 1714 2109 a(u) p Fx 28 w(sub) 5 b(ject) 28 b(to) p Ft 27 w(F) p Fx 12 w(\() p Fs(u) p Fx(\)) c(=) p Ft 23 w(F) p Fr 2532 2121 a(0) p Fx 2588 2109 a(+) p Ft 18 w(u) p Fr 2719 2121 a(1) p Ft 2755 2109 a(F) p Fr 2808 2121 a(1) p Fx 2865 2109 a(+) p Ft 18 w(:) 14 b(:) g(:) f(u) p Fq 3106 2121 a(q) p Ft 3142 2109 a(F) p Fq 3195 2121 a(q) p Fu 3256 2109 a(\026) p Fx 22 w(0) p Ft(:) p Fx 313 w(\(1.11\)) 945 2300 y(The) 21 b(general) e(case) h(of) h (m) n(ultiple) g(LMI) g(constrain) n(ts) e(and) i(a\016ne) f (equalities) h(can) f(b) r(e) h(handled) g(b) n(y) 945 2408 y(elimination) 26 b(of) h(the) g(latter) f(and) g(using) g(blo) r (c) n(k-diagonal) e(matrices) i(to) g(represen) n(t) g(the) h(former) 945 2516 y(as) g(a) g(single) g(LMI.) 1028 2624 y(The) k(classical) e (Lagrange) f(dualit) n(y) j(theory) f(outlined) h(in) g(the) g (previous) f(section) g(do) r(es) h(not) 945 2732 y(directly) 41 b(apply) h(here,) f(since) h(w) n(e) f(are) g(not) h(dealing) f(with) h (\014nitely) h(man) n(y) e(constrain) n(ts) f(in) 945 2839 y(scalar) 21 b(form;) i(as) f(noted) h(earlier,) f(the) h(LMI) g (constrain) n(t) f(in) n(v) n(olv) n(es) f(an) i(in\014nite) g(n) n(um) n(b) r(er) g(of) g(suc) n(h) 945 2947 y(constrain) n(ts,) h(of) h(the) h (form) f(\(1.3\).) g(One) g(w) n(a) n(y) f(to) h(handle) h(suc) n(h) f (constrain) n(ts) f(is) h(to) g(in) n(tro) r(duce) g(a) 945 3055 y(Lagrangian) g(of) i(the) h(form) p Fu 1866 3238 a(L) p Fx(\() p Fs(u) p Ft(;) 14 b(Z) p Fx 6 w(\)) 23 b(=) p Fs 23 w(c) p Fq 2293 3204 a(T) p Fs 2345 3238 a(u) p Fx 19 w(+) 18 b(trace) o(\() p Ft(Z) 6 b(F) p Fx 12 w(\() p Fs(u) p Fx(\)\)) p Ft(;) p Fx 945 3421 a(where) 23 b(the) h(dual) f(v) -5 b(ariable) p Ft 22 w(Z) p Fx 29 w(is) 24 b(no) n(w) f(a) g(symmetric) g(matrix,) g(of) g (same) g(size) g(as) p Ft 23 w(F) p Fx 12 w(\() p Fs(u) p Fx(\).) h(W) -7 b(e) 24 b(can) 945 3529 y(c) n(hec) n(k) j(that) i(suc) n(h) f(a) h(Lagrange) d(function) j(ful\014lls) g(the) g(same) f(role) f (assigned) h(to) g(the) h(function) 945 3636 y(de\014ned) k(in) g (De\014nition) g(1.4) f(for) g(the) h(case) f(with) h(scalar) e (constrain) n(ts.) g(Indeed,) i(if) g(w) n(e) f(de\014ne) p Ft 945 3744 a(h) p Fx(\() p Fs(u) p Fx(\)) 23 b(=) g(max) p Fq 1375 3756 a(Z) p Fk 4 w(\027) p Fr(0) p Fu 1528 3744 a(L) p Fx(\() p Fs(u) p Ft(;) 14 b(Z) p Fx 6 w(\)) 28 b(then) p Ft 1612 3993 a(h) p Fx(\() p Fs(u) p Fx(\)) c(=) e(max) p Fq 1898 4046 a(Z) p Fk 4 w(\027) p Fr(0) p Fu 2057 3993 a(L) p Fx(\() p Fs(u) p Ft(;) 14 b(Z) p Fx 6 w(\)) 23 b(=) p Fm 2442 3851 a(\() p Fs 2550 3933 a(c) p Fq 2592 3903 a(T) p Fs 2645 3933 a(u) p Fx 83 w(if) p Ft 28 w(F) p Fx 12 w(\() p Fs(u) p Fx(\)) p Fu 24 w(\026) p Fx 22 w(0) p Ft(;) p Fx 2550 4063 a(+) p Fu(1) p Fx 83 w(otherwise.) 3721 3993 y(\(1.12\)) 945 4238 y(Th) n(us,) p Ft 35 w(h) p Fx(\() p Fs(u) p Fx(\)) 37 b(is) e(a) h(barrier) d(for) j(the) g (primal) f(SDP) h(\(1.11\),) f(that) h(is,) g(it) g(coincides) f(with) h (the) 945 4346 y(ob) 5 b(jectiv) n(e) 30 b(of) h(\(1.11\)) f(on) h(its) g(feasible) g(set,) g(and) g(is) g(in\014nite) h(otherwise.) e(Notice) h (that) g(to) g(the) 945 4454 y(LMI) d(constrain) n(t) f(w) n(e) h(no) n (w) f(asso) r(ciate) g(a) g(m) n(ultiplier) p Fy 29 w(matrix) p Fx(,) h(whic) n(h) g(will) g(b) r(e) h(constrained) e(to) 945 4562 y(the) h(p) r(ositiv) n(e) f(semide\014nite) h(cone.) 1028 4670 y(In) g(the) g(ab) r(o) n(v) n(e,) e(w) n(e) h(made) g(use) h(of) f (the) h(fact) g(that,) g(for) f(a) g(giv) n(en) g(symmetric) g(matrix) p Ft 27 w(F) p Fx 12 w(,) 2006 4853 y(\002\() p Ft(F) p Fx 12 w(\)) c(:=) k(sup) p Fq 2334 4923 a(Z) p Fk 4 w(\027) p Fr(0) p Fx 2500 4853 a(trace) o(\() p Ft(Z) 6 b(F) p Fx 12 w(\)) p 90 rotate dyy eop %%Page: 12 12 12 11 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(12) 670 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Fx 714 349 a(is) j(+) p Fu(1) p Fx 31 w(if) p Ft 32 w(F) p Fx 44 w(has) g(a) g(p) r(ositiv) n(e) g(eigen) n(v) -5 b(alue,) 30 b(and) i(zero) e(if) p Ft 32 w(F) p Fx 44 w(is) h(negativ) n(e) f(semide\014nite.) j(This) 714 457 y(prop) r(ert) n(y) j(is) h(ob) n(vious) f(for) g(diagonal) g (matrices,) h(since) g(in) g(that) h(case) e(the) i(v) -5 b(ariable) p Ft 36 w(Z) p Fx 43 w(can) 714 565 y(b) r(e) 32 b(constrained) e(to) i(b) r(e) f(diagonal) g(without) h(loss) e(of) i (generalit) n(y) -7 b(.) 30 b(The) i(general) e(case) h(follo) n(ws) 714 673 y(from) 36 b(the) h(fact) f(that) h(if) p Ft 37 w(F) p Fx 49 w(has) f(the) h(eigen) n(v) -5 b(alue) 35 b(decomp) r(osition) p Ft 36 w(F) p Fx 50 w(=) p Ft 38 w(U) p Fx 9 w(\003) p Ft(U) p Fq 3285 643 a(T) p Fx 3336 673 a(,) i(where) f(\003) 714 781 y(is) g(a) f(diagonal) g(matrix) g(con) n(taining) g(the) i(eigen) n (v) -5 b(alues) 34 b(of) p Ft 36 w(F) p Fx 12 w(,) j(and) p Ft 35 w(U) p Fx 45 w(is) f(orthogonal,) e(then) 714 889 y(trace) o(\() p Ft(Z) 6 b(F) p Fx 12 w(\)) 30 b(=) g(trace) o(\() p Ft(Z) p Fk 1486 859 a(0) p Fx 1510 889 a(\003\),) i(where) p Ft 31 w(Z) p Fk 1962 859 a(0) p Fx 2016 889 a(=) p Ft 30 w(U) p Fq 2177 859 a(T) p Ft 2229 889 a(Z) 6 b(U) p Fx 40 w(spans) 31 b(the) i(p) r(ositiv) n(e) e(semide\014nite) i(cone) 714 997 y(whenev) n(er) p Ft 26 w(Z) p Fx 34 w(do) r(es.) 797 1105 y(Using) 41 b(the) h(ab) r(o) n(v) n(e) e(Lagrangian,) e(one) j (can) g(cast) g(the) h(original) e(problem) g(\(1.11\)) h(as) g(an) 714 1213 y(unconstrained) 26 b(optimization) i(problem:) p Ft 1804 1395 a(p) p Fk 1846 1361 a(\003) p Fx 1907 1395 a(=) 22 b(min) p Fo 2043 1445 a(u) p Fx 2147 1395 a(max) p Fq 2157 1449 a(Z) p Fk 4 w(\027) p Fr(0) p Fu 2315 1395 a(L) p Fx(\() p Fs(u) p Ft(;) 14 b(Z) p Fx 6 w(\)) p Ft(:) p Fx 714 1600 a(By) 27 b(w) n(eak) f(dualit) n(y) -7 b(,) 28 b(w) n(e) f(obtain) h(a) f(lo) n(w) n(er) f(b) r(ound) i(on) p Ft 27 w(p) p Fk 2429 1570 a(\003) p Fx 2495 1600 a(b) n(y) f(exc) n (hanging) f(the) i(min) g(and) f(max:) p Ft 1355 1783 a(d) p Fk 1398 1749 a(\003) p Fx 1459 1783 a(=) c(max) p Fq 1557 1837 a(Z) p Fk 4 w(\027) p Fr(0) p Fx 1715 1783 a(min) p Fo 1764 1833 a(u) p Fu 1867 1783 a(L) p Fx(\() p Fs(u) p Ft(;) 14 b(Z) p Fx 6 w(\)) p Fu 24 w(\024) p Fx 23 w(min) p Fo 2301 1833 a(u) p Fx 2405 1783 a(max) p Fq 2415 1837 a(Z) p Fk 4 w(\027) p Fr(0) p Fu 2573 1783 a(L) p Fx(\() p Fs(u) p Ft(;) g(Z) p Fx 6 w(\)) 24 b(=) p Ft 22 w(p) p Fk 3000 1749 a(\003) p Ft 3038 1783 a(:) p Fx 797 1988 a(The) 30 b(inner) g(minimization) g(problem) g(is) g (easily) f(solv) n(ed) g(analytically) -7 b(,) 29 b(due) h(to) g(the) h (sp) r(ecial) 714 2096 y(structure) c(of) g(the) h(SDP) -7 b(.) 28 b(W) -7 b(e) 28 b(obtain) f(a) h(closed) e(form) i(for) f(the) h (\(Lagrange\)) e(dual) h(function:) p Ft 714 2316 a(g) p Fx 3 w(\() p Ft(Z) p Fx 6 w(\)) 22 b(=) h(min) p Fo 1042 2366 a(u) p Fu 1146 2316 a(L) p Fx(\() p Fs(u) p Ft(;) 14 b(Z) p Fx 6 w(\)) 24 b(=) e(min) p Fo 1580 2366 a(u) p Fs 1684 2316 a(c) p Fq 1726 2282 a(T) p Fs 1778 2316 a(u) p Fx 19 w(+) c(trace) o(\() p Ft(Z) 6 b(F) p Fr 2261 2328 a(0) p Fx 2298 2316 a(\)) 19 b(+) p Fq 2476 2208 a(q) p Fm 2432 2237 a(X) p Fq 2438 2414 a(i) p Fr(=1) p Ft 2566 2316 a(u) p Fq 2614 2328 a(i) p Fx 2669 2316 a(trace) o(\() p Ft(Z) 6 b(F) p Fq 2997 2328 a(i) p Fx 3025 2316 a(\)) 1444 2598 y(=) p Fm 1531 2456 a(\() p Fx 1640 2538 a(trace) o(\() p Ft(Z) g(F) p Fr 1968 2550 a(0) p Fx 2005 2538 a(\)) 84 b(if) p Ft 28 w(c) p Fq 2233 2550 a(i) p Fx 2283 2538 a(=) p Fu 23 w(\000) p Fx(trace) o(\() p Ft(Z) 6 b(F) p Fq 2764 2550 a(i) p Fx 2792 2538 a(\)) p Ft(;) 42 b(i) p Fx 22 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(q) p Fu 1640 2667 a(\0001) p Fx 333 w(otherwise) p Ft -1 w(:) p Fx 714 2822 a(The) 27 b(dual) h(problem) f(can) g(b) r(e) h(explicitly) g(stated) f(as) g (follo) n(ws:) p Ft 714 2988 a(d) p Fk 757 2958 a(\003) p Fx 818 2988 a(=) c(max) p Fq 916 3042 a(Z) p Fk 4 w(\027) p Fr(0) p Fx 1074 2988 a(min) p Fo 1122 3038 a(u) p Fu 1226 2988 a(L) p Fx(\() p Fs(u) p Ft(;) 14 b(Z) p Fx 6 w(\)) 777 3148 y(=) 64 b(max) p Fq 958 3202 a(Z) p Fx 1074 3148 a(trace) o(\() p Ft(Z) 6 b(F) p Fr 1402 3160 a(0) p Fx 1440 3148 a(\)) 55 b(sub) 5 b(ject) 28 b(to) p Ft 83 w(Z) p Fu 29 w(\027) p Fx 22 w(0) p Ft(;) 41 b(c) p Fq 2288 3160 a(i) p Fx 2339 3148 a(=) p Fu 23 w(\000) p Fx(trace) n(\() p Ft(Z) 6 b(F) p Fq 2819 3160 a(i) p Fx 2847 3148 a(\)) p Ft(;) 42 b(i) p Fx 23 w(=) 22 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(q) s(:) p Fx 118 w(\(1.13\)) 714 3326 y(W) -7 b(e) 27 b(observ) n(e) f(that) h(the) h (ab) r(o) n(v) n(e) d(problem) i(is) g(an) g(SDP) -7 b(,) 27 b(with) h(a) f(single) f(LMI) h(constrain) n(t) f(and) p Ft 27 w(q) p Fx 714 3434 a(a\016ne) h(equalities) g(in) h(the) g (matrix) f(dual) h(v) -5 b(ariable) p Ft 26 w(Z) p Fx 6 w(.) 797 3541 y(While) 35 b(w) n(eak) e(dualit) n(y) h(alw) n(a) n (ys) e(holds,) i(strong) f(dualit) n(y) h(ma) n(y) g(not,) g(ev) n(en) g (for) g(SDPs.) g(Not) 714 3649 y(surprisingly) -7 b(,) 26 b(a) g(Slater-t) n(yp) r(e) h(condition) g(ensures) f(strong) g(dualit) n(y) -7 b(.) 28 b(Precisely) -7 b(,) 26 b(if) h(the) h(primal) 714 3757 y(SDP) i(\(1.11\)) e(is) i(strictly) g(feasible,) f(that) h(is,) g (there) f(exist) h(a) p Fs 29 w(u) p Fr 2670 3769 a(0) p Fx 2738 3757 a(suc) n(h) f(that) p Ft 30 w(F) p Fx 12 w(\() p Fs(u) p Fr 3259 3769 a(0) p Fx 3296 3757 a(\)) p Fu 27 w(\036) p Fx 27 w(0,) g(then) p Ft 714 3865 a(p) p Fk 756 3835 a(\003) p Fx 830 3865 a(=) p Ft 37 w(d) p Fk 975 3835 a(\003) p Fx 1013 3865 a(.) 36 b(If,) h(in) f(addition,) g (the) g(dual) g(problem) f(is) h(also) f(strictly) g(feasible,) h (meaning) f(that) 714 3973 y(there) e(exist) p Ft 33 w(Z) p Fu 39 w(\037) p Fx 33 w(0) g(suc) n(h) g(that) p Ft 34 w(c) p Fq 1818 3985 a(i) p Fx 1878 3973 a(=) g(trace) o(\() p Ft(Z) 6 b(F) p Fq 2304 3985 a(i) p Fx 2332 3973 a(\),) p Ft 34 w(i) p Fx 32 w(=) 33 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(q) p Fx 3 w(,) 34 b(then) g(b) r(oth) g(primal) f(and) 714 4081 y(dual) f(optimal) h(v) -5 b(alues) 32 b(are) g(attained) h(b) n (y) f(some) g(optimal) h(pair) f(\() p Fs(u) p Fk 2846 4051 a(\003) p Ft 2884 4081 a(;) 14 b(Z) p Fk 2984 4051 a(\003) p Fx 3022 4081 a(\).) 33 b(In) g(that) g(case,) f(w) n(e) 714 4189 y(can) e(c) n(haracterize) f(suc) n(h) i(optimal) f(pairs) g(as) h (follo) n(ws.) f(In) h(view) f(of) h(the) h(equalit) n(y) e(constrain) n (ts) 714 4297 y(of) d(the) h(dual) g(problem,) f(the) h(dualit) n(y) f (gap) g(can) g(b) r(e) h(expressed) f(as) p Ft 714 4463 a(p) p Fk 756 4429 a(\003) p Fu 812 4463 a(\000) p Ft 18 w(d) p Fk 938 4429 a(\003) p Fx 999 4463 a(=) p Fs 23 w(c) p Fq 1129 4429 a(T) p Fs 1182 4463 a(u) p Fk 1235 4429 a(\003) p Fu 1292 4463 a(\000) p Fx 18 w(trace) o(\() p Ft(Z) p Fk 1650 4429 a(\003) p Ft 1688 4463 a(F) p Fr 1741 4475 a(0) p Fx 1778 4463 a(\)) 999 4596 y(=) p Fu 23 w(\000) p Fx(trace) o(\() p Ft(Z) p Fk 1427 4562 a(\003) p Ft 1465 4596 a(F) p Fx 12 w(\() p Fs(u) p Fk 1615 4562 a(\003) p Fx 1653 4596 a(\)\)) p Ft(:) p Fx 714 4762 a(A) e(zero) f(dualit) n(y) g(gap) g(is) h(equiv) -5 b(alen) n(t) 25 b(to) g(trace) o(\() p Ft(Z) p Fk 2253 4732 a(\003) p Ft 2291 4762 a(F) p Fx 12 w(\() p Fs(u) p Fk 2441 4732 a(\003) p Fx 2479 4762 a(\)\)) f(=) e(0,) j(whic) n(h) g (in) g(turn) g(is) g(equiv) -5 b(alen) n(t) 714 4870 y(to) p Ft 27 w(Z) p Fk 878 4840 a(\003) p Ft 915 4870 a(F) p Fx 12 w(\() p Fs(u) p Fk 1065 4840 a(\003) p Fx 1104 4870 a(\)) 23 b(=) p Ft 23 w(O) p Fx 2 w(,) k(where) p Ft 27 w(O) p Fx 30 w(denotes) f(the) i(zero) e(matrix,) g(since) h(the) g(pro) r(duct) g(of) g(a) g(p) r(ositiv) n(e) 714 4978 y(semide\014nite) 33 b(and) h(a) f(negativ) n(e) f(semide\014nite) i (matrix) e(has) h(zero) f(trace) h(if) h(and) f(only) g(if) h(it) g(is) p 90 rotate dyy eop %%Page: 13 13 13 12 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.5) 78 b(Kernel) 28 b(Metho) l(ds) h(for) f(Data) g(F) -6 b(usion) 2389 b(13) p Fx 945 349 a(zero.) 1028 457 y(T) -7 b(o) 33 b(summarize,) f(consider) g(the) i(SDP) f(\(1.11\)) g(and) g (its) h(Lagrange) c(dual) k(\(1.13\).) e(If) i(either) 945 565 y(problem) 18 b(is) h(strictly) g(feasible,) g(then) g(they) g (share) f(the) h(same) g(optimal) g(v) -5 b(alue.) 19 b(If) g(b) r(oth) g(problems) 945 673 y(are) 36 b(strictly) i (feasible,) f(then) h(the) g(optimal) f(v) -5 b(alues) 38 b(of) f(b) r(oth) h(problems) f(are) f(attained) i(and) 945 781 y(coincide.) 27 b(In) h(this) g(case,) f(a) g(primal-dual) f(pair) h (\() p Fs(u) p Fk 2523 751 a(\003) p Ft 2562 781 a(;) 14 b(Z) p Fk 2662 751 a(\003) p Fx 2699 781 a(\)) 28 b(is) g(optimal) f (if) h(and) g(only) f(if) p Ft 945 947 a(F) p Fx 12 w(\() p Fs(u) p Fk 1095 913 a(\003) p Fx 1133 947 a(\)) p Fu 24 w(\026) p Fx 22 w(0) p Ft(;) 945 1080 y(Z) p Fk 1008 1046 a(\003) p Fu 1069 1080 a(\027) p Fx 22 w(0) p Ft(;) 945 1213 y(c) p Fq 981 1225 a(i) p Fx 1031 1213 a(=) p Fu 23 w(\000) p Fx(trace) o(\() p Ft(Z) p Fk 1459 1178 a(\003) p Ft 1497 1213 a(F) p Fq 1550 1225 a(i) p Fx 1578 1213 a(\)) p Ft(;) 60 b(i) p Fx 23 w(=) 22 b(1) p Ft(;) 14 b(:) g(:) g(:) g(;) g(q) s(;) 945 1346 y(Z) p Fk 1008 1311 a(\003) p Ft 1046 1346 a(F) p Fx 12 w(\() p Fs(u) p Fk 1196 1311 a(\003) p Fx 1234 1346 a(\)) 23 b(=) p Ft 23 w(O) r(:) p Fx 945 1512 a(The) k(ab) r(o) n(v) n(e) e(conditions) i (represen) n(t) e(the) j(expression) d(of) i(the) g(general) f (Karush-Kuhn-T) -7 b(uc) n(k) n(er) 945 1620 y(\(KKT\)) 29 b(conditions) g(in) h(the) g(semide\014nite) g(programming) d(setting.) j(The) g(\014rst) f(three) g(sets) h(of) 945 1727 y(conditions) j (express) g(that) p Fs 34 w(u) p Fk 1883 1697 a(\003) p Fx 1955 1727 a(and) p Ft 34 w(Z) p Fk 2186 1697 a(\003) p Fx 2258 1727 a(are) g(feasible) h(for) f(their) h(resp) r(ectiv) n(e) f (problems;) g(the) 945 1835 y(last) 27 b(condition) g(expresses) f(a) i (complemen) n(tarit) n(y) e(condition.) 1028 1943 y(F) -7 b(or) 26 b(a) g(pair) f(of) i(strictly) f(feasible) g(primal-dual) f (SDPs,) i(solving) e(the) i(primal) e(minimization) 945 2051 y(problem) 45 b(is) g(equiv) -5 b(alen) n(t) 45 b(to) g(maximizing) g(the) h(dual) f(problem) g(and) g(b) r(oth) h(can) f(th) n(us) g(b) r(e) 945 2159 y(considered) 22 b(sim) n(ultaneously) -7 b(.) 23 b(Algorithms) g(indeed) h(mak) n(e) f(use) h(of) f(this) h (relationship) f(and) h(use) 945 2267 y(the) f(dualit) n(y) f(gap) g (as) h(a) f(stopping) g(criterion.) g(A) h(general-purp) r(ose) e (program) f(suc) n(h) j(as) f(SeDuMi) 945 2375 y(\(Sturm,) 45 b(1999\)) e(handles) h(those) g(problems) f(e\016cien) n(tly) -7 b(.) 45 b(This) f(co) r(de) g(uses) g(in) n(terior-p) r(oin) n(t) 945 2483 y(metho) r(ds) 31 b(for) f(SDP) h(\(Nestero) n(v) f(and) h(Nemiro) n(vsky,) e(1994\);) g(these) i(metho) r(ds) g(ha) n(v) n(e) f(a) g(w) n (orst-) 945 2591 y(case) h(complexit) n(y) h(of) p Ft 32 w(O) p Fx 2 w(\() p Ft(q) p Fr 1788 2561 a(2) p Ft 1826 2591 a(p) p Fr 1868 2561 a(2) p Fq(:) p Fr(5) p Fx 1958 2591 a(\)) h(for) e(the) i(general) e(problem) h(\(1.11\).) f (In) i(practice,) e(problem) 945 2699 y(structure) 22 b(can) g(b) r(e) h(exploited) g(for) f(great) g(computational) g(sa) n (vings:) f(e.g.,) h(when) p Ft 23 w(F) p Fx 12 w(\() p Fs(u) p Fx(\)) p Fu 24 w(2) p Fp 23 w(R) p Fq 3803 2669 a(p) p Fk(\002) p Fq(p) p Fx 945 2807 a(consists) 28 b(of) p Ft 30 w(L) p Fx 29 w(diagonal) g(blo) r(c) n(ks) g(of) i(size) p Ft 29 w(p) p Fq 2320 2819 a(i) p Ft 2347 2807 a(;) 43 b(i) p Fx 26 w(=) 26 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(L) p Fx(,) 29 b(these) g(metho) r(ds) h(ha) n(v) n(e) e(a) h(w) n(orst-) 945 2915 y(case) d(complexit) n(y) i(of) p Ft 27 w(O) p Fx 2 w(\() p Ft(q) p Fr 1774 2885 a(2) p Fx 1812 2915 a(\() p Fm 1844 2852 a(P) p Fq 1932 2873 a(L) 1932 2940 y(i) p Fr(=1) p Ft 2058 2915 a(p) p Fr 2100 2885 a(2) p Fq 2100 2936 a(i) p Fx 2137 2915 a(\)) p Ft(p) p Fr 2211 2885 a(0) p Fq(:) p Fr(5) p Fx 2301 2915 a(\)) g(\(V) -7 b(anden) n(b) r (erghe) 27 b(and) g(Bo) n(yd,) g(1996\).) p Fv 198 3272 a(1.5) 100 b(Kernel) 35 b(Metho) s(ds) g(for) g(Data) f(F) -9 b(usion) p 198 3139 3736 3 v Fx 945 3488 a(Giv) n(en) 46 b(m) n(ultiple) h(related) f(data) g(sets) g(\(e.g.,) g(gene) g (expression,) f(protein) h(sequence,) g(and) 945 3595 y(protein-protein) f(in) n(teraction) h(data\),) g(eac) n(h) g(k) n (ernel) g(function) h(pro) r(duces,) f(for) g(the) h(y) n(east) 945 3703 y(genome,) g(a) g(square) g(matrix) g(in) h(whic) n(h) g(eac) n(h) f(en) n(try) g(enco) r(des) g(a) h(particular) e(notion) i(of) 945 3811 y(similarit) n(y) 32 b(of) h(one) g(y) n(east) f(protein) h(to) g (another.) f(Implicitly) -7 b(,) 34 b(eac) n(h) e(matrix) h(also) f (de\014nes) h(an) 945 3919 y(em) n(b) r(edding) 26 b(of) g(the) h (proteins) e(in) h(a) g(feature) g(space.) f(Th) n(us,) h(the) h(k) n (ernel) e(represen) n(tation) f(casts) 945 4027 y(heterogeneous) 17 b(data|v) -5 b(ariable-length) 18 b(amino) h(acid) g(strings,) g (real-v) -5 b(alued) 18 b(gene) h(expression) 945 4135 y(data,) 34 b(a) h(graph) f(of) h(protein-protein) f(in) n (teractions|in) n(to) f(the) j(common) e(format) h(of) g(k) n(ernel) 945 4243 y(matrices.) 1028 4351 y(The) e(k) n(ernel) f(formalism) g(also) g (allo) n(ws) g(these) h(v) -5 b(arious) 31 b(matrices) i(to) f(b) r(e) i (com) n(bined.) f(Basic) 945 4459 y(algebraic) d(op) r(erations) h(suc) n(h) i(as) e(addition,) i(m) n(ultiplication) f(and) h(exp) r(onen) n (tiation) e(preserv) n(e) 945 4567 y(the) 26 b(k) n(ey) f(prop) r(ert) n (y) g(of) g(p) r(ositiv) n(e) h(semide\014niteness,) g(and) f(th) n(us) h(allo) n(w) f(a) g(simple) h(but) g(p) r(o) n(w) n(erful) 945 4675 y(algebra) i(of) i(k) n(ernels) f(\(Berg) h(et) g(al.,) g(1984\).) f(F) -7 b(or) 29 b(example,) h(giv) n(en) f(t) n(w) n(o) h(k) n(ernels) p Ft 29 w(K) p Fr 3572 4687 a(1) p Fx 3639 4675 a(and) p Ft 30 w(K) p Fr 3874 4687 a(2) p Fx 3910 4675 a(,) 945 4783 y(inducing) c(the) h(em) n(b) r(eddings) g(\010) p Fr 1934 4795 a(1) p Fx 1971 4783 a(\() p Fs(x) p Fx(\)) h(and) e(\010) p Fr 2333 4795 a(2) p Fx 2371 4783 a(\() p Fs(x) p Fx(\),) h(resp) r (ectiv) n(ely) -7 b(,) 26 b(it) h(is) g(p) r(ossible) f(to) g(de\014ne) h(the) 945 4891 y(k) n(ernel) p Ft 38 w(K) p Fx 47 w(=) p Ft 41 w(K) p Fr 1496 4903 a(1) p Fx 1558 4891 a(+) p Ft 26 w(K) p Fr 1720 4903 a(2) p Fx 1757 4891 a(,) 39 b(inducing) f(the) i(em) n(b) r(edding) e(\010\() p Fs(x) p Fx(\)) 43 b(=) e([\010) p Fr 3157 4903 a(1) p Fx 3194 4891 a(\() p Fs(x) p Fx(\)) p Ft(;) p Fx 14 w(\010) p Fr 3405 4903 a(2) p Fx 3443 4891 a(\() p Fs(x) p Fx(\)].) f(Of) f(ev) n (en) 945 4999 y(greater) d(in) n(terest,) j(w) n(e) f(can) g(consider) f (parameterized) g(com) n(binations) h(of) g(k) n(ernels.) g(In) g(this) p 90 rotate dyy eop %%Page: 14 14 14 13 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(14) 670 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Fx 714 349 a(c) n(hapter,) 39 b(giv) n(en) g(a) h(set) g(of) g(k) n (ernels) p Fu 39 w(K) p Fx 46 w(=) p Fu 43 w(f) p Ft(K) p Fr 2227 361 a(1) p Ft 2263 349 a(;) 14 b(K) p Fr 2371 361 a(2) p Ft 2408 349 a(;) g(:) g(:) g(:) g(;) g(K) p Fq 2664 361 a(m) p Fu 2726 349 a(g) p Fx(,) 40 b(w) n(e) g(will) g (form) g(the) g(linear) 714 457 y(com) n(bination) p Ft 1948 669 a(K) p Fx 29 w(=) p Fq 2166 565 a(m) p Fm 2135 590 a(X) p Fq 2141 767 a(i) p Fr(=1) p Ft 2269 669 a(\026) p Fq 2319 681 a(i) p Ft 2347 669 a(K) p Fq 2418 681 a(i) p Ft 2445 669 a(:) p Fx 1022 w(\(1.14\)) 797 908 y(As) 24 b(w) n(e) g(ha) n(v) n(e) g(discussed,) g(\014tting) h(an) f(SVM) h(to) g(a) f(single) g(data) g(source) f(in) n(v) n(olv) n(es) g (solving) g(the) 714 1016 y(quadratic) g(program) g(\(1.2\)) i(based) f (on) h(the) g(k) n(ernel) f(matrix) g(and) h(the) g(lab) r(els.) g(It) g (is) g(p) r(ossible) g(to) 714 1124 y(extend) g(this) g(optimization) g (problem) g(not) g(only) f(to) h(\014nd) h(optimal) f(discriminan) n(t) f(b) r(oundaries) 714 1231 y(but) g(also) e(to) h(\014nd) h(optimal) f (v) -5 b(alues) 23 b(of) g(the) h(co) r(e\016cien) n(ts) p Ft 23 w(\026) p Fq 2521 1243 a(i) p Fx 2572 1231 a(in) g(\(1.14\)) e (for) h(problems) g(in) n(v) n(olving) 714 1339 y(m) n(ultiple) e(k) n (ernels) f(\(Lanc) n(kriet) f(et) i(al.,) g(2002\).) e(In) i(the) g (case) f(of) h(the) g(1-norm) e(soft) i(margin) f(SVM,) 714 1447 y(w) n(e) 33 b(w) n(an) n(t) f(to) h(minimize) h(the) g(same) e (cost) h(function) h(\(1.1\),) f(no) n(w) g(with) g(resp) r(ect) g(to) g (b) r(oth) h(the) 714 1555 y(discriminan) n(t) f(b) r(oundary) f(and) i (the) p Ft 33 w(\026) p Fq 1943 1567 a(i) p Fx 1971 1555 a(.) g(Since) f(the) h(primal) f(problem) g(\(1.1\)) g(is) h(con) n(v) n (ex) e(and) 714 1663 y(strictly) 27 b(feasible,) g(strong) g(dualit) n (y) g(holds) g(for) g(\(1.1\)) h(and) f(\(1.2\)) h(according) d(to) j (Lemma) f(1.8:) p Ft 714 1879 a(!) p Fq 766 1891 a(S) p Fr 3 w(1) p Fx 847 1879 a(\() p Ft(K) p Fx 6 w(\)) c(=) p Fs 22 w(w) p Fq 1168 1845 a(T) p Fk 1167 1900 a(\003) p Fs 1221 1879 a(w) p Fk 1290 1891 a(\003) p Fx 1346 1879 a(+) p Ft 18 w(C) p Fq 1548 1775 a(n) p Fm 1509 1800 a(X) p Fq 1515 1977 a(i) p Fr(=1) p Ft 1642 1879 a(\030) p Fq 1678 1891 a(i;) p Fk(\003) p Fx 3490 1879 a(\(1.15\)) 1011 2084 y(=) f(max) p Fn 1144 2135 a(\013) p Fx 1290 2084 a(2) p Fn(\013) p Fq 1394 2050 a(T) p Fs 1447 2084 a(e) p Fu 18 w(\000) p Fn 18 w(\013) p Fq 1655 2050 a(T) p Fx 1707 2084 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 53 w(:) p Ft 50 w(C) p Fu 30 w(\025) p Fn 22 w(\013) p Fu 24 w(\025) p Fx 22 w(0) p Ft(;) p Fn 60 w(\013) p Fq 3046 2050 a(T) p Fs 3098 2084 a(y) p Fx 25 w(=) h(0) p Ft(:) p Fx 714 2259 a(where) p Fs 21 w(e) p Fx 23 w(is) f(a) g(v) n (ector) f(con) n(taining) g(ones) h(and) p Fs 22 w(w) p Fk 2204 2271 a(\003) p Fx 2265 2259 a(and) p Ft 22 w(\030) p Fq 2457 2271 a(i;) p Fk(\003) p Fx 2561 2259 a(the) h(optimal) f(v) -5 b(alues) 22 b(of) g(the) h(primal) 714 2367 y(v) -5 b(ariables) p Fs 27 w(w) p Fx 31 w(and) p Ft 28 w(\030) p Fq 1357 2379 a(i) p Fx 1385 2367 a(.) 29 b(T) -7 b(raining) 28 b(an) h(SVM) g(for) g (a) f(giv) n(en) g(k) n(ernel) p Ft 28 w(K) p Fu 31 w(\027) p Fx 25 w(0) g(yields) h(the) g(minimal) 714 2475 y(v) -5 b(alue) 22 b(\(1.15\)) f(of) h(\(1.1\)) g(whic) n(h) g(is) g(ob) n (viously) f(a) g(function) i(of) f(the) g(particular) f(c) n(hoice) g (of) p Ft 22 w(K) p Fx 6 w(,) h(as) g(is) 714 2583 y(expressed) j (explicitly) i(in) g(\(1.15\)) f(as) f(a) i(dual) f(problem.) g(Let) h (us) f(no) n(w) g(optimize) h(this) g(quan) n(tit) n(y) 714 2691 y(with) c(resp) r(ect) f(to) h(the) g(k) n(ernel) f(matrix) p Ft 22 w(K) p Fx 28 w(=) p Fm 2104 2629 a(P) p Fq 2192 2649 a(m) 2192 2716 y(i) p Fr(=1) p Ft 2317 2691 a(\026) p Fq 2367 2703 a(i) p Ft 2395 2691 a(K) p Fq 2466 2703 a(i) p Fx 2493 2691 a(,) h(i.e.,) g(let) g(us) f(try) h(to) f(\014nd) h (the) g(w) n(eigh) n(ts) p Fn 714 2799 a(\026) p Fu 30 w(2) p Fp 32 w(R) p Fq 944 2769 a(m) p Fx 1045 2799 a(for) 32 b(whic) n(h) g(the) h(corresp) r(onding) e(em) n(b) r(edding) h(sho) n (ws) f(minimal) p Ft 33 w(!) p Fq 3153 2811 a(S) p Fr 3 w(1) p Fx 3234 2799 a(\() p Ft(K) p Fx 6 w(\),) h(k) n(eeping) 714 2907 y(the) c(trace) e(of) p Ft 28 w(K) p Fx 33 w(constan) n(t:) 1289 3118 y(min) p Fn 1175 3172 a(\026) p Fk(2) p Fd(R) p Fc 1326 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Fx 3 w(:) 1087 2961 y(min) p Fn 1005 3032 a(\026) p Fq(;t;\025;) p Fn(\027) p Fq 5 w(;) p Fn(\016) p Ft 1473 2961 a(t) p Fx 2218 w(\(1.17\)) 945 3227 y(sub) 5 b(ject) 28 b(to) 165 b(trace) p Fm 1667 3085 a( ) p Fq 1764 3123 a(m) p Fm 1733 3148 a(X) p Fq 1739 3325 a(i) p Fr(=1) p Ft 1867 3227 a(\026) p Fq 1917 3239 a(i) p Ft 1944 3227 a(K) p Fq 2015 3239 a(i) p Fm 2043 3085 a(!) p Fx 2131 3227 a(=) p Ft 23 w(c;) p Fq 1504 3391 a(m) p Fm 1473 3416 a(X) p Fq 1480 3593 a(i) p Fr(=1) p Ft 1607 3495 a(\026) p Fq 1657 3507 a(i) p Ft 1685 3495 a(K) p Fq 1756 3507 a(i) p Fu 1806 3495 a(\027) p Fx 23 w(0) p Ft(;) p Fm 1473 3634 a( ) p Fx 1539 3717 a(diag) o(\() p Fs(y) p Fx 1 w(\)\() p Fm 1838 3655 a(P) p Fq 1927 3675 a(m) 1927 3742 y(i) p Fr(=1) p Ft 2053 3717 a(\026) p Fq 2103 3729 a(i) p Ft 2131 3717 a(K) p Fq 2202 3729 a(i) p Fx 2229 3717 a(\)diag) o(\() p Fs(y) p Fx 1 w(\)) p Fs 85 w(e) p Fx 18 w(+) p Fn 18 w(\027) p Fu 24 w(\000) p Fn 18 w(\016) p Fx 22 w(+) p Ft 18 w(\025) p Fs(y) p Fx 1702 3846 a(\() p Fs(e) p Fx 18 w(+) p Fn 18 w(\027) p Fu 24 w(\000) p Fn 18 w(\016) p Fx 22 w(+) p Ft 18 w(\025) p Fs(y) p Fx 1 w(\)) p Fq 2313 3816 a(T) p Ft 2696 3846 a(t) p Fu 19 w(\000) p Fx 18 w(2) p Ft(C) p Fn 6 w(\016) p Fq 2981 3809 a(T) p Fs 3034 3846 a(e) p Fm 3161 3634 a(!) p Fu 3250 3776 a(\027) p Fx 23 w(0) p Ft(;) p Fn 1473 3968 a(\027) p Ft 6 w(;) p Fn 14 w(\016) p Fu 26 w(\025) p Fx 23 w(0) p Ft(:) p Fz 945 4134 a(Pr) -5 b(o) g(of) p Fx 83 w(After) 28 b(substitution) h(of) p Ft 27 w(!) p Fq 2082 4146 a(S) p Fr 3 w(1) p Fx 2163 4134 a(\() p Ft(K) p Fx 6 w(\)) f(as) f(de\014ned) h(in) f(\(1.15\),) g(\(1.16\)) g (b) r(ecomes:) 1058 4300 y(min) p Fn 945 4354 a(\026) p Fk(2) p Fd(R) p Fc 1096 4337 a(m) p Fq 1145 4354 a(;K) p Fk 4 w(\027) p Fr(0) p Fx 1347 4300 a(max) p Fn 1393 4351 a(\013) p Fx 1667 4300 a(2) p Fn(\013) p Fq 1772 4266 a(T) p Fs 1824 4300 a(e) p Fu 19 w(\000) p Fn 18 w(\013) p Fq 2033 4266 a(T) p Fx 2085 4300 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 1139 4540 a(sub) 5 b(ject) 28 b(to) p Ft 165 w(C) p Fu 30 w(\025) p Fn 22 w(\013) p Fu 24 w(\025) p Fx 22 w(0) p Ft(;) p Fn 60 w(\013) p Fq 2205 4506 a(T) p Fs 2257 4540 a(y) p Fx 25 w(=) 23 b(0) p Ft(;) p Fx 59 w(trace) o(\() p Ft(K) p Fx 6 w(\)) g(=) p Ft 23 w(c;) 60 b(K) p Fx 28 w(=) p Fq 3313 4436 a(m) p Fm 3282 4461 a(X) p Fq 3288 4638 a(i) p Fr(=1) p Ft 3416 4540 a(\026) p Fq 3466 4552 a(i) p Ft 3493 4540 a(K) p Fq 3564 4552 a(i) p Ft 3592 4540 a(;) p Fx 106 w(\(1.18\)) 945 4767 y(with) p Ft 29 w(c) p Fx 29 w(a) 28 b(constan) n(t.) g(Assume) h (that) p Ft 29 w(K) p Fu 31 w(\037) p Fx 24 w(0,) g(hence) g(diag) o (\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fu 26 w(\037) p Fx 25 w(0) f(\(the) i(follo) n (wing) 945 4875 y(can) e(b) r(e) h(extended) g(to) g(the) g(general) e (semide\014nite) j(case\).) e(W) -7 b(e) 29 b(note) g(that) p Ft 29 w(!) p Fq 3354 4887 a(S) p Fr 3 w(1) p Fx 3435 4875 a(\() p Ft(K) p Fx 6 w(\)) f(is) h(con) n(v) n(ex) 945 4983 y(in) p Ft 27 w(K) p Fx 33 w(\(b) r(ecause) f(of) f(Prop) r (osition) f(1.9\)) h(and) g(th) n(us) h(in) p Fn 28 w(\026) p Fx -1 w(,) g(since) p Ft 27 w(K) p Fx 33 w(is) g(a) f(linear) f (function) i(of) p Fn 28 w(\026) p Fx(.) p 90 rotate dyy eop %%Page: 16 16 16 15 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(16) 670 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Fx 714 349 a(Giv) n(en) h(the) g(con) n(v) n(ex) f(constrain) n(ts) g (in) h(\(1.18\),) g(the) g(optimization) g(problem) g(is) g(th) n(us) g (certainly) 714 457 y(con) n(v) n(ex) d(in) p Fn 28 w(\026) p Fx -1 w(.) i(W) -7 b(e) 28 b(write) g(this) f(as:) 849 731 y(min) p Fn 714 785 a(\026) p Fk -1 w(2) p Fd(R) p Fc 864 768 a(m) p Fq 914 785 a(;K) p Fk 4 w(\027) p Fr(0) p Fq(;t) p Ft 1137 731 a(t) p Fx 51 w(:) p Ft 51 w(t) p Fu 23 w(\025) p Fx 23 w(max) p Fn 1478 782 a(\013) p Fx 1624 731 a(2) p Fn(\013) p Fq 1729 697 a(T) p Fs 1781 731 a(e) p Fu 18 w(\000) p Fn 18 w(\013) p Fq 1989 697 a(T) p Fx 2042 731 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Ft 1 w(;) 1292 971 y(C) p Fu 29 w(\025) p Fn 23 w(\013) p Fu 23 w(\025) p Fx 23 w(0) p Ft(;) p Fn 59 w(\013) p Fq 1829 937 a(T) p Fs 1882 971 a(y) p Fx 25 w(=) 22 b(0) p Ft(;) p Fx 60 w(trace) o(\() p Ft(K) p Fx 6 w(\)) h(=) p Ft 23 w(c;) 59 b(K) p Fx 29 w(=) p Fq 2937 868 a(m) p Fm 2907 893 a(X) p Fq 2913 1069 a(i) p Fr(=1) p Ft 3040 971 a(\026) p Fq 3090 983 a(i) p Ft 3118 971 a(K) p Fq 3189 983 a(i) p Ft 3216 971 a(:) p Fx 251 w(\(1.19\)) 714 1206 y(W) -7 b(e) 25 b(will) g(no) n(w) f(express) p Ft 24 w(t) p Fu 23 w(\025) p Fx 23 w(max) p Fn 1760 1218 a(\013) p Fx 1865 1206 a(2) p Fn(\013) p Fq 1969 1176 a(T) p Fs 2021 1206 a(e) p Fu 13 w(\000) p Fn 13 w(\013) p Fq 2219 1176 a(T) p Fx 2271 1206 a(diag\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 26 w(as) g(an) h(LMI) g(using) g(du-) 714 1314 y(alit) n(y) -7 b(.) 23 b(In) i(particular,) e(w) n(e) h(express) f (the) h(constrain) n(t) f(using) h(the) h(dual) f(minimization) g (problem.) 714 1422 y(This) j(will) h(allo) n(w) e(us) h(to) h(drop) f (the) g(minimization) h(and) f(use) h(the) f(Sc) n(h) n(ur) g (complemen) n(t) h(lemma) 714 1529 y(to) f(obtain) g(an) h(LMI.) f(W) -7 b(e) 28 b(explain) g(this) g(no) n(w) f(in) g(more) g(detail.) 797 1637 y(De\014ne) h(the) g(Lagrangian) d(of) i(the) h(maximization) f (problem) g(\(1.2\)) h(b) n(y) p Fu 747 1820 a(L) p Fx(\() p Fn(\013) p Ft(;) p Fn 14 w(\027) p Ft 6 w(;) 14 b(\025;) p Fn 14 w(\016) p Fx 4 w(\)) 23 b(=) g(2) p Fn(\013) p Fq 1406 1786 a(T) p Fs 1458 1820 a(e) p Fu 19 w(\000) p Fn 18 w(\013) p Fq 1667 1786 a(T) p Fx 1719 1820 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 20 w(+) 18 b(2) p Fn(\027) p Fq 2592 1786 a(T) p Fn 2645 1820 a(\013) p Fx 18 w(+) g(2) p Ft(\025) p Fs(y) p Fq 2950 1786 a(T) p Fn 3003 1820 a(\013) p Fx 19 w(+) g(2) p Fn(\016) p Fq 3256 1783 a(T) p Fx 3308 1820 a(\() p Ft(C) p Fs 6 w(e) p Fu 19 w(\000) p Fn 18 w(\013) p Fx(\)) p Ft(;) p Fx 714 2003 a(where) p Ft 27 w(\025) p Fu 23 w(2) p Fp 24 w(R) p Fx 33 w(and) p Fn 28 w(\027) p Ft 5 w(;) p Fn 14 w(\016) p Fu 27 w(2) p Fp 23 w(R) p Fq 1645 1973 a(n) p Fx 1696 2003 a(.) 28 b(By) f(dualit) n(y) -7 b(,) 28 b(w) n(e) f(ha) n(v) n(e) p Ft 927 2185 a(!) p Fq 979 2197 a(S) p Fr 3 w(1) p Fx 1060 2185 a(\() p Ft(K) p Fx 6 w(\)) c(=) g(max) p Fn 1358 2236 a(\013) p Fx 1609 2185 a(min) p Fn 1503 2257 a(\027) p Fk 6 w(\025) p Fr(0) p Fq(;) p Fn(\016) p Fk 3 w(\025) p Fr(0) p Fq(;\025) p Fu 1889 2185 a(L) p Fx(\() p Fn(\013) p Ft 1 w(;) p Fn 14 w(\027) p Ft 5 w(;) 14 b(\025;) p Fn 14 w(\016) p Fx 4 w(\)) 24 b(=) 127 b(min) p Fn 2444 2257 a(\027) p Fk 6 w(\025) p Fr(0) p Fq(;) p Fn(\016) p Fk 3 w(\025) p Fr(0) p Fq(;\025) p Fx 2830 2185 a(max) p Fn 2876 2236 a(\013) p Fu 3021 2185 a(L) p Fx(\() p Fn(\013) p Ft 1 w(;) p Fn 14 w(\027) p Ft 6 w(;) 14 b(\025;) p Fn 14 w(\016) p Fx 3 w(\)) p Ft(;) p Fx 714 2413 a(where) p Fn 27 w(\027) p Fu 29 w(\025) p Fx 23 w(0) p Fu 22 w(,) p Ft 24 w(\027) p Fq 1331 2425 a(i) p Fu 1381 2413 a(\025) p Fx 23 w(0) 28 b(for) p Ft 27 w(i) p Fx 23 w(=) 22 b(1) p Ft(;) 14 b(:) g(:) g(:) g(;) g(n) p Fx(,) 27 b(similarly) g(for) p Fn 27 w(\016) p Fx 4 w(.) g(Since) h(diag\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fu 25 w(\037) p Fx 23 w(0,) 714 2521 y(at) f(the) h(optim) n (um,) g(w) n(e) f(ha) n(v) n(e) p Fn 1413 2704 a(\013) p Fx 23 w(=) 22 b(\() q(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)\)) p Fk 2264 2662 a(\000) p Fr(1) p Fx 2367 2704 a(\() p Fs(e) p Fx 19 w(+) p Fn 18 w(\027) p Fu 24 w(\000) p Fn 18 w(\016) p Fx 22 w(+) p Ft 18 w(\025) p Fs(y) p Fx 1 w(\)) p Ft(;) p Fx 714 2887 a(and) 27 b(can) g(form) h(the) g(dual) f(problem) p Ft 714 3069 a(!) p Fq 766 3081 a(S) p Fr 3 w(1) p Fx 847 3069 a(\() p Ft(K) p Fx 6 w(\)) c(=) 61 b(min) p Fn 1098 3141 a(\027) p Fq 6 w(;) p Fn 18 w(\016) p Fq 4 w(;) 19 b(\025) p Fx 1338 3069 a(\() p Fs(e) p Fx(+) p Fn(\027) p Fu 5 w(\000) p Fn(\016) p Fx 3 w(+) p Ft(\025) p Fs(y) p Fx 1 w(\)) p Fq 1839 3035 a(T) p Fx 1907 3069 a(\(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)\)) p Fk 2585 3028 a(\000) p Fr(1) p Fx 2687 3069 a(\() p Fs(e) p Fx(+) p Fn(\027) p Fu 6 w(\000) p Fn(\016) p Fx 3 w(+) p Ft(\025) p Fs(y) p Fx 1 w(\)+2) p Ft(C) p Fn 6 w(\016) p Fq 3408 3032 a(T) p Fs 3460 3069 a(e) p Fx 41 w(:) p Fn 42 w(\027) p Fu 29 w(\025) p Fx 22 w(0) p Ft(;) p Fn 32 w(\016) p Fu 27 w(\025) p Fx 22 w(0) p Ft(:) p Fx 714 3297 a(W) -7 b(e) 26 b(obtain) g(that) g(for) f(an) n(y) p Ft 26 w(t) e(>) p Fx 22 w(0,) j(the) g(constrain) n(t) p Ft 25 w(!) p Fq 2384 3309 a(S) p Fr 3 w(1) p Fx 2465 3297 a(\() p Ft(K) p Fx 6 w(\)) p Fu 23 w(\024) p Ft 23 w(t) p Fx 26 w(is) f(true) h(if) h (and) f(only) f(if) i(there) 714 3405 y(exist) p Fn 27 w(\027) p Fu 29 w(\025) p Fx 22 w(0,) p Fn 28 w(\016) p Ft 26 w(>) p Fx 23 w(0) g(and) p Ft 28 w(\025) p Fx 28 w(suc) n(h) g(that) 914 3588 y(\() p Fs(e) p Fx 19 w(+) p Fn 18 w(\027) p Fu 24 w(\000) p Fn 18 w(\016) p Fx 22 w(+) p Ft 18 w(\025) p Fs(y) p Fx 1 w(\)) p Fq 1526 3554 a(T) p Fx 1593 3588 a(\() q(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)\)) p Fk 2271 3546 a(\000) p Fr(1) p Fx 2374 3588 a(\() p Fs(e) p Fx 19 w(+) p Fn 18 w(\027) p Fu 24 w(\000) p Fn 18 w(\016) p Fx 22 w(+) p Ft 18 w(\025) p Fs(y) p Fx 1 w(\)) 20 b(+) e(2) p Ft(C) p Fn 6 w(\016) p Fq 3242 3551 a(T) p Fs 3294 3588 a(e) p Fu 23 w(\024) p Ft 23 w(t;) p Fx 714 3771 a(or,) 26 b(equiv) -5 b(alen) n(tly) 28 b(\(using) f(the) h(Sc) n(h) n(ur) f (complemen) n(t) h(lemma\),) f(suc) n(h) h(that) p Fm 1376 3874 a( ) p Fx 1509 3957 a(diag\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fs 110 w(e) p Fx 18 w(+) p Fn 18 w(\027) p Fu 24 w(\000) p Fn 18 w(\016) p Fx 22 w(+) p Ft 18 w(\025) p Fs(y) p Fx 1484 4086 a(\() p Fs(e) p Fx 18 w(+) p Fn 18 w(\027) p Fu 24 w(\000) p Fn 18 w(\016) p Fx 22 w(+) p Ft 18 w(\025) p Fs(y) p Fx 1 w(\)) p Fq 2095 4056 a(T) p Ft 2316 4086 a(t) p Fu 18 w(\000) p Fx 18 w(2) p Ft(C) p Fn 6 w(\016) p Fq 2600 4049 a(T) p Fs 2653 4086 a(e) p Fm 2822 3874 a(!) p Fu 2911 4016 a(\027) p Fx 22 w(0) p 90 rotate dyy eop %%Page: 17 17 17 16 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.5) 78 b(Kernel) 28 b(Metho) l(ds) h(for) f(Data) g(F) -6 b(usion) 2389 b(17) p Fx 945 349 a(holds.) 27 b(T) -7 b(aking) 27 b(this) h(in) n(to) f(accoun) n(t,) g(\(1.19\)) g(can) g(b) r(e) h(expressed) e(as:) 1136 515 y(min) p Fn 945 587 a(\026) p Fk(2) p Fd(R) p Fc 1096 570 a(m) p Fq 1145 587 a(;K;t;\025;) p Fn(\027) p Fq 6 w(;) p Fn(\016) p Ft 1659 515 a(t) p Fx 2032 w(\(1.20\)) 1102 698 y(sub) 5 b(ject) 28 b(to) 194 b(trace) o(\() p Ft(K) p Fx 6 w(\)) 23 b(=) p Ft 22 w(c;) 1631 881 y(K) p Fx 28 w(=) p Fq 1849 777 a(m) p Fm 1818 802 a(X) p Fq 1824 979 a(i) p Fr(=1) p Ft 1952 881 a(\026) p Fq 2002 893 a(i) p Ft 2029 881 a(K) p Fq 2100 893 a(i) p Fu 2151 881 a(\027) p Fx 22 w(0) p Ft(;) p Fm 1631 1021 a( ) p Fx 1723 1103 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fs 110 w(e) p Fx 18 w(+) p Fn 18 w(\027) p Fu 24 w(\000) p Fn 18 w(\016) p Fx 22 w(+) p Ft 18 w(\025) p Fs(y) p Fx 1697 1233 a(\() p Fs(e) p Fx 18 w(+) p Fn 18 w(\027) p Fu 24 w(\000) p Fn 18 w(\016) p Fx 22 w(+) p Ft 18 w(\025) p Fs(y) p Fx 1 w(\)) p Fq 2308 1202 a(T) p Ft 2529 1233 a(t) p Fu 18 w(\000) p Fx 18 w(2) p Ft(C) p Fn 6 w(\016) p Fq 2814 1195 a(T) p Fs 2866 1233 a(e) p Fm 2993 1021 a(!) p Fu 3082 1163 a(\027) p Fx 23 w(0) p Ft(;) p Fn 1631 1350 a(\027) p Fu 29 w(\025) p Fx 22 w(0) p Ft(;) p Fn 1631 1483 a(\016) p Fu 26 w(\025) p Fx 23 w(0) p Ft(;) p Fx 945 1649 a(whic) n(h) 28 b(yields) h(\(1.17\)) f (after) g(substituting) p Ft 29 w(K) p Fx 31 w(=) p Fm 2510 1586 a(P) p Fq 2598 1607 a(m) 2598 1674 y(i) p Fr(=1) p Ft 2723 1649 a(\026) p Fq 2773 1661 a(i) p Ft 2801 1649 a(K) p Fq 2872 1661 a(i) p Fx 2928 1649 a(to) h(eliminate) p Ft 28 w(K) p Fx 6 w(.) g(Notice) f(that) p Fn 945 1757 a(\027) p Fu 28 w(\025) p Fx 23 w(0) p Fu 23 w(,) p Fx 23 w(diag) o(\() p Fn(\027) p Fx 6 w(\)) p Fu 23 w(\027) p Fx 23 w(0,) f(and) h(th) n(us) f(an) g(LMI;) h(similarly) f(for) p Fn 27 w(\016) p Fu 27 w(\025) p Fx 22 w(0.) p 3193 1757 42 42 v 1028 1923 a(Notice) 38 b(that) h(the) f(optimization) g (problem) g(\(1.17\)) f(is) h(an) g(SDP) h(in) f(the) h(standard) e (form) 945 2031 y(\(1.4\).) 22 b(This) h(leads) e(to) i(a) f(general) f (metho) r(d) i(for) f(learning) f(the) i(optimal) f(com) n(bination) g (of) h(k) n(ernel) 945 2138 y(matrices) k(as) g(a) g(semide\014nite) h (programming) e(problem,) h(whic) n(h) h(can) f(b) r(e) i(solv) n(ed) d (via) i(e\016cien) n(t) 945 2246 y(in) n(terior-p) r(oin) n(t) 21 b(algorithms) h(\(V) -7 b(anden) n(b) r(erghe) 23 b(and) g(Bo) n(yd,) f (1996\).) g(Although) h(e\016cien) n(t,) h(these) 945 2354 y(algorithms) 32 b(will) i(still) f(ha) n(v) n(e) g(a) g(w) n (orst-case) e(complexit) n(y) p Ft 33 w(O) p Fx 2 w(\() p Ft(n) p Fr 2946 2324 a(4) p Fq(:) p Fr(5) p Fx 3037 2354 a(\)) j(in) f(this) h(particular) e(case,) 945 2462 y(according) 26 b(to) h(the) h(complexit) n(y) f(results) g(men) n(tioned) h(in) g (Section) f(1.4.3.) 1028 2570 y(In) j(this) g(discussion,) f(the) p Ft 30 w(K) p Fq 1928 2582 a(i) p Fx 1985 2570 a(are) g(p) r(ositiv) n (e) g(semide\014nite) h(b) n(y) g(construction;) f(th) n(us) p Ft 30 w(K) p Fu 32 w(\027) p Fx 26 w(0) 945 2678 y(is) 42 b(automatically) f(satis\014ed) h(if) h(the) g(w) n(eigh) n(ts) p Ft 41 w(\026) p Fq 2524 2690 a(i) p Fx 2594 2678 a(are) e(constrained) g (to) i(b) r(e) f(non-negativ) n(e.) 945 2786 y(W) -7 b(e) 37 b(will) f(no) n(w) g(p) r(oin) n(t) g(out) h(an) f(additional) g (adv) -5 b(an) n(tage) 34 b(of) j(the) g(restriction) p Fn 35 w(\026) p Fu 37 w(\025) p Fx 38 w(0:) f(it) g(will) 945 2894 y(allo) n(w) c(us) i(to) g(cast) f(the) h(SDP) g(\(1.17\)) f(as) g (a) p Fy 33 w(quadr) l(atic) l(al) t(ly) 38 b(c) l(onstr) l(aine) l(d) e (quadr) l(atic) g(pr) l(o) l(gr) l(am) 945 3002 y(\(QCQP\)) p Fx(,) 20 b(whic) n(h) h(has) e(b) r(ene\014cial) i(computational) f (e\013ects) g(b) n(y) g(lo) n(w) n(ering) f(the) i(e\016ciency) f(of) g (the) 945 3110 y(computation) 28 b(to) p Ft 29 w(O) p Fx 2 w(\() p Ft(n) p Fr 1680 3080 a(3) p Fx 1718 3110 a(\)) i(in) f(terms) f(of) h(the) h(n) n(um) n(b) r(er) e(of) h(data) g (p) r(oin) n(ts.) g(Also,) f(the) i(constrain) n(t) 945 3218 y(can) k(result) f(in) i(impro) n(v) n(ed) e(n) n(umerical) g (stabilit) n(y|it) h(prev) n(en) n(ts) g(the) g(algorithm) f(from) h (using) 945 3326 y(large) 28 b(w) n(eigh) n(ts) i(with) g(opp) r(osite) g(sign) g(that) g(cancel.) g(Finally) -7 b(,) 30 b(Lanc) n(kriet) f(et) h(al.) g(\(2002\)) f(sho) n(w) 945 3434 y(that) j(the) h(constrain) n (t) e(also) g(yields) i(b) r(etter) f(estimates) g(of) h(the) f (generalization) f(p) r(erformance) 945 3542 y(of) c(these) h (algorithms.) 1028 3649 y(Solving) 21 b(the) i(original) d(learning) h (problem) h(\(1.16\)) f(sub) 5 b(ject) 22 b(to) g(the) h(extra) e (constrain) n(t) p Fn 21 w(\026) p Fu 22 w(\025) p Fx 23 w(0) 945 3757 y(yields:) 1016 3923 y(min) p Fn 945 3977 a(\026) p Fk(2) p Fd(R) p Fc 1096 3961 a(m) p Fq 1145 3977 a(;K) p Fx 1483 3923 a(max) p Fn 1239 3982 a(\013) p Fr 28 w(:) p Fq 27 w(C) p Fk 4 w(\025) p Fn(\013) p Fk(\025) p Fr(0) p Fq(;) p Fn(\013) p Fc 1711 3966 a(T) p Fo 1757 3982 a(y) p Fr 1 w(=0) p Fx 2047 3923 a(2) p Fn(\013) p Fq 2152 3889 a(T) p Fs 2204 3923 a(e) p Fu 19 w(\000) p Fn 18 w(\013) p Fq 2413 3889 a(T) p Fx 2465 3923 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fx 6 w(diag\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 1519 4089 a(sub) 5 b(ject) 28 b(to) 165 b(trace) o(\() p Ft(K) p Fx 6 w(\)) 24 b(=) p Ft 22 w(c;) 2047 4222 y(K) p Fu 29 w(\027) p Fx 23 w(0) p Ft(;) 2047 4400 y(K) p Fx 29 w(=) p Fq 2265 4296 a(m) p Fm 2235 4321 a(X) p Fq 2241 4498 a(i) p Fr(=1) p Ft 2368 4400 a(\026) p Fq 2418 4412 a(i) p Ft 2446 4400 a(K) p Fq 2517 4412 a(i) p Ft 2544 4400 a(;) p Fn 2047 4585 a(\026) p Fu 23 w(\025) p Fx 23 w(0) p Ft(;) p Fx 945 4751 a(when) p Ft 23 w(!) p Fq 1209 4763 a(S) p Fr 3 w(1) p Fx 1290 4751 a(\() p Ft(K) p Fx 6 w(\)) e(is) h(expressed) f(using) g(\(1.15\).) g(W) -7 b(e) 23 b(can) f(omit) h(the) g(second) f(constrain) n(t,) g(b) r (ecause) 945 4859 y(this) 31 b(is) f(implied) h(b) n(y) f(the) h(last) g (t) n(w) n(o) e(constrain) n(ts,) h(since) p Ft 30 w(K) p Fq 2805 4871 a(i) p Fu 2860 4859 a(\027) p Fx 28 w(0.) g(If) h(w) n(e) f (let) h(trace) o(\() p Ft(K) p Fq 3665 4871 a(i) p Fx 3693 4859 a(\)) d(=) p Ft 27 w(r) p Fq 3882 4871 a(i) p Fx 3910 4859 a(,) p 90 rotate dyy eop %%Page: 18 18 18 17 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(18) 670 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Fx 714 349 a(where) p Fs 27 w(r) p Fu 23 w(2) p Fp 23 w(R) p Fq 1148 319 a(m) p Fx 1218 349 a(,) f(the) h(problem) f (reduces) g(to:) 714 569 y(min) p Fn 753 620 a(\026) p Fx 1110 569 a(max) p Fn 866 628 a(\013) p Fr 28 w(:) p Fq 27 w(C) p Fk 4 w(\025) p Fn(\013) p Fk(\025) p Fr(0) p Fq(;) p Fn(\013) p Fc 1338 611 a(T) p Fo 1384 628 a(y) p Fr 1 w(=0) p Fx 1674 569 a(2) p Fn(\013) p Fq 1779 535 a(T) p Fs 1831 569 a(e) p Fu 19 w(\000) p Fn 18 w(\013) p Fq 2040 535 a(T) p Fx 2092 569 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fm 2373 427 a( ) p Fq 2471 465 a(m) p Fm 2440 490 a(X) p Fq 2446 667 a(i) p Fr(=1) p Ft 2574 569 a(\026) p Fq 2624 581 a(i) p Ft 2652 569 a(K) p Fq 2723 581 a(i) p Fm 2750 427 a(!) p Fx 2829 569 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 1146 774 a(sub) 5 b(ject) 28 b(to) p Fn 165 w(\026) p Fq 1733 740 a(T) p Fs 1786 774 a(r) p Fx 23 w(=) p Ft 23 w(c;) p Fn 1674 907 a(\026) p Fu 23 w(\025) p Fx 23 w(0) p Ft(:) p Fx 714 1073 a(W) -7 b(e) 28 b(can) f(write) g(this) h(as:) 901 1276 y(min) p Fn 714 1335 a(\026) p Fr 27 w(:) p Fn 28 w(\026) p Fk -1 w(\025) p Fr(0) p Fq(;) p Fn(\026) p Fc 1069 1318 a(T) p Fo 1115 1335 a(r) p Fr(=) p Fq(c) p Fx 1540 1276 a(max) p Fn 1296 1335 a(\013) p Fr 27 w(:) p Fq 28 w(C) p Fk 4 w(\025) p Fn(\013) p Fk(\025) p Fr(0) p Fq(;) p Fn(\013) p Fc 1767 1318 a(T) p Fo 1813 1335 a(y) p Fr 1 w(=0) p Fx 1952 1276 a(2) p Fn(\013) p Fq 2057 1242 a(T) p Fs 2109 1276 a(e) p Fu 18 w(\000) p Fn 18 w(\013) p Fq 2317 1242 a(T) p Fx 2370 1276 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fm 2651 1134 a( ) p Fq 2748 1172 a(m) p Fm 2718 1197 a(X) p Fq 2724 1374 a(i) p Fr(=1) p Ft 2851 1276 a(\026) p Fq 2901 1288 a(i) p Ft 2929 1276 a(K) p Fq 3000 1288 a(i) p Fm 3027 1134 a(!) p Fx 3107 1276 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 714 1545 a(=) 209 b(min) p Fn 801 1603 a(\026) p Fr 28 w(:) p Fn 27 w(\026) p Fk(\025) p Fr(0) p Fq(;) p Fn(\026) p Fc 1156 1587 a(T) p Fo 1202 1603 a(r) p Fr(=) p Fq(c) p Fx 1627 1545 a(max) p Fn 1383 1603 a(\013) p Fr 28 w(:) p Fq 28 w(C) p Fk 4 w(\025) p Fn(\013) p Fk -1 w(\025) p Fr(0) p Fq(;) p Fn(\013) p Fc 1855 1587 a(T) p Fo 1901 1603 a(y) p Fr 1 w(=0) p Fx 2040 1545 a(2) p Fn(\013) p Fq 2144 1510 a(T) p Fs 2197 1545 a(e) p Fu 18 w(\000) p Fq 2372 1441 a(m) p Fm 2342 1466 a(X) p Fq 2348 1642 a(i) p Fr(=1) p Ft 2476 1545 a(\026) p Fq 2526 1557 a(i) p Fn 2553 1545 a(\013) p Fq 2616 1510 a(T) p Fx 2669 1545 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fq 3007 1557 a(i) p Fx 3035 1545 a(diag\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 714 1813 a(=) 266 b(max) p Fn 801 1872 a(\013) p Fr 28 w(:) p Fq 28 w(C) p Fk 4 w(\025) p Fn(\013) p Fk -1 w(\025) p Fr(0) p Fq(;) p Fn(\013) p Fc 1273 1855 a(T) p Fo 1319 1872 a(y) p Fr 1 w(=0) p Fx 1700 1813 a(min) p Fn 1513 1872 a(\026) p Fr 27 w(:) p Fn 28 w(\026) p Fk(\025) p Fr(0) p Fq(;) p Fn(\026) p Fc 1868 1855 a(T) p Fo 1914 1872 a(r) p Fr(=) p Fq(c) p Fx 2040 1813 a(2) p Fn(\013) p Fq 2144 1779 a(T) p Fs 2197 1813 a(e) p Fu 18 w(\000) p Fq 2372 1709 a(m) p Fm 2342 1734 a(X) p Fq 2348 1911 a(i) p Fr(=1) p Ft 2476 1813 a(\026) p Fq 2526 1825 a(i) p Fn 2553 1813 a(\013) p Fq 2616 1779 a(T) p Fx 2669 1813 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fq 3007 1825 a(i) p Fx 3035 1813 a(diag\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Ft 1 w(;) p Fx 714 2035 a(The) 26 b(in) n(terc) n(hange) f(of) h(the) h(order) e(of) h(the) h (minimization) f(and) g(the) h(maximization) f(is) g(justi\014ed) 714 2143 y(b) n(y) 18 b(standard) h(results) f(in) h(con) n(v) n(ex) f (optimization) h(\(see,) g(e.g.,) g(Bo) n(yd) f(and) h(V) -7 b(anden) n(b) r(erghe,) 18 b(2001\)) 714 2251 y(since) 26 b(the) h(ob) 5 b(jectiv) n(e) 26 b(is) h(con) n(v) n(ex) e(in) p Fn 27 w(\026) p Fx 27 w(\(it) i(is) g(linear) f(in) p Fn 27 w(\026) p Fx(\)) h(and) f(conca) n(v) n(e) f(in) p Fn 27 w(\013) p Fx(,) i(and) g(b) r(ecause) 714 2359 y(the) 33 b(minimization) g(problem) f(is) h(strictly) f(feasible) h (in) p Fn 33 w(\026) p Fx 32 w(and) g(the) g(maximization) f(problem) 714 2467 y(strictly) 27 b(feasible) g(in) p Fn 28 w(\013) p Fx(.) h(W) -7 b(e) 28 b(th) n(us) g(obtain:) 958 2678 y(max) p Fn 714 2737 a(\013) p Fr 27 w(:) p Fq 28 w(C) p Fk 4 w(\025) p Fn(\013) p Fk(\025) p Fr(0) p Fq(;) p Fn(\013) p Fc 1186 2720 a(T) p Fo 1231 2737 a(y) p Fr 1 w(=0) p Fx 1613 2678 a(min) p Fn 1425 2737 a(\026) p Fr 28 w(:) p Fn 27 w(\026) p Fk(\025) p Fr(0) p Fq(;) p Fn(\026) p Fc 1781 2720 a(T) p Fo 1826 2737 a(r) p Fr(=) p Fq(c) p Fx 1952 2678 a(2) p Fn(\013) p Fq 2057 2644 a(T) p Fs 2109 2678 a(e) p Fu 18 w(\000) p Fq 2285 2574 a(m) p Fm 2254 2599 a(X) p Fq 2260 2776 a(i) p Fr(=1) p Ft 2388 2678 a(\026) p Fq 2438 2690 a(i) p Fn 2466 2678 a(\013) p Fq 2529 2644 a(T) p Fx 2581 2678 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fq 2919 2690 a(i) p Fx 2948 2678 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fx 714 2954 a(=) 266 b(max) p Fn 801 3013 a(\013) p Fr 28 w(:) p Fq 28 w(C) p Fk 4 w(\025) p Fn(\013) p Fk -1 w(\025) p Fr(0) p Fq(;) p Fn(\013) p Fc 1273 2997 a(T) p Fo 1319 3013 a(y) p Fr 1 w(=0) p Fm 1458 2812 a(") p Fx 1506 2954 a(2) p Fn(\013) p Fq 1611 2920 a(T) p Fs 1663 2954 a(e) p Fu 18 w(\000) p Fx 197 w(max) p Fn 1808 3013 a(\026) p Fr 28 w(:) p Fn 27 w(\026) p Fk(\025) p Fr(0) p Fq(;) p Fn(\026) p Fc 2163 2997 a(T) p Fo 2209 3013 a(r) p Fr(=) p Fq(c) p Fm 2335 2812 a( ) p Fq 2431 2851 a(m) p Fm 2401 2875 a(X) p Fq 2407 3052 a(i) p Fr(=1) p Ft 2534 2954 a(\026) p Fq 2584 2966 a(i) p Fn 2612 2954 a(\013) p Fq 2675 2920 a(T) p Fx 2727 2954 a(diag\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fq 3066 2966 a(i) p Fx 3094 2954 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fm 3426 2812 a(!) o(#) p Fx 714 3206 a(=) g(max) p Fn 801 3265 a(\013) p Fr 28 w(:) p Fq 28 w(C) p Fk 4 w(\025) p Fn(\013) p Fk -1 w(\025) p Fr(0) p Fq(;) p Fn(\013) p Fc 1273 3248 a(T) p Fo 1319 3265 a(y) p Fr 1 w(=0) p Fm 1458 3089 a(\024) p Fx 1501 3206 a(2) p Fn(\013) p Fq 1606 3172 a(T) p Fs 1658 3206 a(e) p Fu 19 w(\000) p Fx 18 w(max) p Fq 1869 3258 a(i) p Fm 1972 3089 a(\022) p Ft 2058 3150 a(c) p 2043 3187 66 4 v 2043 3263 a(r) p Fq 2080 3275 a(i) p Fn 2118 3206 a(\013) p Fq 2181 3172 a(T) p Fx 2234 3206 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fq 2572 3218 a(i) p Fx 2600 3206 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Fm 2932 3089 a(\023\025) p Ft 3051 3206 a(:) p Fx 714 3401 a(Finally) -7 b(,) 27 b(this) h(can) f(b) r (e) h(reform) n(ulated) f(as) g(follo) n(ws:) 922 3567 y(max) p Fn 945 3617 a(\013) p Fq(;t) p Fx 1242 3567 a(2) p Fn(\013) p Fq 1347 3532 a(T) p Fs 1399 3567 a(e) p Fu 18 w(\000) p Ft 18 w(ct) p Fx 1880 w(\(1.21\)) 714 3771 y(sub) 5 b(ject) 27 b(to) p Ft 166 w(t) p Fu 23 w(\025) p Fx 1405 3715 a(1) p 1393 3752 V Ft 1393 3828 a(r) p Fq 1430 3840 a(i) p Fn 1468 3771 a(\013) p Fq 1531 3737 a(T) p Fx 1583 3771 a(diag\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fq 1922 3783 a(i) p Fx 1950 3771 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Ft 2 w(;) 69 b(i) p Fx 23 w(=) 22 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(m) p Fn 1242 3940 a(\013) p Fq 1305 3906 a(T) p Fs 1358 3940 a(y) p Fx 24 w(=) 23 b(0) p Ft(;) 1242 4073 y(C) p Fu 30 w(\025) p Fn 22 w(\013) p Fu 23 w(\025) p Fx 23 w(0) p Ft(;) p Fx 714 4239 a(or,) j(when) i(the) p Ft 28 w(K) p Fq 1269 4251 a(i) p Fx 1324 4239 a(are) f(normalized) f(\([) p Ft(K) p Fq 2013 4251 a(i) p Fx 2041 4239 a(]) p Fq 2064 4251 a(j) s(j) p Fx 2153 4239 a(=) c(1,) p Ft 27 w(j) p Fx 28 w(=) h(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(n) p Fx(,) 28 b(suc) n(h) f(that) p Ft 28 w(r) p Fq 3213 4251 a(i) p Fx 3264 4239 a(=) p Ft 23 w(n) p Fx(\):) 922 4405 y(max) p Fn 945 4455 a(\013) p Fq(;t) p Fx 1242 4405 a(2) p Fn(\013) p Fq 1347 4370 a(T) p Fs 1399 4405 a(e) p Fu 18 w(\000) p Ft 18 w(ct) p Fx 1880 w(\(1.22\)) 714 4610 y(sub) 5 b(ject) 27 b(to) p Ft 166 w(t) p Fu 23 w(\025) p Fx 1397 4553 a(1) p 1393 4591 50 4 v Ft 1393 4667 a(n) p Fn 1453 4610 a(\013) p Fq 1516 4575 a(T) p Fx 1568 4610 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Ft(K) p Fq 1906 4622 a(i) p Fx 1935 4610 a(diag) o(\() p Fs(y) p Fx 1 w(\)) p Fn(\013) p Ft 1 w(;) 70 b(i) p Fx 22 w(=) 23 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h(m) p Fn 1242 4765 a(\013) p Fq 1305 4731 a(T) p Fs 1358 4765 a(y) p Fx 24 w(=) 23 b(0) p Ft(;) 1242 4898 y(C) p Fu 30 w(\025) p Fn 22 w(\013) p Fu 23 w(\025) p Fx 23 w(0) p Ft(:) p 90 rotate dyy eop %%Page: 19 19 19 18 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.6) 78 b(Two) 27 b(Biolo) l(gic) l(al) g(Applic) l(ations) 2546 b(19) p Fx 945 349 a(This) 42 b(problem) g(is) g(a) g(con) n(v) n(ex) f (optimization) h(problem,) g(more) f(precisely) g(a) p Fy 42 w(quadr) l(atic) l(al) t(ly) 945 457 y(c) l(onstr) l(aine) l(d) h (quadr) l(atic) h(pr) l(o) l(gr) l(am) f(\(QCQP\)) p Fx 44 w(\(Bo) n(yd) e(and) h(V) -7 b(anden) n(b) r(erghe,) 40 b(2001\).) f(Th) n(us,) 945 565 y(the) k(SDP) h(\(1.17\)) e(can) h(b) r (e) h(cast) e(as) h(a) g(QCQP) -7 b(,) 42 b(whic) n(h) h(impro) n(v) n (es) e(the) j(e\016ciency) f(of) g(the) 945 673 y(computation) 26 b(to) p Ft 27 w(O) p Fx 2 w(\() p Ft(n) p Fr 1676 643 a(3) p Fx 1714 673 a(\)) h(in) h(terms) e(of) h(the) g(n) n(um) n(b) r (er) g(of) g(data) f(p) r(oin) n(ts.) h(The) g(optimal) g(w) n(eigh) n (ts) p Ft 945 781 a(\026) p Fq 995 793 a(i) p Ft 1022 781 a(;) 58 b(i) p Fx 23 w(=) 22 b(1) p Ft(;) 14 b(:) g(:) g(:) f(;) h (m) p Fx(,) 22 b(can) f(b) r(e) h(reco) n(v) n(ered) e(from) h(the) h (primal-dual) f(solution) g(found) h(b) n(y) g(standard) 945 889 y(soft) n(w) n(are) k(suc) n(h) h(as) g(SeDuMi) h(\(Sturm,) g (1999\).) 1028 997 y(Th) n(us,) 44 b(b) n(y) h(solving) f(a) g(QCQP) -7 b(,) 44 b(w) n(e) g(are) g(able) g(to) h(\014nd) g(an) g(adaptiv) n(e) f (com) n(bination) g(of) 945 1105 y(k) n(ernel) 31 b(matrices|and) g(th) n(us) h(an) f(adaptiv) n(e) g(com) n(bination) g(of) h(heterogeneous) e (information) 945 1213 y(sources|that) h(solv) n(es) g(our) h (classi\014cation) f(problem.) i(The) f(output) h(of) g(our) f(pro) r (cedure) g(is) g(a) 945 1321 y(set) h(of) h(w) n(eigh) n(ts) p Ft 33 w(\026) p Fq 1533 1333 a(i) p Fx 1594 1321 a(and) g(a) f (discriminan) n(t) g(function) h(based) f(on) h(these) f(w) n(eigh) n (ts.) g(W) -7 b(e) 34 b(obtain) 945 1429 y(a) 42 b(classi\014cation) f (decision) i(that) g(merges) e(information) h(enco) r(ded) h(in) g(the) g(v) -5 b(arious) 42 b(k) n(ernel) 945 1536 y(matrices,) 36 b(and) g(w) n(e) g(obtain) g(w) n(eigh) n(ts) p Ft 36 w(\026) p Fq 2233 1548 a(i) p Fx 2297 1536 a(that) h(re\015ect) f(the) h (relativ) n(e) e(imp) r(ortance) h(of) h(these) 945 1644 y(information) 27 b(sources.) p Fv 198 1977 a(1.6) 100 b(Tw) m(o) 34 b(Biological) j(Applications) p 198 1844 3736 3 v Fx 945 2192 a(In) g(this) g(section,) g(w) n(e) g(illustrate) g (the) g(k) n(ernel-based) f(approac) n(h) f(for) h(fusing) i (heterogeneous) 945 2300 y(genomic) 25 b(data) g(using) h (semide\014nite) g(programming) e(for) h(t) n(w) n(o) g(biologically) f (imp) r(ortan) n(t) i(prob-) 945 2408 y(lems:) i(mem) n(brane) g (protein) g(prediction) g(and) h(protein) f(function) h(prediction) f (in) h(y) n(east.) f(More) 945 2516 y(details) j(can) g(b) r(e) h (found) g(in) g(Lanc) n(kriet) e(et) i(al.) g(\(2003\)) e(for) h(mem) n (brane) g(protein) g(recognition,) 945 2624 y(and) c(in) h(Lanc) n (kriet) e(et) i(al.) g(\(2004\)) e(for) h(the) h(protein) f(function) h (classi\014cation.) p Fs 945 2840 a(1.6.1) 98 b(Mem) m(brane) 30 b(Protein) i(Classi\014cation) p Fx 945 3056 a(Mem) n(brane) 19 b(proteins) h(are) f(proteins) h(that) g(anc) n(hor) f(in) h(one) g(of) g(v) -5 b(arious) 19 b(mem) n(branes) g(in) i(the) g(cell.) 945 3164 y(Man) n(y) d(mem) n(brane) g(proteins) g(serv) n(e) g(imp) r (ortan) n(t) g(comm) n(unicativ) n(e) g(functions.) h(Generally) -7 b(,) 18 b(eac) n(h) 945 3272 y(mem) n(brane) i(protein) g(passes) g (through) g(the) h(mem) n(brane) f(sev) n(eral) e(times.) j(The) g (transmem) n(brane) 945 3380 y(regions) 35 b(of) j(the) f(amino) g (acid) g(sequence) g(are) f(t) n(ypically) h(h) n(ydrophobic,) f (whereas) g(the) h(non-) 945 3488 y(mem) n(brane) 43 b(p) r(ortions) h(are) f(h) n(ydrophilic.) g(This) h(sp) r(eci\014c) h (h) n(ydrophobicit) n(y) e(pro\014le) g(of) h(the) 945 3595 y(protein) 27 b(allo) n(ws) f(it) i(to) g(anc) n(hor) e(itself) i (in) g(the) g(cell) f(mem) n(brane.) 1028 3703 y(Because) 46 b(the) h(h) n(ydrophobicit) n(y) f(pro\014le) h(of) g(a) g(mem) n (brane) f(protein) h(is) g(critical) f(to) h(its) 945 3811 y(function,) 31 b(this) g(pro\014le) f(is) h(b) r(etter) g (conserv) n(ed) e(in) h(ev) n(olution) g(than) h(the) g(sp) r(eci\014c) g(amino) f(acid) 945 3919 y(sequence.) h(Therefore,) f(classical) g (metho) r(ds) h(for) g(determining) g(whether) h(a) e(protein) h(spans) g(a) 945 4027 y(mem) n(brane) 19 b(\(Chen) i(and) g(Rost,) f(2002\)) f (dep) r(end) i(up) r(on) g(a) p Fy 20 w(hydr) l(op) l(athy) k(pr) l (o\014le) p Fx(,) d(whic) n(h) e(plots) g(the) 945 4135 y(h) n(ydrophobicit) n(y) 26 b(of) h(the) h(amino) e(acids) h(along) f (the) i(protein) f(\(Engleman) g(et) g(al.,) g(1986;) f(Blac) n(k) 945 4243 y(and) 33 b(Mould,) g(1991;) e(Hopp) i(and) g(W) -7 b(o) r(o) r(ds,) 33 b(1981\).) f(In) h(this) h(subsection,) e(w) n(e) h (build) h(on) f(these) 945 4351 y(classical) 19 b(metho) r(ds) i(b) n (y) g(dev) n(eloping) f(a) g(k) n(ernel) g(function) h(that) g(is) g (based) f(on) h(the) g(lo) n(w-frequency) 945 4459 y(alternation) f(of) i(h) n(ydrophobic) e(and) h(h) n(ydrophilic) g(regions) e(in) j(mem) n (brane) f(proteins.) g(Ho) n(w) n(ev) n(er,) 945 4567 y(w) n(e) 26 b(also) g(demonstrate) g(that) h(the) h(h) n(ydropath) n (y) d(pro\014le) h(pro) n(vides) g(only) g(partial) g(evidence) h(for) 945 4675 y(transmem) n(brane) 21 b(regions.) g(Additional) i (information) f(is) g(gleaned) g(from) g(sequence) g(homology) 945 4783 y(and) 27 b(from) g(protein-protein) g(in) n(teractions.) 1028 4891 y(Note) f(that,) h(in) f(general,) f(mem) n(brane) h(protein) g (prediction) g(consists) f(of) i(predicting) f(the) g(lo-) 945 4999 y(cations) d(of) h(m) n(ultiple) h(transmem) n(brane) e(regions) f (within) j(a) f(single) f(protein.) h(In) h(this) f(example,) p 90 rotate dyy eop %%Page: 20 20 20 19 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(20) 670 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Ff 714 332 a(T) -7 b(able) 35 b(1.1) 83 b(Kernel) 37 b(functions.) p Fx 35 w(The) d(table) g(lists) g(the) g(sev) n(en) f(k) n(ernels) g(used) h(to) g(compare) 714 440 y(proteins,) 44 b(the) i(data) e(on) h(whic) n(h) g(they) h(are) e(de\014ned,) i(and) f (the) g(metho) r(d) h(for) f(computing) 714 548 y(similarities.) 28 b(The) i(\014nal) f(k) n(ernel,) p Fj 28 w(K) p Fg 1852 556 a(RN) 5 b(D) p Fx 2007 548 a(,) 30 b(is) f(included) h(as) e(a) h (con) n(trol.) f(All) i(k) n(ernels) e(matrices,) 714 656 y(along) 34 b(with) i(the) h(data) e(from) g(whic) n(h) h(they) g (w) n(ere) f(generated,) f(are) h(a) n(v) -5 b(ailable) 35 b(at) p Fb 35 w(noble.gs.) 714 764 y(washington.edu/sdp-) t(svm) p Fx(.) p Ff 1287 915 a(Kernel) 100 b(Data) 567 b(Similarit) n(y) 32 b(measure) p 1237 952 1942 4 v Fj 1287 1028 a(K) p Fg 1352 1036 a(S) s(W) p Fl 1647 1028 a(protein) 26 b(sequences) 167 b(Smith-W) -6 b(aterman) p Fj 1287 1138 a(K) p Fg 1352 1146 a(B) p Fl 1647 1138 a(protein) 26 b(sequences) 167 b(BLAST) p Fj 1287 1247 a(K) p Fg 1352 1255 a(H) t(M) 5 b(M) p Fl 1647 1247 a(protein) 26 b(sequences) 167 b(Pfam) 26 b(HMM) p Fj 1287 1357 a(K) p Fg 1352 1365 a(F) 8 b(F) g(T) p Fl 1647 1357 a(h) n(ydropath) n(y) 24 b(pro\014le) 136 b(FFT) p Fj 1287 1467 a(K) p Fg 1352 1475 a(LI) p Fl 1647 1467 a(protein) 26 b(in) n(teractions) 100 b(linear) 27 b(k) n(ernel) p Fj 1287 1576 a(K) p Fg 1352 1584 a(D) p Fl 1647 1576 a(protein) f(in) n(teractions) 100 b(di\013usion) 26 b(k) n(ernel) p Fj 1287 1686 a(K) p Fg 1352 1694 a(E) p Fl 1647 1686 a(gene) g(expression) 236 b(radial) 27 b(basis) g(k) n (ernel) p Fj 1287 1795 a(K) p Fg 1352 1803 a(RN) 5 b(D) p Fl 1647 1795 a(random) 25 b(n) n(um) n(b) r(ers) 187 b(radial) 27 b(basis) g(k) n(ernel) p Fx 714 2111 a(ho) n(w) n(ev) n (er,) 17 b(for) j(the) g(purp) r(oses) f(of) h(demonstrating) f(the) h (SDP) g(metho) r(d,) h(w) n(e) e(fo) r(cus) h(on) g(the) g(binary) 714 2218 y(prediction) 27 b(task) g(of) h(di\013eren) n(tiating) f(b) r(et) n(w) n(een) g(mem) n(brane) g(and) h(non-mem) n(brane) e(proteins.) p Fz 714 2434 a(1.6.1.1) 98 b(Kernels) 35 b(for) f(membr) -5 b(ane) 36 b(pr) -5 b(otein) 34 b(pr) -5 b(e) g(diction) p Fx 714 2650 a(F) e(or) 33 b(the) h(task) f(of) h(mem) n(brane) f (protein) g(classi\014cation) f(w) n(e) i(exp) r(erimen) n(t) f(with) h (sev) n(en) f(k) n(ernel) 714 2758 y(matrices) 28 b(deriv) n(ed) g (from) g(three) h(di\013eren) n(t) g(t) n(yp) r(es) g(of) g(data:) f (four) g(from) h(the) g(primary) f(protein) 714 2866 y(sequence,) h(t) n(w) n(o) f(from) h(protein-protein) f(in) n (teraction) h(data,) f(and) i(one) f(from) g(mRNA) h(expres-) 714 2974 y(sion) d(data.) g(These) g(are) g(summarized) g(in) h(T) -7 b(able) 27 b(1.1.) p Fz 714 3190 a(1.6.1.2) 98 b(Pr) -5 b(otein) 34 b(se) -5 b(quenc) g(e:) 36 b(Smith-Waterman,) f(BLAST) f (and) g(Pfam) i(HMM) 1098 3298 y(kernels) p Fx 714 3514 a(A) f(homolog) e(of) i(a) g(mem) n(brane) f(protein) g(is) h(lik) n (ely) f(also) g(to) h(b) r(e) g(lo) r(cated) g(in) g(the) g(mem) n (brane.) 714 3622 y(Therefore,) 20 b(w) n(e) i(de\014ne) g(three) f(k) n (ernel) g(matrices) g(based) g(up) r(on) h(standard) f(homology) f (detection) 714 3729 y(metho) r(ds.) 29 b(The) h(\014rst) f(t) n(w) n (o) g(sequence-based) f(k) n(ernel) g(matrices) h(\() p Ft(K) p Fq 2839 3741 a(S) s(W) p Fx 2988 3729 a(and) p Ft 29 w(K) p Fq 3222 3741 a(B) p Fx 3279 3729 a(\)) g(are) g(gener-) 714 3837 y(ated) h(using) g(the) h(BLAST) g(\(Altsc) n(h) n(ul) g(et) g (al.,) f(1990\)) g(and) g(Smith-W) -7 b(aterman) 31 b(\(SW\)) g (\(Smith) 714 3945 y(and) e(W) -7 b(aterman,) 29 b(1981\)) g(pairwise) f (sequence) h(comparison) f(algorithms,) h(as) g(describ) r(ed) g(pre-) 714 4053 y(viously) i(\(Liao) g(and) g(Noble,) h(2002\).) e(Because) h (matrices) g(of) h(BLAST) g(or) f(Smith-W) -7 b(aterman) 714 4161 y(scores) 25 b(are) h(not) h(necessarily) e(p) r(ositiv) n(e) i (semide\014nite,) g(w) n(e) g(represen) n(t) f(eac) n(h) g(protein) h (as) f(a) g(v) n(ec-) 714 4269 y(tor) d(of) g(scores) g(\(BLAST) h(and) f(SW) h(log) f(E-v) -5 b(alues,) 23 b(resp) r(ectiv) n(ely\)) g (against) g(all) g(other) g(proteins.) 714 4377 y(De\014ning) 36 b(the) g(similarit) n(y) g(b) r(et) n(w) n(een) g(proteins) f(as) g (the) i(inner) e(pro) r(duct) h(b) r(et) n(w) n(een) g(the) h(score) 714 4485 y(v) n(ectors) c(\(the) i(so-called) e(empirical) h(k) n(ernel) f (map,) i(Tsuda,) f(1999\)) f(leads) g(to) i(a) f(v) -5 b(alid) 34 b(k) n(ernel) 714 4593 y(matrix,) 25 b(one) h(for) f(the) h (BLAST) g(score) f(and) h(one) f(for) h(the) g(SW) g(score.) f(Note) h (that) g(including) g(in) 714 4701 y(the) c(comparison) e(set) i (proteins) g(with) g(unkno) n(wn) g(sub) r(cellular) f(lo) r(cations) g (allo) n(ws) g(the) i(k) n(ernel) e(to) 714 4809 y(exploit) i(this) i (unlab) r(eled) f(data.) f(The) h(third) h(k) n(ernel) e(matrix) g(\() p Ft(K) p Fq 2705 4821 a(H) t(M) 6 b(M) p Fx 2907 4809 a(\)) 24 b(is) g(a) g(generalization) e(of) 714 4917 y(the) j(previous) f(pairwise) g(comparison-based) e(matrices) j(in) g (whic) n(h) g(the) g(pairwise) f(comparison) p 90 rotate dyy eop %%Page: 21 21 21 20 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.6) 78 b(Two) 27 b(Biolo) l(gic) l(al) g(Applic) l(ations) 2546 b(21) p Fx 945 349 a(scores) 31 b(are) i(replaced) f(b) n(y) h(exp) r (ectation) g(v) -5 b(alues) 32 b(deriv) n(ed) h(from) g(hidden) g(Mark) n(o) n(v) e(mo) r(dels) i(in) 945 457 y(the) e(Pfam) f(database) g (\(Sonnhammer) g(et) h(al.,) g(1997\).) e(These) i(similarit) n(y) e (measures) h(are) g(not) 945 565 y(sp) r(eci\014c) e(to) f(the) h(mem) n (brane) f(protein) g(classi\014cation) f(task.) p Fz 945 781 a(1.6.1.3) 98 b(Pr) -5 b(otein) 34 b(se) -5 b(quenc) g(e:) 35 b(FFT) h(kernel) p Fx 945 997 a(In) 27 b(con) n(trast,) e(the) i (fourth) f(sequence-based) f(k) n(ernel) h(matrix) g(\() p Ft(K) p Fq 2973 1009 a(F) 9 b(F) g(T) p Fx 3127 997 a(\)) 27 b(directly) g(incorp) r(orates) 945 1105 y(information) h(ab) r(out) g (h) n(ydrophobicit) n(y) f(patterns,) h(whic) n(h) h(are) e(kno) n(wn) h (to) g(b) r(e) h(useful) g(in) g(iden-) 945 1213 y(tifying) 40 b(mem) n(brane) f(proteins.) h(The) g(k) n(ernel) f(uses) h(h) n (ydropath) n(y) e(pro\014les) h(generated) g(from) 945 1321 y(the) 27 b(Kyte-Do) r(olittle) f(index) g(\(Kyte) h(and) f(Do) r (olittle,) h(1982\).) e(This) h(k) n(ernel) f(compares) g(the) i(fre-) 945 1429 y(quency) 21 b(con) n(ten) n(t) g(of) g(the) g(h) n(ydropath) n (y) f(pro\014les) g(of) i(the) f(t) n(w) n(o) g(proteins.) f(After) i (pre-\014ltering) e(the) 945 1536 y(h) n(ydropath) n(y) h(pro\014les,) g (their) h(F) -7 b(ourier) 22 b(transforms) f(\(describing) h(the) g (frequency) g(con) n(ten) n(t\)) h(are) 945 1644 y(computed) j(using) f (an) h(FFT) g(algorithm.) f(The) h(frequency) g(con) n(ten) n(ts) f(of) h(di\013eren) n(t) g(pro\014les) f(are) 945 1752 y(compared) 18 b(b) n(y) h(applying) g(a) f(Gaussian) h(k) n(ernel) f(function,) p Ft 20 w(k) p Fx 3 w(\() p Fs(x) p Fr 2866 1764 a(1) p Ft 2904 1752 a(;) p Fs 14 w(x) p Fr 2991 1764 a(2) p Fx 3028 1752 a(\)) 24 b(=) e(exp\() p Fu(\000jj) p Fs(x) p Fr 3491 1764 a(1) p Fu 3530 1752 a(\000) p Fs 2 w(x) p Fr 3647 1764 a(2) p Fu 3684 1752 a(jj) p Fr 3730 1722 a(2) p Ft 3768 1752 a(=) p Fx(2) p Ft(\033) p Fx 3 w(\)) 945 1860 y(with) j(width) p Ft 24 w(\033) p Fx 27 w(=) e(10,) g(to) h(the) h (corresp) r(onding) d(v) n(ectors) h(of) h(FFT) h(v) -5 b(alues.) 24 b(This) g(k) n(ernel) f(detects) 945 1968 y(p) r(erio) r(dicities) e(in) g(the) g(h) n(ydropath) n(y) e (pro\014le,) i(a) f(feature) h(that) g(is) g(relev) -5 b(an) n(t) 20 b(to) h(the) g(iden) n(ti\014cation) 945 2076 y(of) h(mem) n(brane) f(proteins) h(and) g(complemen) n(tary) f (to) h(the) g(previous,) f(homology-based) f(k) n(ernels.) p Fz 945 2292 a(1.6.1.4) 98 b(Pr) -5 b(otein) 34 b(inter) -5 b(actions:) 34 b(line) -5 b(ar) 34 b(and) g(di\013usion) f(kernels) p Fx 945 2508 a(W) -7 b(e) 26 b(exp) r(ect) h(information) f(ab) r(out) g (protein-protein) f(in) n(teractions) g(to) h(b) r(e) g(informativ) n (e) g(in) g(this) 945 2616 y(con) n(text) i(for) f(t) n(w) n(o) h (reasons.) e(First,) i(h) n(ydrophobic) f(molecules) h(or) f(regions) f (of) j(molecules) e(tend) 945 2724 y(to) h(in) n(teract) g(with) h(eac) n(h) e(other.) h(Second,) h(transmem) n(brane) d(proteins) i(are) g (often) g(in) n(v) n(olv) n(ed) f(in) 945 2832 y(signaling) 22 b(path) n(w) n(a) n(ys,) f(and) i(therefore) f(di\013eren) n(t) i(mem) n (brane) e(proteins) g(are) h(lik) n(ely) f(to) h(in) n(teract) 945 2940 y(with) h(a) f(similar) f(class) h(of) g(molecules) g(upstream) g (and) g(do) n(wnstream) f(in) i(these) f(path) n(w) n(a) n(ys) f (\(e.g.,) 945 3047 y(hormones) 33 b(upstream) h(or) g(kinases) f(do) n (wnstream\).) h(The) g(t) n(w) n(o) g(protein) g(in) n(teraction) g(k) n (ernels) 945 3155 y(are) c(generated) f(using) i(medium-) g(and) g (high-con\014dence) f(in) n(teractions) f(from) i(a) f(database) g(of) 945 3263 y(kno) n(wn) 22 b(in) n(teractions) g(\(v) n(on) g(Mering) g (et) i(al.,) e(2002\).) g(These) g(in) n(teractions) g(can) h(b) r(e) g (represen) n(ted) 945 3371 y(as) 31 b(an) g(in) n(teraction) g(matrix,) g(in) h(whic) n(h) g(ro) n(ws) e(and) i(columns) f(corresp) r(ond) f (to) i(proteins,) f(and) 945 3479 y(binary) c(en) n(tries) f(indicate) i (whether) g(the) g(t) n(w) n(o) e(proteins) h(in) n(teract.) 1028 3587 y(The) f(\014rst) f(in) n(teraction) g(k) n(ernel) g(matrix) g(\() p Ft(K) p Fq 2396 3599 a(LI) p Fx 2480 3587 a(\)) h(is) g(comprised) f (of) h(linear) f(in) n(teractions,) f(i.e.,) 945 3695 y(inner) 29 b(pro) r(ducts) h(of) g(ro) n(ws) e(and) i(columns) f(from) h(the) g(cen) n(tered,) f(binary) g(in) n(teraction) g(matrix.) 945 3803 y(The) g(more) g(similar) g(the) h(in) n(teraction) f(pattern) g (\(corresp) r(onding) f(to) i(a) f(ro) n(w) g(or) f(column) i(from) 945 3911 y(the) i(in) n(teraction) g(matrix\)) f(is) h(for) g(a) g(pair) f (of) h(proteins,) g(the) h(larger) d(the) i(inner) g(pro) r(duct) g (will) 945 4019 y(b) r(e.) 1028 4127 y(An) 40 b(alternativ) n(e) e(w) n (a) n(y) h(to) g(represen) n(t) g(the) h(same) f(in) n(teraction) f (data) i(is) f(to) h(consider) e(the) 945 4235 y(proteins) 23 b(as) h(no) r(des) g(in) g(a) g(large) e(graph.) h(In) i(this) f (graph,) f(t) n(w) n(o) h(proteins) f(are) g(link) n(ed) h(when) g (they) 945 4343 y(in) n(teract) k(and) g(otherwise) g(not.) g(Kondor) f (and) i(La\013ert) n(y) e(\(2002\)) h(prop) r(ose) f(a) h(general) f (metho) r(d) 945 4451 y(for) e(establishing) h(similarities) g(b) r(et) n(w) n(een) g(the) g(no) r(des) g(of) h(a) f(graph,) f(based) g(on) h (a) g(random) g(w) n(alk) 945 4558 y(on) 36 b(the) h(graph.) f(This) g (metho) r(d) h(e\016cien) n(tly) g(accoun) n(ts) e(for) h(all) g(p) r (ossible) h(paths) f(connecting) 945 4666 y(t) n(w) n(o) 25 b(no) r(des,) g(and) g(for) g(the) h(lengths) f(of) h(those) f(paths.) h (No) r(des) f(that) h(are) e(connected) i(b) n(y) f(shorter) 945 4774 y(paths) g(or) f(b) n(y) h(man) n(y) g(paths) g(are) f(considered) h(more) f(similar.) h(The) g(resulting) p Fy 25 w(di\013usion) j (kernel) p Fx 945 4882 a(generates) e(the) i(second) f(in) n(teraction) f(k) n(ernel) h(matrix) g(\() p Ft(K) p Fq 2756 4894 a(D) p Fx 2816 4882 a(\).) 1028 4990 y(An) e(app) r(ealing) f(c) n (haracteristic) f(of) h(the) h(di\013usion) g(k) n(ernel) f(is) h(its) f (abilit) n(y) -7 b(,) 25 b(lik) n(e) f(the) h(empirical) p 90 rotate dyy eop %%Page: 22 22 22 21 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(22) 670 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) p Fx 714 349 a(k) n(ernel) 34 b(map,) i(to) f(exploit) h(unlab) r(eled) f(data.) h(In) f(order) g(to) g(compute) h(the) g(di\013usion) f(k) n (ernel,) 714 457 y(a) 47 b(graph) f(is) h(constructed) g(using) h(all) f (kno) n(wn) g(protein-protein) f(in) n(teractions,) g(including) 714 565 y(in) n(teractions) 22 b(in) n(v) n(olving) f(proteins) i(whose) f (sub) r(cellular) h(lo) r(cations) f(are) g(unkno) n(wn.) h(Therefore,) 714 673 y(the) e(di\013usion) f(pro) r(cess) g(includes) g(in) n (teractions) g(in) n(v) n(olving) f(unlab) r(eled) i(proteins,) f(ev) n (en) g(though) 714 781 y(the) k(k) n(ernel) f(matrix) g(only) h(con) n (tains) e(en) n(tries) i(for) f(lab) r(eled) h(proteins.) f(This) h (allo) n(ws) e(t) n(w) n(o) h(lab) r(eled) 714 889 y(proteins) 18 b(to) g(b) r(e) h(considered) e(close) h(to) h(one) f(another) f(if) i (they) g(b) r(oth) g(in) n(teract) f(with) h(an) f(unlab) r(eled) 714 997 y(protein.) p Fz 714 1213 a(1.6.1.5) 98 b(Gene) 35 b(expr) -5 b(ession:) 34 b(r) -5 b(adial) 34 b(b) -5 b(asis) 35 b(kernel) p Fx 714 1429 a(Finally) -7 b(,) 26 b(w) n(e) h(also) f(include) h(a) f(k) n(ernel) g(constructed) g(en) n (tirely) g(from) h(microarra) n(y) c(gene) k(expres-) 714 1536 y(sion) g(measuremen) n(ts.) g(A) h(collection) g(of) g(441) e (distinct) j(exp) r(erimen) n(ts) e(w) n(as) g(do) n(wnloaded) g(from) 714 1644 y(the) 63 b(Stanford) g(Microarra) n(y) d(Database) i(\() p Fa(genome-) t(www.stanfo) o(rd) o(.e) o(du/) o(mi) o(cro) o(ar) o(ra) o (y) p Fx(\).) 714 1752 y(This) 39 b(data) g(pro) n(vides) e(us) j(with) f(a) g(441-elemen) n(t) f(expression) g(v) n(ector) f(c) n (haracterizing) g(eac) n(h) 714 1860 y(gene.) 28 b(A) h(Gaussian) f(k) n (ernel) f(matrix) h(\() p Ft(K) p Fq 1995 1872 a(E) p Fx 2051 1860 a(\)) h(is) g(computed) g(from) f(these) h(v) n(ectors) e (b) n(y) h(applying) 714 1968 y(a) c(Gaussian) g(k) n(ernel) f (function) j(with) f(width) p Ft 25 w(\033) p Fx 27 w(=) d(100) i(to) g (eac) n(h) g(pair) g(of) h(441-elemen) n(t) e(v) n(ectors,) 714 2076 y(c) n(haracterizing) 33 b(a) i(pair) g(of) h(genes.) f(Note) h (that) g(w) n(e) f(do) h(not) g(exp) r(ect) g(that) g(gene) f (expression) 714 2184 y(will) h(b) r(e) g(particularly) e(useful) j (for) e(the) h(mem) n(brane) f(classi\014cation) g(task.) g(W) -7 b(e) 36 b(do) g(not) f(need) 714 2292 y(to) 30 b(mak) n(e) g(this) h (decision) p Fy 30 w(a) i(priori) p Fx(,) g(ho) n(w) n(ev) n(er;) c(as) h(explained) g(in) h(the) g(follo) n(wing) f(section,) g(our) 714 2400 y(metho) r(d) j(is) f(able) h(to) f(pro) n(vide) f(an) p Fy 33 w(a) j(p) l(osteriori) p Fx 35 w(measure) d(of) i(ho) n(w) f (useful) h(a) f(data) g(source) g(is) 714 2508 y(relativ) n(e) c(to) h (the) g(other) g(sources) e(of) i(data.) g(W) -7 b(e) 30 b(th) n(us) f(include) g(the) h(expression) e(k) n(ernel) g(in) h(our) 714 2616 y(exp) r(erimen) n(ts) e(to) g(test) h(this) g(asp) r(ect) g (of) f(the) h(metho) r(d.) p Fz 714 2832 a(1.6.1.6) 98 b(Exp) -5 b(erimental) 34 b(design) p Fx 714 3047 a(In) 40 b(order) f(to) i(test) f(our) g(k) n(ernel-based) f(approac) n(h) f(in) j(the) f(setting) h(of) f(mem) n(brane) g(protein) 714 3155 y(classi\014cation,) 30 b(w) n(e) h(use) h(as) f(a) g(gold) g (standard) g(the) h(annotations) f(pro) n(vided) g(b) n(y) g(the) h (Munic) n(h) 714 3263 y(Information) 25 b(Cen) n(ter) g(for) g(Protein) g(Sequences) g(Comprehensiv) n(e) f(Y) -7 b(east) 26 b(Genome) f(Database) 714 3371 y(\(CYGD\)) 38 b(\(Mew) n(es) e(et) h (al.,) f(2000\).) g(The) g(CYGD) i(assigns) d(sub) r(cellular) h(lo) r (cations) g(to) g(2318) 714 3479 y(y) n(east) 42 b(proteins,) i(of) g (whic) n(h) f(497) g(b) r(elong) g(to) h(v) -5 b(arious) 42 b(mem) n(brane) h(protein) h(classes.) e(The) 714 3587 y(remaining) k(appro) n(ximately) g(4000) g(y) n(east) g(proteins) h (ha) n(v) n(e) g(uncertain) g(lo) r(cation) g(and) h(are) 714 3695 y(therefore) 26 b(not) i(used) f(in) h(these) g(exp) r(erimen) n (ts.) 797 3803 y(The) 44 b(primary) f(input) i(to) f(the) g (classi\014cation) f(algorithm) g(is) h(the) h(collection) e(of) h(k) n (ernel) 714 3911 y(matrices) 34 b(from) i(T) -7 b(able) 35 b(1.1.) g(Using) g(the) h(SDP) g(tec) n(hniques) g(describ) r(ed) f(ab) r(o) n(v) n(e,) f(w) n(e) i(\014nd) g(an) 714 4019 y(optimal) 26 b(com) n(bination) g(of) h(the) g(sev) n(en) f(k) n(ernel) g(matrices,) g(and) g(the) i(resulting) e(matrix) g(is) h(used) 714 4127 y(to) g(train) g(an) h(SVM) g(classi\014er.) 797 4235 y(F) -7 b(or) 44 b(comparison) f(with) i(the) h(SDP/SVM) f (learning) e(algorithm,) h(w) n(e) g(consider) g(sev) n(eral) 714 4343 y(classical) 24 b(biological) g(metho) r(ds) h(that) h(are) f (commonly) f(used) i(to) f(determine) h(whether) f(a) g(Kyte-) 714 4451 y(Do) r(olittle) 41 b(plot) f(corresp) r(onds) f(to) i(a) f(mem) n (brane) g(protein,) g(as) g(w) n(ell) g(as) g(a) g(state-of-the-art) 714 4558 y(tec) n(hnique) 27 b(using) g(hidden) h(Mark) n(o) n(v) d(mo) r (dels) i(\(HMMs\)) h(to) f(predict) g(transmem) n(brane) f(helices) 714 4666 y(in) 31 b(proteins) f(\(Krogh) f(et) i(al.,) g(2001;) e(Chen) i (and) f(Rost,) h(2002\).) e(The) i(\014rst) f(metho) r(d) i(relies) e (on) 714 4774 y(the) c(observ) -5 b(ation) 25 b(that) i(the) g(a) n(v) n (erage) c(h) n(ydrophobicit) n(y) i(of) h(mem) n(brane) g(proteins) f (tends) i(to) f(b) r(e) 714 4882 y(higher) 19 b(than) i(that) f(of) h (non-mem) n(brane) e(proteins,) g(b) r(ecause) h(the) h(transmem) n (brane) e(regions) g(are) 714 4990 y(more) f(h) n(ydrophobic.) g(W) -7 b(e) 20 b(therefore) f(de\014ne) p Ft 19 w(f) p Fr 2151 5002 a(1) p Fx 2208 4990 a(as) f(the) i(a) n(v) n(erage) c(h) n (ydrophobicit) n(y) -7 b(,) 19 b(normalized) p 90 rotate dyy eop %%Page: 23 23 23 22 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.6) 78 b(Two) 27 b(Biolo) l(gic) l(al) g(Applic) l(ations) 2546 b(23) p Fx 945 349 a(b) n(y) 25 b(the) i(length) f(of) g(the) g (protein.) g(W) -7 b(e) 26 b(will) h(compare) d(the) j (classi\014cation) d(p) r(erformance) h(of) h(our) 945 457 y(statistical) h(learning) f(algorithm) h(with) h(this) g(metric.) 1028 565 y(Clearly) -7 b(,) 29 b(ho) n(w) n(ev) n(er,) p Ft 28 w(f) p Fr 1720 577 a(1) p Fx 1787 565 a(is) h(to) r(o) f (simplistic.) h(F) -7 b(or) 30 b(example,) f(protein) h(regions) e (that) j(are) e(not) 945 673 y(transmem) n(brane) 39 b(only) h(induce) h(noise) e(in) p Ft 41 w(f) p Fr 2389 685 a(1) p Fx 2426 673 a(.) i(Therefore,) e(an) h(alternativ) n(e) g (metric) g(\014lters) 945 781 y(the) 29 b(h) n(ydrophobicit) n(y) e (plot) i(with) g(a) f(lo) n(w-pass) e(\014lter) j(and) f(then) h (computes) g(the) g(n) n(um) n(b) r(er,) f(the) 945 889 y(heigh) n(t) 35 b(and) h(the) g(width) h(of) f(those) f(p) r(eaks) g (ab) r(o) n(v) n(e) g(a) g(certain) h(threshold) f(\(Chen) h(and) g (Rost,) 945 997 y(2002\).) 22 b(The) h(\014lter) g(is) h(in) n(tended) f (to) g(smo) r(oth) h(out) f(p) r(erio) r(dic) g(e\013ects.) h(W) -7 b(e) 23 b(implemen) n(t) h(t) n(w) n(o) f(suc) n(h) 945 1105 y(\014lters,) 35 b(c) n(ho) r(osing) e(v) -5 b(alues) 35 b(for) g(the) g(\014lter) g(order) f(and) h(the) g(threshold) g(based) g (on) g(Chen) g(and) 945 1213 y(Rost) d(\(2002\).) f(In) i(particular,) e (w) n(e) h(de\014ne) p Ft 33 w(f) p Fr 2364 1225 a(2) p Fx 2433 1213 a(as) g(the) h(area) e(under) h(the) h(7th-order) e(lo) n (w-pass) 945 1321 y(\014ltered) 22 b(Kyte-Do) r(olittle) g(plot) h(and) f(ab) r(o) n(v) n(e) f(a) h(threshold) g(v) -5 b(alue) 22 b(2,) g(normalized) g(b) n(y) g(the) h(length) 945 1429 y(of) k(the) i(protein.) e(Similarly) -7 b(,) p Ft 27 w(f) p Fr 1902 1441 a(3) p Fx 1967 1429 a(is) 28 b(the) g(corresp) r (onding) e(area) g(using) h(a) h(20th-order) d(\014lter) j(and) 945 1536 y(a) f(threshold) g(of) h(1.6.) 1028 1644 y(Finally) -7 b(,) 38 b(the) h(T) -7 b(ransmem) n(brane) 36 b(HMM) j(\(TMHMM\)) g(w) n (eb) f(serv) n(er) e(\() p Fa(www.cbs.dtu.dk/) 945 1752 y(services/TMHMM) p Fx(\)) d(is) 39 b(used) g(to) g(mak) n(e) f (predictions) h(for) f(eac) n(h) g(protein.) h(In) g(Krogh) f(et) h (al.) 945 1860 y(\(2001\),) c(transmem) n(brane) f(proteins) h(are) g (iden) n(ti\014ed) i(b) n(y) e(TMHMM) i(using) f(three) g(di\013eren) n (t) 945 1968 y(metrics:) 22 b(the) g(exp) r(ected) h(n) n(um) n(b) r (er) f(of) g(amino) g(acids) g(in) g(transmem) n(brane) f(helices,) h (the) h(n) n(um) n(b) r(er) 945 2076 y(of) 39 b(transmem) n(brane) e (helices) i(predicted) g(b) n(y) f(the) p Ft 40 w(N) p Fx 9 w(-b) r(est) g(algorithm,) g(and) h(the) g(exp) r(ected) 945 2184 y(n) n(um) n(b) r(er) 26 b(of) h(transmem) n(brane) e(helices.) h (Only) h(the) g(\014rst) f(t) n(w) n(o) g(of) h(these) g(metrics) f (are) g(pro) n(vided) 945 2292 y(in) 31 b(the) f(TMHMM) h(output.) h (Accordingly) -7 b(,) 29 b(w) n(e) h(pro) r(duce) g(t) n(w) n(o) g (lists) g(of) h(proteins,) f(rank) n(ed) f(b) n(y) 945 2400 y(the) 22 b(n) n(um) n(b) r(er) g(of) f(predicted) h(transmem) n (brane) e(helices) i(\() p Ft(T) p Fq 2748 2412 a(P) 9 b(H) p Fx 2862 2400 a(\)) 22 b(and) g(b) n(y) f(the) i(exp) r(ected) f (n) n(um) n(b) r(er) 945 2508 y(of) 27 b(residues) g(in) h(transmem) n (brane) e(helices) h(\() p Ft(T) p Fq 2384 2520 a(E) s(N) 6 b(R) p Fx 2549 2508 a(\).) 1028 2616 y(Eac) n(h) 30 b(algorithm's) f(p) r(erformance) h(is) h(measured) f(b) n(y) h(splitting) g(the) h(data) e (in) n(to) h(a) f(training) 945 2724 y(and) 37 b(test) g(set) g(in) g (a) f(ratio) g(of) h(80/20.) e(W) -7 b(e) 37 b(rep) r(ort) f(the) i (receiv) n(er) d(op) r(erating) h(c) n(haracteristic) 945 2832 y(\(R) n(OC\)) g(score,) f(whic) n(h) h(is) h(the) f(area) f (under) h(a) g(curv) n(e) g(that) g(plots) h(true) f(p) r(ositiv) n(e) g (rate) f(as) h(a) 945 2940 y(function) k(of) g(false) f(p) r(ositiv) n (e) g(rate) g(for) g(di\013ering) h(classi\014cation) e(thresholds) h (\(Hanley) h(and) 945 3047 y(McNeil,) 26 b(1982;) f(Gribsk) n(o) n(v) f (and) i(Robinson,) g(1996\).) f(The) h(R) n(OC) f(score) g(measures) g (the) i(o) n(v) n(erall) 945 3155 y(qualit) n(y) h(of) g(the) h (ranking) f(induced) h(b) n(y) f(the) h(classi\014er,) e(rather) h (than) g(the) h(qualit) n(y) f(of) h(a) f(single) 945 3263 y(p) r(oin) n(t) 22 b(in) h(that) f(ranking.) f(An) i(R) n(OC) e (score) g(of) h(0.5) f(corresp) r(onds) g(to) h(random) f(guessing,) g (and) h(an) 945 3371 y(R) n(OC) h(score) g(of) h(1.0) f(implies) h (that) h(the) f(algorithm) f(succeeded) h(in) g(putting) h(all) f(of) g (the) g(p) r(ositiv) n(e) 945 3479 y(examples) 34 b(b) r(efore) h(all) g (of) g(the) h(negativ) n(es.) e(Eac) n(h) g(exp) r(erimen) n(t) h(is) g (rep) r(eated) f(30) h(times) g(with) 945 3587 y(di\013eren) n(t) 25 b(random) f(splits) g(in) i(order) d(to) i(estimate) f(the) i(v) -5 b(ariance) 23 b(of) i(the) g(p) r(erformance) f(v) -5 b(alues.) p Fz 945 3803 a(1.6.1.7) 98 b(R) -5 b(esults) p Fx 945 4019 a(W) e(e) 43 b(p) r(erformed) g(computational) g(exp) r (erimen) n(ts) g(whic) n(h) g(study) g(the) h(p) r(erformance) e(of) h (the) 945 4127 y(SDP/SVM) c(approac) n(h) d(as) j(a) f(function) h(of) g (the) g(n) n(um) n(b) r(er) g(of) f(data) g(sources,) g(compare) f(the) 945 4235 y(p) r(erformance) 42 b(of) h(the) h(metho) r(d) f(to) g (classical) f(biological) g(metho) r(ds) h(and) g(state-of-the-art) 945 4343 y(tec) n(hniques) 38 b(for) f(mem) n(brane) h(protein) g (classi\014cation,) f(and) h(study) g(the) h(robustness) e(of) h(the) 945 4451 y(metho) r(d) 28 b(to) f(the) h(presence) f(of) h(noise.) 1028 4558 y(The) i(results) f(from) g(the) h(\014rst) f(three) h(exp) r (erimen) n(ts) f(are) g(summarized) g(in) h(Figure) f(1.2.) g(The) 945 4666 y(plot) j(illustrates) f(that) h(SDP/SVM) g(learns) f (signi\014can) n(tly) g(b) r(etter) h(from) f(the) i(heterogeneous) 945 4774 y(data) d(than) g(from) g(an) n(y) g(single) g(data) g(t) n(yp) r (e.) h(The) f(mean) h(R) n(OC) e(score) h(using) g(all) g(sev) n(en) g (k) n(ernel) 945 4882 y(matrices) j(\(0) p Ft(:) p Fx(9174) p Fu 21 w(\006) p Fx 23 w(0) p Ft(:) p Fx(0025\)) g(is) h(signi\014can) n (tly) g(higher) f(than) i(the) g(b) r(est) g(R) n(OC) f(score) f(using) 945 4990 y(only) k(one) h(matrix) f(\(0) p Ft(:) p Fx(8487) p Fu 23 w(\006) p Fx 25 w(0) p Ft(:) p Fx(0039) f(using) h(the) i (di\013usion) f(k) n(ernel\).) f(This) h(impro) n(v) n(emen) n(t) p 90 rotate dyy eop %%Page: 24 24 24 23 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(24) 670 b(Kernel-b) l(ase) l(d) 30 b(Inte) l(gr) l(ation) f(of) e (Genomic) h(Data) g(using) g(Semide\014nite) g(Pr) l(o) l(gr) l(amming) 863 2365 y @beginspecial 55 @llx 221 @lly 542 @urx 589 @ury 3227 @rwi @setspecial %%BeginDocument: comparison_with_bars_membr_gray.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: MATLAB, The Mathworks, Inc. %%Title: C:\research\Bioinformatics\project_Nello_Bill\PNAS\figures\comparison_with_bars_membr_gray.eps %%CreationDate: 09/26/2003 21:10:05 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%LanguageLevel: 2 %%Pages: 1 %%BoundingBox: 55 221 542 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinecap 1 cap 3 1 roll 2 copy mt lineto stroke cap PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile 3 index 0 eq {/ASCIIHexDecode filter} {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } ifelse exch readstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 55 221 542 589 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0204 7344 csm 460 274 5845 4412 rc 85 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5185 rf 6 w 0 1783 5356 0 0 -1783 899 2172 4 MP PP -5356 0 0 1783 5356 0 0 -1783 899 2172 5 MP stroke 4 w DO SO 6 w 0 sg 899 389 mt 6255 389 L 899 2172 mt 6255 2172 L 6255 2172 mt 6255 389 L 899 2172 mt 899 389 L 899 2172 mt 6255 2172 L 899 2172 mt 899 389 L 1105 2172 mt 1105 2118 L 1105 389 mt 1105 442 L %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 120 FMSR 1065 2317 mt (B) s 1517 2172 mt 1517 2118 L 1517 389 mt 1517 442 L 1421 2317 mt (SW) s 1929 2172 mt 1929 2118 L 1929 389 mt 1929 442 L 1786 2317 mt (HMM) s 2341 2172 mt 2341 2118 L 2341 389 mt 2341 442 L 2232 2317 mt (FFT) s 2753 2172 mt 2753 2118 L 2753 389 mt 2753 442 L 2703 2317 mt (LI) s 3165 2172 mt 3165 2118 L 3165 389 mt 3165 442 L 3122 2317 mt (D) s 3577 2172 mt 3577 2118 L 3577 389 mt 3577 442 L 3537 2317 mt (E) s 3988 2172 mt 3988 2118 L 3988 389 mt 3988 442 L 3928 2317 mt (all) s 4401 2172 mt 4401 2118 L 4401 389 mt 4401 442 L 4351 2317 mt (f1) s 4813 2172 mt 4813 2118 L 4813 389 mt 4813 442 L 4763 2317 mt (f2) s 5225 2172 mt 5225 2118 L 5225 389 mt 5225 442 L 5175 2317 mt (f3) s 5637 2172 mt 5637 2118 L 5637 389 mt 5637 442 L 5554 2317 mt (PH) s 6049 2172 mt 6049 2118 L 6049 389 mt 6049 442 L 5923 2317 mt (ENR) s 899 2172 mt 952 2172 L 6255 2172 mt 6201 2172 L 631 2216 mt (0.70) s 899 1874 mt 952 1874 L 6255 1874 mt 6201 1874 L 631 1918 mt (0.75) s 899 1577 mt 952 1577 L 6255 1577 mt 6201 1577 L 631 1621 mt (0.80) s 899 1280 mt 952 1280 L 6255 1280 mt 6201 1280 L 631 1324 mt (0.85) s 899 983 mt 952 983 L 6255 983 mt 6201 983 L 631 1027 mt (0.90) s 899 686 mt 952 686 L 6255 686 mt 6201 686 L 631 730 mt (0.95) s 899 389 mt 952 389 L 6255 389 mt 6201 389 L 631 433 mt (1.00) s 899 389 mt 6255 389 L 899 2172 mt 6255 2172 L 6255 2172 mt 6255 389 L 899 2172 mt 899 389 L gs 899 389 5357 1784 rc 1 sg 0 536 329 0 0 -536 940 2172 4 MP PP 0 sg -329 0 0 536 329 0 0 -536 940 2172 5 MP stroke 1 sg 0 652 329 0 0 -652 1352 2172 4 MP PP 0 sg -329 0 0 652 329 0 0 -652 1352 2172 5 MP stroke 1 sg 0 822 329 0 0 -822 1764 2172 4 MP PP 0 sg -329 0 0 822 329 0 0 -822 1764 2172 5 MP stroke 1 sg 0 431 329 0 0 -431 2176 2172 4 MP PP 0 sg -329 0 0 431 329 0 0 -431 2176 2172 5 MP stroke 1 sg 0 198 329 0 0 -198 2588 2172 4 MP PP 0 sg -329 0 0 198 329 0 0 -198 2588 2172 5 MP stroke 1 sg 0 884 329 0 0 -884 3000 2172 4 MP PP 0 sg -329 0 0 884 329 0 0 -884 3000 2172 5 MP stroke 1 sg 0 311 329 0 0 -311 3412 2172 4 MP PP 0 sg -329 0 0 311 329 0 0 -311 3412 2172 5 MP stroke 1 sg 0 1292 329 0 0 -1292 3824 2172 4 MP PP 0 sg -329 0 0 1292 329 0 0 -1292 3824 2172 5 MP stroke 1 sg 0 205 329 0 0 -205 4236 2172 4 MP PP 0 sg -329 0 0 205 329 0 0 -205 4236 2172 5 MP stroke 1 sg 0 300 329 0 0 -300 4648 2172 4 MP PP 0 sg -329 0 0 300 329 0 0 -300 4648 2172 5 MP stroke 1 sg 0 523 329 0 0 -523 5060 2172 4 MP PP 0 sg -329 0 0 523 329 0 0 -523 5060 2172 5 MP stroke 1 sg 0 216 329 0 0 -216 5472 2172 4 MP PP 0 sg -329 0 0 216 329 0 0 -216 5472 2172 5 MP stroke 1 sg 0 605 329 0 0 -605 5884 2172 4 MP PP 0 sg -329 0 0 605 329 0 0 -605 5884 2172 5 MP stroke gr 576 1414 mt -90 rotate (ROC) s 90 rotate gs 899 389 5357 1784 rc 0 51 1105 1610 2 MP stroke 57 0 1076 1610 2 MP stroke 57 0 1076 1661 2 MP stroke 0 39 1517 1501 2 MP stroke 57 0 1488 1501 2 MP stroke 57 0 1488 1540 2 MP stroke 0 45 1929 1328 2 MP stroke 57 0 1900 1328 2 MP stroke 57 0 1900 1373 2 MP stroke 0 57 2341 1712 2 MP stroke 57 0 2312 1712 2 MP stroke 57 0 2312 1769 2 MP stroke 0 55 2753 1947 2 MP stroke 57 0 2724 1947 2 MP stroke 57 0 2724 2002 2 MP stroke 0 46 3165 1265 2 MP stroke 57 0 3136 1265 2 MP stroke 57 0 3136 1311 2 MP stroke 0 53 3577 1835 2 MP stroke 57 0 3548 1835 2 MP stroke 57 0 3548 1888 2 MP stroke 0 30 3989 865 2 MP stroke 57 0 3960 865 2 MP stroke 57 0 3960 895 2 MP stroke gr 24 W 1105 1636 PD 24 W 1517 1520 PD 24 W 1929 1350 PD 24 W 2341 1741 PD 24 W 2753 1974 PD 24 W 3165 1288 PD 24 W 3577 1861 PD 24 W 3989 880 PD gs 899 389 5357 1784 rc gr 1 sg 0 1783 5356 0 0 -1783 899 4614 4 MP PP -5356 0 0 1783 5356 0 0 -1783 899 4614 5 MP stroke 4 w DO SO 6 w 0 sg 899 2831 mt 6255 2831 L 899 4614 mt 6255 4614 L 6255 4614 mt 6255 2831 L 899 4614 mt 899 2831 L 899 4614 mt 6255 4614 L 899 4614 mt 899 2831 L 1105 4614 mt 1105 4560 L 1105 2831 mt 1105 2884 L 1517 4614 mt 1517 4560 L 1517 2831 mt 1517 2884 L 1929 4614 mt 1929 4560 L 1929 2831 mt 1929 2884 L 2341 4614 mt 2341 4560 L 2341 2831 mt 2341 2884 L 2753 4614 mt 2753 4560 L 2753 2831 mt 2753 2884 L 3165 4614 mt 3165 4560 L 3165 2831 mt 3165 2884 L 3577 4614 mt 3577 4560 L 3577 2831 mt 3577 2884 L 3988 4614 mt 3988 4560 L 3988 2831 mt 3988 2884 L 4401 4614 mt 4401 4560 L 4401 2831 mt 4401 2884 L 4813 4614 mt 4813 4560 L 4813 2831 mt 4813 2884 L 5225 4614 mt 5225 4560 L 5225 2831 mt 5225 2884 L 5637 4614 mt 5637 4560 L 5637 2831 mt 5637 2884 L 6049 4614 mt 6049 4560 L 6049 2831 mt 6049 2884 L 899 4614 mt 952 4614 L 6255 4614 mt 6201 4614 L 798 4658 mt (0) s 899 4257 mt 952 4257 L 6255 4257 mt 6201 4257 L 698 4301 mt (0.2) s 899 3900 mt 952 3900 L 6255 3900 mt 6201 3900 L 698 3944 mt (0.4) s 899 3544 mt 952 3544 L 6255 3544 mt 6201 3544 L 698 3588 mt (0.6) s 899 3187 mt 952 3187 L 6255 3187 mt 6201 3187 L 698 3231 mt (0.8) s 899 2831 mt 952 2831 L 6255 2831 mt 6201 2831 L 798 2875 mt (1) s 899 2831 mt 6255 2831 L 899 4614 mt 6255 4614 L 6255 4614 mt 6255 2831 L 899 4614 mt 899 2831 L gs 899 2831 5357 1784 rc 0 1783 329 0 0 -1783 940 4614 4 MP PP -329 0 0 1783 329 0 0 -1783 940 4614 5 MP stroke 0 0 329 0 0 0 1352 4614 4 MP PP -329 0 0 0 329 0 0 0 1352 4614 5 MP stroke 0 0 329 0 0 0 1764 4614 4 MP PP -329 0 0 0 329 0 0 0 1764 4614 5 MP stroke 0 0 329 0 0 0 2176 4614 4 MP PP -329 0 0 0 329 0 0 0 2176 4614 5 MP stroke 0 0 329 0 0 0 2588 4614 4 MP PP -329 0 0 0 329 0 0 0 2588 4614 5 MP stroke 0 0 329 0 0 0 3000 4614 4 MP PP -329 0 0 0 329 0 0 0 3000 4614 5 MP stroke 0 0 329 0 0 0 3412 4614 4 MP PP -329 0 0 0 329 0 0 0 3412 4614 5 MP stroke 0 423 329 0 0 -423 3824 4614 4 MP PP -329 0 0 423 329 0 0 -423 3824 4614 5 MP stroke 0 0 329 0 0 0 4236 4614 4 MP PP -329 0 0 0 329 0 0 0 4236 4614 5 MP stroke 0 0 329 0 0 0 4648 4614 4 MP PP -329 0 0 0 329 0 0 0 4648 4614 5 MP stroke 0 0 329 0 0 0 5060 4614 4 MP PP -329 0 0 0 329 0 0 0 5060 4614 5 MP stroke 0 0 329 0 0 0 5472 4614 4 MP PP -329 0 0 0 329 0 0 0 5472 4614 5 MP stroke 0 0 329 0 0 0 5884 4614 4 MP PP -329 0 0 0 329 0 0 0 5884 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 940 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 940 2831 5 MP stroke 0.15873 sg 0 1783 329 0 0 -1783 1352 4614 4 MP PP 0 sg -329 0 0 1783 329 0 0 -1783 1352 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 1764 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1764 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 2176 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2176 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 2588 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2588 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 3000 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3000 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 3412 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3412 4614 5 MP stroke 0.15873 sg 0 465 329 0 0 -465 3824 4191 4 MP PP 0 sg -329 0 0 465 329 0 0 -465 3824 4191 5 MP stroke 0.15873 sg 0 0 329 0 0 0 4236 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4236 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 4648 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4648 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 5060 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5060 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 5472 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5472 4614 5 MP stroke 0.15873 sg 0 0 329 0 0 0 5884 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5884 4614 5 MP stroke 0.333333 sg 0 0 329 0 0 0 940 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 940 2831 5 MP stroke 0.333333 sg 0 0 329 0 0 0 1352 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1352 2831 5 MP stroke 0.333333 sg 0 1783 329 0 0 -1783 1764 4614 4 MP PP 0 sg -329 0 0 1783 329 0 0 -1783 1764 4614 5 MP stroke 0.333333 sg 0 0 329 0 0 0 2176 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2176 4614 5 MP stroke 0.333333 sg 0 0 329 0 0 0 2588 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2588 4614 5 MP stroke 0.333333 sg 0 0 329 0 0 0 3000 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3000 4614 5 MP stroke 0.333333 sg 0 0 329 0 0 0 3412 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3412 4614 5 MP stroke 0.333333 sg 0 237 329 0 0 -237 3824 3726 4 MP PP 0 sg -329 0 0 237 329 0 0 -237 3824 3726 5 MP stroke 0.333333 sg 0 0 329 0 0 0 4236 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4236 4614 5 MP stroke 0.333333 sg 0 0 329 0 0 0 4648 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4648 4614 5 MP stroke 0.333333 sg 0 0 329 0 0 0 5060 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5060 4614 5 MP stroke 0.333333 sg 0 0 329 0 0 0 5472 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5472 4614 5 MP stroke 0.333333 sg 0 0 329 0 0 0 5884 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5884 4614 5 MP stroke 0.507937 sg 0 0 329 0 0 0 940 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 940 2831 5 MP stroke 0.507937 sg 0 0 329 0 0 0 1352 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1352 2831 5 MP stroke 0.507937 sg 0 0 329 0 0 0 1764 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1764 2831 5 MP stroke 0.507937 sg 0 1783 329 0 0 -1783 2176 4614 4 MP PP 0 sg -329 0 0 1783 329 0 0 -1783 2176 4614 5 MP stroke 0.507937 sg 0 0 329 0 0 0 2588 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2588 4614 5 MP stroke 0.507937 sg 0 0 329 0 0 0 3000 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3000 4614 5 MP stroke 0.507937 sg 0 0 329 0 0 0 3412 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3412 4614 5 MP stroke 0.507937 sg 0 99 329 0 0 -99 3824 3489 4 MP PP 0 sg -329 0 0 99 329 0 0 -99 3824 3489 5 MP stroke 0.507937 sg 0 0 329 0 0 0 4236 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4236 4614 5 MP stroke 0.507937 sg 0 0 329 0 0 0 4648 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4648 4614 5 MP stroke 0.507937 sg 0 0 329 0 0 0 5060 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5060 4614 5 MP stroke 0.507937 sg 0 0 329 0 0 0 5472 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5472 4614 5 MP stroke 0.507937 sg 0 0 329 0 0 0 5884 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5884 4614 5 MP stroke 0.666667 sg 0 0 329 0 0 0 940 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 940 2831 5 MP stroke 0.666667 sg 0 0 329 0 0 0 1352 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1352 2831 5 MP stroke 0.666667 sg 0 0 329 0 0 0 1764 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1764 2831 5 MP stroke 0.666667 sg 0 0 329 0 0 0 2176 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2176 2831 5 MP stroke 0.666667 sg 0 1783 329 0 0 -1783 2588 4614 4 MP PP 0 sg -329 0 0 1783 329 0 0 -1783 2588 4614 5 MP stroke 0.666667 sg 0 0 329 0 0 0 3000 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3000 4614 5 MP stroke 0.666667 sg 0 0 329 0 0 0 3412 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3412 4614 5 MP stroke 0.666667 sg 0 2 329 0 0 -2 3824 3390 4 MP PP 0 sg -329 0 0 2 329 0 0 -2 3824 3390 5 MP stroke 0.666667 sg 0 0 329 0 0 0 4236 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4236 4614 5 MP stroke 0.666667 sg 0 0 329 0 0 0 4648 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4648 4614 5 MP stroke 0.666667 sg 0 0 329 0 0 0 5060 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5060 4614 5 MP stroke 0.666667 sg 0 0 329 0 0 0 5472 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5472 4614 5 MP stroke 0.666667 sg 0 0 329 0 0 0 5884 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5884 4614 5 MP stroke 0.84127 sg 0 0 329 0 0 0 940 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 940 2831 5 MP stroke 0.84127 sg 0 0 329 0 0 0 1352 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1352 2831 5 MP stroke 0.84127 sg 0 0 329 0 0 0 1764 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1764 2831 5 MP stroke 0.84127 sg 0 0 329 0 0 0 2176 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2176 2831 5 MP stroke 0.84127 sg 0 0 329 0 0 0 2588 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2588 2831 5 MP stroke 0.84127 sg 0 1783 329 0 0 -1783 3000 4614 4 MP PP 0 sg -329 0 0 1783 329 0 0 -1783 3000 4614 5 MP stroke 0.84127 sg 0 0 329 0 0 0 3412 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3412 4614 5 MP stroke 0.84127 sg 0 349 329 0 0 -349 3824 3388 4 MP PP 0 sg -329 0 0 349 329 0 0 -349 3824 3388 5 MP stroke 0.84127 sg 0 0 329 0 0 0 4236 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4236 4614 5 MP stroke 0.84127 sg 0 0 329 0 0 0 4648 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4648 4614 5 MP stroke 0.84127 sg 0 0 329 0 0 0 5060 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5060 4614 5 MP stroke 0.84127 sg 0 0 329 0 0 0 5472 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5472 4614 5 MP stroke 0.84127 sg 0 0 329 0 0 0 5884 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5884 4614 5 MP stroke 1 sg 0 0 329 0 0 0 940 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 940 2831 5 MP stroke 1 sg 0 0 329 0 0 0 1352 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1352 2831 5 MP stroke 1 sg 0 0 329 0 0 0 1764 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 1764 2831 5 MP stroke 1 sg 0 0 329 0 0 0 2176 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2176 2831 5 MP stroke 1 sg 0 0 329 0 0 0 2588 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 2588 2831 5 MP stroke 1 sg 0 0 329 0 0 0 3000 2831 4 MP PP 0 sg -329 0 0 0 329 0 0 0 3000 2831 5 MP stroke 1 sg 0 1783 329 0 0 -1783 3412 4614 4 MP PP 0 sg -329 0 0 1783 329 0 0 -1783 3412 4614 5 MP stroke 1 sg 0 208 329 0 0 -208 3824 3039 4 MP PP 0 sg -329 0 0 208 329 0 0 -208 3824 3039 5 MP stroke 1 sg 0 0 329 0 0 0 4236 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4236 4614 5 MP stroke 1 sg 0 0 329 0 0 0 4648 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 4648 4614 5 MP stroke 1 sg 0 0 329 0 0 0 5060 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5060 4614 5 MP stroke 1 sg 0 0 329 0 0 0 5472 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5472 4614 5 MP stroke 1 sg 0 0 329 0 0 0 5884 4614 4 MP PP 0 sg -329 0 0 0 329 0 0 0 5884 4614 5 MP stroke gr 643 3938 mt -90 rotate (Weights) s 90 rotate end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial Ff 714 2523 a(Figure) f(1.2) 83 b(Com) n(bining) 28 b(data) g(sets) f(yields) g(b) r(etter) h(classi\014cation) i(p) r (erformance.) p Fx 26 w(The) 714 2630 y(heigh) n(t) 25 b(of) g(eac) n(h) f(bar) h(is) g(prop) r(ortional) e(to) i(the) h(R) n (OC) e(score) g(of) h(the) h(giv) n(en) e(mem) n(brane) h(protein) 714 2738 y(classi\014cation) 32 b(metho) r(d.) i(The) f(bars) f(lab) r (eled) p Fj 34 w(B) p Fx 37 w(to) p Fj 33 w(E) p Fx 38 w(and) p Fj 33 w(al) q(l) p Fx 34 w(corresp) r(ond) g(to) h(SDP/SVM) 714 2846 y(metho) r(ds,) 20 b(the) h(bars) e(lab) r(eled) p Fj 20 w(f) p Fx 29 w(are) g(h) n(ydropath) n(y) f(pro\014le) i (metrics,) f(and) h(the) h(bars) e(lab) r(eled) p Fj 20 w(P) 11 b(H) p Fx 714 2954 a(and) p Fj 23 w(E) t(N) d(R) p Fx 25 w(refer) 23 b(to) h(the) g(TMHMM) h(metho) r(ds) f(as) f (de\014ned) h(in) h(the) f(text.) g(Error) e(bars) g(indicate) 714 3062 y(standard) g(error) f(across) g(30) h(random) g(train/test) h (splits.) g(The) g(heigh) n(ts) g(of) g(the) g(grey) f(lev) n(el) h (bars) 714 3170 y(b) r(elo) n(w) 29 b(eac) n(h) g(plot) g(indicate) h (the) g(relativ) n(e) e(w) n(eigh) n(t) h(of) h(the) g(di\013eren) n(t) g(k) n(ernel) e(matrices) h(in) h(the) 714 3278 y(optimal) d(linear) g (com) n(bination.) 714 3518 y(corresp) r(onds) e(to) j(a) f(c) n(hange) f(in) i(test) g(set) g(accuracy) e(of) h(7) p Ft(:) p Fx(3\045,) g(from) g(81) p Ft(:) p Fx(3\045) g(to) g(88) p Ft(:) p Fx(6\045.) 797 3626 y(As) c(exp) r(ected,) h(the) h (sequence-based) d(k) n(ernels) g(yield) i(go) r(o) r(d) f(individual) h (p) r(erformance.) f(This) 714 3734 y(is) 28 b(eviden) n(t) g(from) g (the) g(R) n(OC) g(scores.) e(F) -7 b(urthermore,) 27 b(when) i(all) f(sev) n(en) f(matrices) g(are) h(used) g(at) 714 3842 y(once,) c(the) h(SDP) g(assigns) e(relativ) n(ely) g(large) h(w) n (eigh) n(ts) f(to) i(the) g(sequence-based) e(k) n(ernels.) h(These) 714 3950 y(w) n(eigh) n(ts) 37 b(are) g(as) h(follo) n(ws:) p Ft 37 w(\026) p Fq 1638 3962 a(B) p Fx 1736 3950 a(=) i(1) p Ft(:) p Fx(66,) p Ft 37 w(\026) p Fq 2100 3962 a(S) s(W) p Fx 2260 3950 a(=) g(1) p Ft(:) p Fx(83,) p Ft 37 w(\026) p Fq 2624 3962 a(H) t(M) 6 b(M) p Fx 2867 3950 a(=) 40 b(0) p Ft(:) p Fx(93,) p Ft 37 w(\026) p Fq 3231 3962 a(F) 9 b(F) g(T) p Fx 3426 3950 a(=) 41 b(0) p Ft(:) p Fx(39,) p Ft 714 4058 a(\026) p Fq 764 4070 a(LI) p Fx 870 4058 a(=) 23 b(0) p Ft(:) p Fx(01,) p Ft 22 w(\026) p Fq 1202 4070 a(D) p Fx 1286 4058 a(=) f(1) p Ft(:) p Fx(37) h(and) p Ft 23 w(\026) p Fq 1752 4070 a(E) p Fx 1832 4058 a(=) f(0) p Ft(:) p Fx(82) h(\(note) g(that) h(for) g(ease) f (of) g(in) n(terpretation,) g(w) n(e) g(scale) 714 4166 y(the) 39 b(w) n(eigh) n(ts) f(suc) n(h) h(that) g(their) g(sum) g(is) g (equal) f(to) h(the) g(n) n(um) n(b) r(er) p Ft 39 w(m) p Fx 39 w(of) g(k) n(ernel) f(matrices\).) 714 4274 y(Th) n(us,) 43 b(t) n(w) n(o) h(of) g(the) g(three) g(k) n(ernel) f(matrices) g(that) i (receiv) n(e) d(w) n(eigh) n(ts) i(larger) e(than) i(1) g(are) 714 4382 y(deriv) n(ed) 31 b(from) h(the) h(amino) f(acid) g(sequence.) g (The) h(Smith-W) -7 b(aterman) 32 b(k) n(ernel) g(yields) g(b) r(etter) 714 4490 y(results) 23 b(than) g(the) h(BLAST) f(k) n(ernel,) g (re\015ecting) g(the) h(fact) f(that) h(BLAST) f(is) h(a) f(heuristic) g (searc) n(h) 714 4598 y(pro) r(cedure,) h(whereas) h(the) h(Smith-W) -7 b(aterman) 25 b(algorithm) g(guaran) n(tees) e(\014nding) j(the) g (optimal) 714 4706 y(lo) r(cal) h(alignmen) n(t) g(of) g(t) n(w) n(o) g (sequences.) 797 4814 y(The) c(results) f(also) g(sho) n(w) g(that) i (the) f(in) n(teraction-based) e(di\013usion) i(k) n(ernel) f(is) h (more) f(informa-) 714 4921 y(tiv) n(e) f(than) g(the) h(expression) e (k) n(ernel.) g(Not) h(only) g(has) g(the) h(di\013usion) f(k) n(ernel) g(an) g(individual) g(R) n(OC) p 90 rotate dyy eop %%Page: 25 25 25 24 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.6) 78 b(Two) 27 b(Biolo) l(gic) l(al) g(Applic) l(ations) 2546 b(25) p Fx 945 349 a(score) 25 b(whic) n(h) h(is) g(signi\014can) n (tly) g(higher) g(than) g(the) h(expression) e(k) n(ernel,) g(the) i (SDP) g(also) e(assigns) 945 457 y(a) 34 b(w) n(eigh) n(t) g(of) g (1.37) g(to) g(the) h(di\013usion) g(k) n(ernel,) f(whereas) f(the) i (expression) e(k) n(ernel) h(receiv) n(es) f(a) 945 565 y(w) n(eigh) n(t) e(of) g(0.82.) f(Accordingly) -7 b(,) 31 b(remo) n(ving) e(the) j(di\013usion) g(k) n(ernel) f(reduces) f(the) i (R) n(OC) f(score) 945 673 y(from) 25 b(0.9174) d(to) k(0.8984,) c (whereas) i(remo) n(ving) g(the) h(expression) f(k) n(ernel) g(has) h (a) g(smaller) f(e\013ect,) 945 781 y(leading) h(to) h(a) f(R) n(OC) h (score) e(of) i(0.9033.) e(F) -7 b(urther) 26 b(description) f(of) h (the) g(results) g(obtained) f(when) 945 889 y(v) -5 b(arious) 26 b(subsets) h(of) h(k) n(ernels) e(are) h(used) h(is) f (pro) n(vided) g(in) h(Lanc) n(kriet) e(et) i(al.) f(\(2003\).) 1028 997 y(Figure) 40 b(1.2) g(also) f(compares) g(the) j(mem) n(brane) d (protein) i(classi\014cation) e(p) r(erformance) h(of) 945 1105 y(the) 34 b(SDP/SVM) f(metho) r(d) h(with) g(that) g(of) g (previously) e(describ) r(ed) h(tec) n(hniques.) h(The) f(results) 945 1213 y(con\014rm) 19 b(that) 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f(that) i (separates) e(p) r(ositiv) n(e) h(and) g(negativ) n(e) f(instances) h (of) g(mem) n(brane) 945 1968 y(proteins,) 41 b(the) h(TMHMM) h(is) e (a) h(generativ) n(e) e(metho) r(d) i(that) h(simply) e(attempts) i(to) e(mo) r(del) 945 2076 y(the) 34 b(mem) n(brane) f(proteins.) g(As) h (an) f(illustration) h(of) f(the) i(di\013erence,) e(it) h(is) g(kno) n (wn) f(that) h(the) 945 2184 y(TMHMM) d(tends) g(to) g(yield) f(false) h (p) r(ositiv) n(es) f(for) g(sequences) g(con) n(taining) f(signal) h (p) r(eptides|) 945 2292 y(h) n(ydrophobic) 44 b(sequences) g(in) i (the) g(N-terminal) f(regions) f(of) h(proteins) g(\(Chen) h(and) f (Rost,) 945 2400 y(2002\).) 23 b(The) i(SDP/SVM) g(approac) n(h) e (tends) j(to) f(a) n(v) n(oid) e(these) i(false) g(p) r(ositiv) n(es,) f (b) r(ecause) h(signal) 945 2508 y(p) r(eptides) g(app) r(ear) f(among) f(the) i(negativ) n(e) f(instances) g(in) h(the) g(training) f(set.) h (Indeed,) f(as) g(sho) n(wn) 945 2616 y(in) d(Lanc) n(kriet) e(et) i (al.) f(\(2003\),) g(signal) g(p) r(eptides) h(tend) g(to) g(b) r(e) g (highly) f(rank) n(ed) g(b) n(y) g(the) h(TMHMM,) 945 2724 y(and) 27 b(are) g(more) g(uniformly) g(spread) g(within) h(the) g (SDP/SVM) g(rankings.) 1028 2832 y(Finally) -7 b(,) 37 b(in) g(order) f(to) h(test) h(the) f(robustness) g(of) g(our) f (approac) n(h,) f(a) i(second) g(exp) r(erimen) n(t) 945 2940 y(w) n(as) 29 b(p) r(erformed) h(in) h(whic) n(h) g(a) f(randomly) f(generated) h(k) n(ernel) f(matrix) p Ft 30 w(K) p Fq 3267 2952 a(RN) 6 b(D) p Fx 3467 2940 a(w) n(as) 29 b(included) 945 3047 y(among) d(the) i(k) n(ernel) f(matrices) g(used) h(as) f(input) h (to) g(our) f(algorithm.) f(This) i(k) n(ernel) f(matrix) g(w) n(as) 945 3155 y(generated) 22 b(b) n(y) h(sampling) f(100-elemen) n(t) g(v) n (ectors) g(for) g(eac) n(h) h(protein,) g(where) f(eac) n(h) h(comp) r (onen) n(t) 945 3263 y(of) 37 b(eac) n(h) f(v) n(ector) g(w) n(as) g (sampled) h(indep) r(enden) n(tly) h(from) f(a) g(standard) f(normal) g (distribution,) 945 3371 y(and) j(then) h(computing) g(inner) f(pro) r (ducts) g(of) h(the) g(100-elemen) n(t) e(v) n(ectors) g(to) h(form) p Ft 40 w(K) p Fq 3742 3383 a(RN) 6 b(D) p Fx 3910 3371 a(.) 945 3479 y(A) 33 b(con) n(trol) f(classi\014er) g(trained) h (using) g(only) g(the) g(random) g(k) n(ernel) f(yields) h(an) g(R) n (OC) g(score) f(of) 945 3587 y(0.5,) 23 b(indicating) i(that) p Ft 25 w(K) p Fq 1731 3599 a(RN) 6 b(D) p Fx 1924 3587 a(is) 24 b(indeed) h(uninformativ) n(e) f(for) g(the) h (classi\014cation) e(problem) h(at) 945 3695 y(hand.) f(More) f(imp) r (ortan) n(tly) -7 b(,) 22 b(when) h(a) f(classi\014er) g(is) h(trained) f(using) g(all) h(sev) n(en) f(real) g(k) n(ernels) g(plus) p Ft 945 3803 a(K) p Fq 1016 3815 a(RN) 6 b(D) p Fx 1185 3803 a(,) 34 b(SDP) g(assigns) f(the) h(random) f(k) n(ernel) h(a) f(w) n(eigh) n(t) h(that) g(is) g(close) f(to) h(zero.) f(Th) n(us,) h(the) 945 3911 y(R) n(OC) 28 b(score) f(deriv) n(ed) h(from) g(sev) n(en) g (matrices) g(do) 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b(All) f(three) g(data) g (sets) g(can) g(b) r(e) g(represen) n(ted) f(as) h(graphs,) f(with) i (proteins) e(as) h(no) r(des) 945 457 y(and) j(in) n(teractions) e(as) i (edges.) f(As) h(explained) g(b) r(efore,) f(eac) n(h) g(in) n (teraction) g(graph) g(allo) n(ws) g(to) 945 565 y(establish) 31 b(similarities) g(among) g(proteins) g(through) g(the) h(construction) f (of) g(a) h(corresp) r(onding) 945 673 y(di\013usion) 23 b(k) n(ernel.) g(This) g(generates) f(three) h(in) n(teraction) g(k) n (ernel) f(matrices,) p Ft 23 w(K) p Fq 3370 685 a(Gen) p Fx 3498 673 a(,) p Ft 23 w(K) p Fq 3615 685 a(P) 9 b(hy) r(s) p Fx 3800 673 a(and) p Ft 945 781 a(K) p Fq 1016 793 a(T) g(AP) p Fx 1169 781 a(.) 28 b(Because) f(direct) i(ph) n(ysical) e(in) n (teraction) g(is) h(not) h(necessarily) d(guaran) n(teed) h(when) h(t) n (w) n(o) 945 889 y(proteins) j(participate) g(in) g(a) h(complex,) f(a) g(smaller) g(di\013usion) h(constan) n(t|this) f(parameter) f(is) 945 997 y(required) i(to) g(construct) 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r(ect) 945 2832 y(to) 27 b(the) h(Pfam) f(database.) p Fz 945 3047 a(1.6.2.2) 98 b(R) -5 b(esults) p Fx 945 3263 a(Eac) n(h) 37 b(algorithm's) g(p) r(erformance) g(is) h(measured) f(b) n(y) h(p) r (erforming) f(5-fold) h(cross-v) -5 b(alidation) 945 3371 y(three) 35 b(times.) h(F) -7 b(or) 36 b(a) f(giv) n(en) g(split,) h(w) n(e) g(again) e(ev) -5 b(aluate) 36 b(eac) n(h) f(classi\014er) f (b) n(y) i(rep) r(orting) e(the) 945 3479 y(receiv) n(er) 19 b(op) r(erating) h(c) n(haracteristic) f(\(R) n(OC\)) i(score) f(on) h (the) h(test) f(set.) g(F) -7 b(or) 21 b(eac) n(h) f(classi\014cation,) 945 3587 y(w) n(e) 34 b(measure) f(15) h(R) n(OC) g(scores) e(\(three) j (5-fold) f(splits\),) h(whic) n(h) f(allo) n(ws) f(us) i(to) f (estimate) g(the) 945 3695 y(v) -5 b(ariance) 26 b(of) i(the) g(score.) 1028 3803 y(The) 20 b(exp) r(erimen) n(tal) h(results) f(are) f (summarized) h(in) h(Figure) f(1.3.) g(The) h(\014gure) f(sho) n(ws) f (that,) i(for) 945 3911 y(eac) n(h) e(of) h(the) h(13) e (classi\014cations,) f(the) j(R) n(OC) e(score) g(of) h(the) g(SDP/SVM) h(metho) r(d) f(is) g(b) r(etter) h(than) 945 4019 y(that) 32 b(of) h(the) g(MRF) f(metho) r(d.) h(Ov) n(erall,) e(the) i(mean) f(R) n (OC) g(impro) n(v) n(es) f(from) h(0.715) e(to) i(0.854.) 945 4127 y(The) 26 b(impro) n(v) n(emen) n(t) e(is) i(consisten) n(t) f (and) h(statistically) f(signi\014can) n(t) g(across) f(all) i(13) f (classes.) f(An) 945 4235 y(additional) i(impro) n(v) n(emen) n(t,) f (though) h(not) g(as) g(large,) f(is) h(gained) f(b) n(y) h(replacing) f (the) i(expression) 945 4343 y(and) 34 b(Pfam) g(k) n(ernels) g(with) h (their) f(enric) n(hed) g(v) n(ersions.) f(The) h(most) h(impro) n(v) n (emen) n(t) e(is) h(o\013ered) 945 4451 y(b) n(y) i(using) h(the) g (enric) n(hed) f(Pfam) g(k) n(ernel) g(and) g(replacing) g(the) h (expression) e(k) n(ernel) h(with) h(the) 945 4558 y(Smith-W) -7 b(aterman) 20 b(k) n(ernel.) f(The) h(resulting) f(mean) h(R) n(OC) f (is) h(0.870.) e(Again,) i(the) 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/bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile 3 index 0 eq {/ASCIIHexDecode filter} {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } ifelse exch readstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup 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8839 5 MP stroke 0.507937 sg 0 7685 73 0 0 -7685 2716 8839 4 MP PP 0 sg -73 0 0 7685 73 0 0 -7685 2716 8839 5 MP stroke 0.507937 sg 0 7387 73 0 0 -7387 3128 8839 4 MP PP 0 sg -73 0 0 7387 73 0 0 -7387 3128 8839 5 MP stroke 0.507937 sg 0 7731 73 0 0 -7731 3540 8839 4 MP PP 0 sg -73 0 0 7731 73 0 0 -7731 3540 8839 5 MP stroke 0.507937 sg 0 6780 73 0 0 -6780 3952 8839 4 MP PP 0 sg -73 0 0 6780 73 0 0 -6780 3952 8839 5 MP stroke 0.507937 sg 0 7287 73 0 0 -7287 4364 8839 4 MP PP 0 sg -73 0 0 7287 73 0 0 -7287 4364 8839 5 MP stroke 0.507937 sg 0 6862 73 0 0 -6862 4776 8839 4 MP PP 0 sg -73 0 0 6862 73 0 0 -6862 4776 8839 5 MP stroke 0.507937 sg 0 5556 73 0 0 -5556 5188 8839 4 MP PP 0 sg -73 0 0 5556 73 0 0 -5556 5188 8839 5 MP stroke 0.507937 sg 0 8175 73 0 0 -8175 5600 8839 4 MP PP 0 sg -73 0 0 8175 73 0 0 -8175 5600 8839 5 MP stroke 0.507937 sg 0 6831 73 0 0 -6831 6012 8839 4 MP PP 0 sg -73 0 0 6831 73 0 0 -6831 6012 8839 5 MP stroke 1 sg 0 7624 74 0 0 -7624 1159 8839 4 MP PP 0 sg -74 0 0 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g(bars) 714 3065 y(corresp) r(ond) 26 b(to) j(the) g(SDP/SVM) f(metho) r(d) h (using) f(\014v) n(e) g(k) n(ernels) f(computed) i(on) f(binary) g (data,) 714 3173 y(and) j(white) h(bars) e(corresp) r(ond) g(to) h(the) g(SDP/SVM) h(using) f(the) h(enric) n(hed) e(Pfam) h(k) n(ernel) g(and) 714 3281 y(replacing) 26 b(the) i(expression) e(k) n(ernel) h(with) h (the) g(SW) g(k) n(ernel.) p Ff 714 3568 a(T) -7 b(able) 25 b(1.3) 83 b(Kernel) 26 b(w) n(eigh) n(ts) g(and) g(R) n(OC) g(scores) g (for) h(the) e(transp) r(ort) k(facilitation) g(class.) p Fx 714 3676 a(The) 19 b(table) g(sho) n(ws,) f(for) h(b) r(oth) h(exp) r (erimen) n(ts,) f(the) g(mean) h(w) n(eigh) n(t) e(asso) r(ciated) g (with) i(eac) n(h) e(k) n(ernel,) 714 3784 y(as) 28 b(w) n(ell) h(as) f (the) h(R) n(OC) g(score) e(resulting) i(from) f(learning) g(the) h (classi\014cation) f(using) h(only) f(that) 714 3892 y(k) n(ernel.) e(The) i(\014nal) g(ro) n(w) e(lists) h(the) h(R) n(OC) f (score) g(using) g(all) g(k) n(ernels.) p 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-649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(1.7) 78 b(Discussion) 3120 b(29) p Fx 945 349 a(tance) 28 b(that) i(the) f(SDP/SVM) g(pro) r(cedure) f(assigns) f(to) i(that) g (k) n(ernel.) f(The) h(Pfam) f(and) h(Smith-) 945 457 y(W) -7 b(aterman) 28 b(k) n(ernels) g(yield) i(the) f(largest) f(w) n (eigh) n(ts,) g(as) g(w) n(ell) h(as) g(the) g(largest) f(individual) h (R) n(OC) 945 565 y(scores.) 24 b(Note) h(that) h(the) g(com) n (bination) f(of) h(k) n(ernels) e(p) r(erforms) h(signi\014can) n(tly) g (b) r(etter) h(than) f(an) n(y) 945 673 y(single) i(k) n(ernel.) g (Results) g(for) g(the) h(other) f(t) n(w) n(elv) n(e) g (classi\014cations) f(are) h(similar.) p Fv 198 1021 a(1.7) 100 b(Discussion) p 198 888 3736 3 v Fx 945 1237 a(W) -7 b(e) 39 b(ha) n(v) n(e) f(describ) r(ed) h(a) f(general) g (metho) r(d) h(for) f(com) n(bining) h(heterogeneous) e(genome-wide) 945 1345 y(data) 42 b(sets) h(in) g(the) g(setting) f(of) h(k) n (ernel-based) e(statistical) h(learning) g(algorithms,) f(and) i(w) n (e) 945 1453 y(ha) n(v) n(e) 36 b(demonstrated) h(an) g(application) g (of) g(this) h(metho) r(d) g(to) f(the) h(problems) e(of) i (classifying) 945 1561 y(y) n(east) 33 b(mem) n(brane) h(proteins) g (and) h(protein) f(function) h(prediction) g(in) g(y) n(east.) e(The) i (resulting) 945 1669 y(SDP/SVM) i(algorithm) e(yields) h(signi\014can) n (t) g(impro) n(v) n(emen) n(t) g(relativ) n(e) f(to) i(an) f(SVM) h (trained) 945 1777 y(from) e(an) n(y) g(single) g(data) h(t) n(yp) r (e,) g(relativ) n(e) e(to) i(b) r(oth) g(state-of-the-art) e(and) i (classical) e(biologi-) 945 1885 y(cal) i(metho) r(ds) h(for) f(mem) n (brane) f(protein) i(prediction) f(as) g(w) n(ell) g(as) g(relativ) n (e) f(to) i(a) f(previously) 945 1993 y(prop) r(osed) 28 b(graphical) g(mo) r(del) h(approac) n(h) f(for) h(fusing) g (heterogeneous) e(genomic) i(data.) g(More-) 945 2101 y(o) n(v) n(er,) k(the) i(p) r(erformance) e(of) i(the) g(algorithm) e (impro) n(v) n(es) g(consisten) n(tly) h(in) h(our) f(exp) r(erimen) n (ts) 945 2209 y(as) 28 b(additional) h(genome-wide) f(data) h(sets) g (are) f(added) i(to) f(the) g(k) n(ernel) g(represen) n(tation,) e(if) j (the) 945 2316 y(additional) d(data) g(con) n(tain) g(complemen) n (tary) f(information.) 1028 2424 y(Kernel-based) 34 b(statistical) h (learning) g(metho) r(ds) h(ha) n(v) n(e) e(a) i(n) n(um) n(b) r(er) f (of) h(general) e(virtues) h(as) 945 2532 y(to) r(ols) f(for) g (biological) f(data) h(analysis.) f(First,) h(the) h(k) n(ernel) f (framew) n(ork) e(accommo) r(dates) i(not) 945 2640 y(only) 25 b(the) h(v) n(ectorial) e(and) i(matrix) f(data) g(that) h(are) f (familiar) g(in) h(classical) f(statistical) g(analysis,) 945 2748 y(but) 42 b(also) e(more) g(exotic) h(data) g(t) n(yp) r(es) g (suc) n(h) g(as) g(strings,) f(trees) h(and) g(graphs.) f(The) h (abilit) n(y) 945 2856 y(to) 34 b(handle) h(suc) n(h) f(data) g(is) g (clearly) f(essen) n(tial) h(in) h(the) g(biological) d(domain.) j (Second,) f(k) n(ernels) 945 2964 y(pro) n(vide) g(signi\014can) n(t) h (opp) r(ortunities) g(for) g(the) h(incorp) r(oration) e(of) h(more) g (sp) r(eci\014c) g(biological) 945 3072 y(kno) n(wledge,) 19 b(as) g(w) n(e) g(ha) n(v) n(e) g(seen) h(with) h(the) f(FFT) g(k) n (ernel) f(and) h(the) h(Pfam) e(k) n(ernel,) g(and) h(unlab) r(eled) 945 3180 y(data,) 36 b(as) h(in) g(the) g(di\013usion) h(and) e(Smith-W) -7 b(aterman) 37 b(k) n(ernels.) f(Third,) h(the) g(gro) n(wing) e(suite) 945 3288 y(of) h(k) n(ernel-based) f(data) i(analysis) e(algorithms) g (require) h(only) g(that) i(data) e(b) r(e) h(reduced) f(to) h(a) 945 3396 y(k) n(ernel) d(matrix;) g(this) i(creates) d(opp) r(ortunities) i (for) f(standardization.) g(Finally) -7 b(,) 35 b(as) f(w) n(e) h(ha) n (v) n(e) 945 3504 y(sho) n(wn) d(here,) h(the) h(reduction) f(of) g (heterogeneous) f(data) h(t) n(yp) r(es) g(to) g(the) h(common) f (format) g(of) 945 3612 y(k) n(ernel) 40 b(matrices) g(allo) n(ws) g (the) h(dev) n(elopmen) n(t) f(of) h(general) f(to) r(ols) g(for) h (com) n(bining) f(m) n(ultiple) 945 3720 y(data) 28 b(t) n(yp) r(es.) h (Kernel) f(matrices) h(are) f(required) g(only) g(to) h(resp) r(ect) g (the) g(constrain) n(t) f(of) h(p) r(ositiv) n(e) 945 3827 y(semide\014niteness,) c(and) f(th) n(us) h(the) g(p) r(o) n(w) n (erful) f(tec) n(hnique) h(of) f(semide\014nite) h(programming) e(can) 945 3935 y(b) r(e) 43 b(exploited) f(to) h(deriv) n(e) e(general) h (pro) r(cedures) f(for) h(com) n(bining) g(data) g(of) h(heterogeneous) 945 4043 y(format) 27 b(and) g(origin.) 1028 4151 y(W) -7 b(e) 43 b(th) n(us) f(en) n(vision) g(the) h(dev) n(elopmen) n(t) f(of) h(general) e(libraries) g(of) i(k) n(ernel) e(matrices) h(for) 945 4259 y(biological) 35 b(data,) i(suc) n(h) f(as) g(those) h(that) g(w) n (e) g(ha) n(v) n(e) e(pro) n(vided) h(at) p Fa 37 w(noble.gs.washing) o (to) o(n.) 945 4367 y(edu/sdp-) t(svm) p Fx(,) 29 b(that) 34 b(summarize) f(the) h(statistically-relev) -5 b(an) n(t) 32 b(features) h(of) h(primary) e(data,) 945 4475 y(encapsulate) 18 b(biological) e(kno) n(wledge,) i(and) g(serv) n(e) f(as) h(inputs) h (to) f(a) g(wide) h(v) -5 b(ariet) n(y) 18 b(of) g(subsequen) n(t) 945 4583 y(data) 39 b(analyses.) f(Indeed,) i(giv) n(en) e(the) i (appropriate) e(k) n(ernel) h(matrices,) g(the) h(metho) r(ds) f(that) 945 4691 y(w) n(e) 48 b(ha) n(v) n(e) f(describ) r(ed) h(here) g(are) f (applicable) h(to) g(problems) f(suc) n(h) h(as) g(the) h(prediction) f (of) 945 4799 y(protein) 23 b(metab) r(olic,) g(regulatory) e(and) i (other) f(functional) i(classes,) e(the) h(prediction) g(of) g(protein) 945 4907 y(sub) r(cellular) k(lo) r(cations,) g(and) g(the) h (prediction) f(of) h(protein-protein) e(in) n(teractions.) p 90 rotate dyy eop %%Page: 30 30 30 29 bop 90 rotate dyy eop %%Page: 31 31 31 30 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p 198 407 3736 50 v FA 945 656 a(References) p Fx 945 1337 a(B.) g(Alb) r(erts,) h (D.) g(Bra) n(y) -7 b(,) 34 b(A.) i(Johnson,) e(J.) h(Lewis,) g(M.) h (Ra\013,) f(K.) g(Rob) r(erts,) g(and) h(P) -7 b(.) 35 b(W) -7 b(alter.) p Fy 1028 1445 a(Essential) 24 b(c) l(el) t(l) g (biolo) l(gy:) i(an) e(intr) l(o) l(duction) f(to) h(the) f(mole) l (cular) i(biolo) l(gy) g(of) f(the) g(c) l(el) t(l) p Fx(.) j(Garland) 1028 1553 y(Science) g(Publishing,) h(1998.) 945 1686 y(S.) k(F.) g(Altsc) n(h) n(ul,) g(W.) h(Gish,) f(W.) g(Miller,) g (E.) f(W.) i(My) n(ers,) e(and) g(D.) i(J.) f(Lipman.) 49 b(Basic) 31 b(lo) r(cal) 1028 1794 y(alignmen) n(t) c(searc) n(h) f(to) r(ol.) p Fy 37 w(Journal) j(of) i(Mole) l(cular) g(Biolo) l(gy) p Fx(,) f(215:403{410,) 23 b(1990.) 945 1927 y(C.) 40 b(Berg,) e(C.) i (J.) g(Christensen,) f(and) h(P) -7 b(.) 39 b(Ressel.) p Fy 72 w(Harmonic) j(A) n(nalysis) g(on) f(Semigr) l(oups:) 1028 2035 y(The) l(ory) f(of) g(Positive) g(De\014nite) e(and) h(R) l(elate) l(d) g(F) -6 b(unctions) p Fx(.) 65 b(Springer,) 37 b(New) g(Y) -7 b(ork,) 37 b(NY,) 1028 2143 y(1984.) 945 2275 y(S.) 20 b(D.) g(Blac) n(k) f(and) g(D.) i(R.) f(Mould.) k(Dev) n(elopmen) n(t) 19 b(of) h(h) n(ydrophobicit) n(y) e(parameters) g(to) i(analyze) 1028 2383 y(proteins) 25 b(whic) n(h) i(b) r(ear) e(p) r(ost-) h(or) f (cotranslational) f(mo) r(di\014cations.) p Fy 35 w(A) n(nal.) k(Bio) l (chem.) p Fx(,) h(193:) 1028 2491 y(72{82,) c(1991.) 945 2624 y(B.) 33 b(E.) g(Boser,) g(I.) g(Guy) n(on,) g(and) h(V.) g(V) -7 b(apnik.) 54 b(A) 34 b(training) f(algorithm) f(for) h(optimal) h (margin) 1028 2732 y(classi\014ers.) h(In) p Fy 28 w(Computational) c (L) l(e) l(aring) f(The) l(ory) p Fx(,) g(pages) c(144{152,) e(1992.) 945 2865 y(S.) d(Bo) n(yd,) e(L.) i(El) f(Ghaoui,) g(E.) g(F) -7 b(eron,) 20 b(and) h(V.) g(Balakrishnan.) p Fy 23 w(Line) l(ar) j (Matrix) g(Ine) l(qualities) f(in) 1028 2973 y(System) f(and) h(Contr) l (ol) g(The) l(ory) p Fx(.) j(SIAM,) 20 b(Philadelphia,) f(P) -7 b(A,) 20 b(1994.) j(ISBN) d(0-89871-334-X.) 945 3106 y(S.) h(Bo) n(yd) g(and) g(L.) h(V) -7 b(anden) n(b) r(erghe.) 26 b(Con) n(v) n(ex) 20 b(optimization.) 26 b(Course) 21 b(notes) g(for) g(EE364,) e(Stan-) 1028 3214 y(ford) 30 b(Univ) n(ersit) n(y) -7 b(.) 29 b(Av) -5 b(ailable) 30 b(at) p Fa 31 w(http://www.stan) o(fo) o(rd.) o(ed) o(u/c) o(la) o(ss/) o(ee) o(36) o(4) p Fx(,) 25 b(August) 1028 3321 y(2001.) 945 3454 y(M.) 36 b(P) -7 b(.) 36 b(S.) h(Bro) n(wn,) e(W.) i(N.) f(Grundy) -7 b(,) 37 b(D.) g(Lin,) f(N.) h(Cristianini,) f(C.) g(Sugnet,) h(T.) f (S.) h(F) -7 b(urey) g(,) 1028 3562 y(Jr.) 39 b(M.) i(Ares,) f(and) g (D.) h(Haussler.) 74 b(Kno) n(wledge-based) 38 b(analysis) h(of) h (microarra) n(y) d(gene) 1028 3670 y(expression) 26 b(data) h(using) g (supp) r(ort) h(v) n(ector) e(mac) n(hines.) p Fy 36 w(PNAS) p Fx(,) h(97\(1\):262{267,) d(2000.) 945 3803 y(C.) j(P) -7 b(.) 27 b(Chen) g(and) g(B.) g(Rost.) 35 b(State-of-the-art) 26 b(in) h(mem) n(brane) g(protein) f(prediction.) p Fy 36 w(Applie) l(d) 1028 3911 y(Bioinformatics) p Fx(,) k (1\(1\):21{35,) 25 b(2002.) 945 4044 y(M.) 32 b(Deng,) h(T.) f(Chen,) h (and) f(F.) g(Sun.) 52 b(An) 32 b(in) n(tegrated) g(probabilistic) f (mo) r(del) i(for) e(functional) 1028 4152 y(prediction) c(of) h (proteins.) 36 b(In) p Fy 28 w(RECOMB) p Fx(,) 28 b(pages) f(95{103,) d (2003a.) 945 4285 y(M.) 36 b(Deng,) g(F.) g(Sun,) h(and) f(T.) g(Chen.) 62 b(Assessmen) n(t) 35 b(of) h(the) h(reliabilit) n(y) e(of) h (protein-protein) 1028 4392 y(in) n(teractions) 26 b(and) i(protein) f (function) h(prediction.) 36 b(In) p Fy 28 w(PSB) p Fx(,) 28 b(pages) f(140{151,) d(2003b.) 945 4525 y(A.) 31 b(Dra) n(wid) g(and) f (M.) i(Gerstein.) 46 b(A) 32 b(Ba) n(y) n(esian) d(system) h(in) n (tegrating) g(expression) g(data) g(with) 1028 4633 y(sequence) 25 b(patterns) h(for) f(lo) r(calizing) g(proteins:) h(comprehensiv) n(e) e (application) h(to) h(the) h(y) n(east) 1028 4741 y(genome.) p Fy 36 w(J.) j(Mol.) h(Biol.) p Fx(,) f(301:1059{1075,) 21 b(2000.) 945 4874 y(D.) 27 b(M.) g(Engleman,) f(T.) h(A.) g(Steitz,) h (and) f(A.) g(Goldman.) 35 b(Iden) n(tifying) 27 b(nonp) r(olar) f (transbila) n(y) n(er) 1028 4982 y(helices) 43 b(in) h(amino) f(acid) g (sequences) g(of) g(mem) n(brane) g(proteins.) p Fy 83 w(A) n(nn.) h(R) l(ev.) h(Biophys.) p 90 rotate dyy eop %%Page: 32 32 32 31 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(32) 670 b(R) l(efer) l(enc) l(es) p Fy 797 349 a(Biophys.) 32 b(Chem.) p Fx(,) d(15:321{353,) 24 b(1986.) 714 482 y(T.) e(S.) g(F) -7 b(urey) g(,) 22 b(N.) g(Cristianini,) f(N.) i (Du\013y) -7 b(,) 23 b(D.) f(W.) h(Bednarski,) d(M.) i(Sc) n(h) n (ummer,) g(and) g(D.) g(Haus-) 797 590 y(sler.) j(Supp) r(ort) 20 b(v) n(ector) g(mac) n(hine) g(classi\014cation) f(and) i(v) -5 b(alidation) 20 b(of) h(cancer) f(tissue) g(samples) 797 698 y(using) 27 b(microarra) n(y) d(expression) i(data.) p Fy 37 w(Bioinformatics) p Fx(,) k(16\(10\):906{914,) 23 b(2000.) 714 831 y(H.) 31 b(Ge,) h(Z.) f(Liu,) g(G.) h(Ch) n(urc) n(h,) f(and) g(M.) g(Vidal.) 48 b(Correlation) 29 b(b) r(et) n(w) n(een) j (transcriptome) e(and) 797 939 y(in) n(teractome) d(mapping) i(data) f (from) p Fy 29 w(Sac) l(char) l(omyc) l(es) k(c) l(er) l(evisiae) p Fx(.) p Fy 43 w(Natur) l(e) e(Genetics) p Fx(,) f(29:) 797 1047 y(482{486,) 24 b(2001.) 714 1179 y(M.) 31 b(Gribsk) n(o) n(v) e (and) i(N.) h(L.) f(Robinson.) 47 b(Use) 31 b(of) g(receiv) n(er) e(op) r(erating) h(c) n(haracteristic) g(\(R) n(OC\)) 797 1287 y(analysis) 24 b(to) h(ev) -5 b(aluate) 25 b(sequence) g(matc) n(hing.) p Fy 33 w(Computers) j(and) h(Chemistry) p Fx(,) e(20\(1\):25{33,) 797 1395 y(1996.) 714 1528 y(A.) k(Grigoriev.) 44 b(A) 31 b(relationship) f(b) r(et) n(w) n(een) h(gene) f(expression) f(and) i (protein) f(in) n(teractions) g(on) 797 1636 y(the) 20 b(proteome) e(scale:) h(analysis) f(of) i(the) f(bacteriophage) f(T7) h (and) g(the) h(y) n(east) p Fy 19 w(S) p Fx(acc) n(harom) n(yces) 797 1744 y(cerevisiae.) p Fy 35 w(Nucleic) 30 b(A) l(cids) g(R) l(es.) p Fx(,) e(29:3513{3519,) 22 b(2001.) 714 1877 y(J.) 28 b(A.) g(Hanley) g(and) g(B.) h(J.) f(McNeil.) 39 b(The) 28 b(meaning) g(and) g(use) g(of) g(the) h(area) d(under) i(a) g(receiv) n (er) 797 1985 y(op) r(erating) e(c) n(haracteristic) g(\(R) n(OC\)) h (curv) n(e.) p Fy 36 w(R) l(adiolo) l(gy) p Fx(,) j(143:29{36,) 24 b(1982.) 714 2118 y(I.) h(Holmes) g(and) g(W.) h(J.) f(Bruno.) 32 b(Finding) 26 b(regulatory) d(elemen) n(ts) i(using) g(join) n(t) g (lik) n(eliho) r(o) r(ds) g(for) 797 2226 y(sequence) i(and) g (expression) f(pro\014le) h(data.) 37 b(In) p Fy 27 w(ISMB) p Fx(,) 29 b(pages) d(202{210,) e(2000.) 714 2358 y(T.) 31 b(P) -7 b(.) 31 b(Hopp) h(and) f(K.) h(R.) f(W) -7 b(o) r(o) r(ds.) 49 b(Prediction) 30 b(of) i(protein) f(an) n(tigenic) g(determinan) n(ts) g (from) 797 2466 y(amino) c(acid) g(sequences.) p Fy 36 w(Pr) l(o) l(c.) k(Natl.) f(A) l(c) l(ad.) g(Sci.) h(USA) p Fx(,) c(78:3824{3828,) 22 b(1981.) 714 2599 y(T.) 35 b(Jaakk) n(ola,) d(M.) k(Diekhans,) f(and) g(D.) g(Haussler.) 59 b(Using) 34 b(the) i(Fisher) f(k) n(ernel) f(metho) r(d) i(to) 797 2707 y(detect) f(remote) f(protein) g(homologies.) 55 b(In) p Fy 35 w(ISMB) p Fx(,) 35 b(pages) f(149{158,) d(Menlo) j(P) n (ark,) f(CA,) 797 2815 y(1999.) 25 b(AAAI) k(Press.) 714 2948 y(R.) 39 b(Jansen,) f(N.) i(Lan,) f(J.) g(Qian,) f(and) h(M.) g (Gerstein.) 71 b(In) n(tegration) 38 b(of) h(genomic) f(datasets) 797 3056 y(to) j(predict) h(protein) f(complexes) g(in) h(y) n(east.) p Fy 78 w(Journal) g(of) i(Structur) l(al) d(and) i(F) -6 b(unctional) 797 3164 y(Genomics) p Fx(,) 28 b(2:71{81,) d(2002.) 714 3297 y(R.) j(I.) h(Kondor) e(and) h(J.) g(La\013ert) n(y) -7 b(.) 39 b(Di\013usion) 29 b(k) n(ernels) e(on) h(graphs) g(and) g (other) g(discrete) g(input) 797 3404 y(spaces.) g(In) 23 b(C.) f(Samm) n(ut) h(and) g(A.) g(Ho\013mann,) g(editors,) p Fy 22 w(Pr) l(o) l(c) l(e) l(e) l(dings) j(of) h(the) e(International) 797 3512 y(Confer) l(enc) l(e) 30 b(on) g(Machine) i(L) l(e) l(arning) p Fx(.) c(Morgan) e(Kaufmann,) h(2002.) 714 3645 y(A.) 43 b(Krogh,) f(B.) i(Larsson,) d(G.) j(v) n(on) f(Heijne,) h(and) f(E.) g (L.) h(L.) f(Sonnhammer.) 83 b(Predicting) 797 3753 y(transmem) n (brane) 33 b(protein) i(top) r(ology) g(with) g(a) g(hidden) h(mark) n (o) n(v) e(mo) r(del:) h(Application) h(to) 797 3861 y(complete) 27 b(genomes.) p Fy 36 w(Journal) j(of) g(Mole) l(cular) h (Biolo) l(gy) p Fx(,) f(305\(3\):567{580,) 23 b(2001.) 714 3994 y(J.) 40 b(Kyte) h(and) g(R.) g(F.) g(Do) r(olittle.) 77 b(A) 41 b(simple) g(metho) r(d) g(for) g(displa) n(ying) e(the) j(h) n (ydropathic) 797 4102 y(c) n(haracter) 25 b(of) j(a) f(protein.) p Fy 36 w(Journal) j(of) h(Mole) l(cular) g(Biolo) l(gy) p Fx(,) f(157:105{132,) 23 b(1982.) 714 4235 y(G.) 38 b(R.) h(G.) f(Lanc) n(kriet,) f(T.) i(De) f(Bie,) h(N.) f(Cristianini,) g(M.) h(I.) f (Jordan,) f(and) h(W.) h(S.) g(Noble.) 797 4343 y(A) 33 b(framew) n(ork) d(for) i(genomic) g(data) h(fusion) f(and) h(its) g (application) f(to) g(mem) n(brane) g(protein) 797 4451 y(prediction.) 27 b(T) -7 b(ec) n(hnical) 21 b(Rep) r(ort) h(03-1273,) d (Univ) n(ersit) n(y) i(of) g(California,) g(Berk) n(eley) -7 b(,) 21 b(Division) 797 4558 y(of) 27 b(Computer) g(Science,) h(2003.) 714 4691 y(G.) 34 b(R.) h(G.) g(Lanc) n(kriet,) e(N.) i(Cristianini,) g (P) -7 b(.) 34 b(Bartlett,) g(L.) h(El) f(Ghaoui,) g(and) g(M.) h(I.) g (Jordan.) 797 4799 y(Learning) 41 b(the) i(k) n(ernel) f(matrix) g (with) h(semi-de\014nite) g(programming.) 80 b(In) 43 b(C.) f(Samm) n(ut) 797 4907 y(and) 32 b(A.) g(Ho\013mann,) h(editors,) p Fy 31 w(Pr) l(o) l(c) l(e) l(e) l(dings) i(of) f(the) h(19th) f (International) h(Confer) l(enc) l(e) f(on) 797 5015 y(Machine) d(L) l(e) l(arning) p Fx(.) d(Morgan) e(Kaufmann,) h(2002.) p 90 rotate dyy eop %%Page: 33 33 33 32 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw 198 100 a(R) l(efer) l(enc) l(es) 3308 b(33) p Fx 945 349 a(G.) 21 b(R.) h(G.) f(Lanc) n(kriet,) f(M.) h(Deng,) g(N.) h (Cristianini,) f(M.) g(I.) g(Jordan,) f(and) h(W.) g(S.) h(Noble.) k (Kernel-) 1028 457 y(based) i(data) g(fusion) h(and) f(its) h (application) f(to) g(protein) g(function) i(prediction) e(in) h(y) n (east.) 38 b(In) p Fy 1028 565 a(PSB) p Fx(,) 28 b(2004.) 945 698 y(L.) 38 b(Liao) f(and) h(W.) g(S.) h(Noble.) 68 b(Com) n(bining) 37 b(pairwise) g(sequence) h(similarit) n(y) f(and) h (supp) r(ort) 1028 806 y(v) n(ector) f(mac) n(hines) g(for) g(remote) h (protein) f(homology) g(detection.) 68 b(In) p Fy 38 w(RECOMB) p Fx(,) 39 b(pages) 1028 914 y(225{232,) 24 b(2002.) 945 1047 y(E.) 30 b(M.) h(Marcotte,) e(M.) i(P) n(ellegrini,) e (M.) i(J.) f(Thompson,) g(T.) h(O.) f(Y) -7 b(eates,) 31 b(and) f(D.) h(Eisen) n(b) r(erg.) 1028 1155 y(A) h(com) n(bined) g (algorithm) e(for) i(genome-wide) e(prediction) i(of) g(protein) f (function.) p Fy 50 w(Natur) l(e) p Fx(,) 1028 1263 y (402\(6757\):83{86,) 22 b(1999.) 945 1395 y(H.) 30 b(W.) g(Mew) n(es,) f (D.) h(F) -7 b(rishman,) 30 b(C.) g(Grub) r(er,) f(B.) h(Geier,) f(D.) h (Haase,) f(A.) h(Kaps,) f(K.) h(Lemc) n(k) n(e,) 1028 1503 y(G.) 22 b(Mannhaupt,) g(F.) g(Pfei\013er,) f(C) h(Sc) n(h) r (\177) -44 b(uller,) 21 b(S.) h(Sto) r(c) n(k) n(er,) f(and) h(B.) f(W) -7 b(eil.) 28 b(MIPS:) 22 b(a) f(database) 1028 1611 y(for) 27 b(genomes) f(and) i(protein) f(sequences.) p Fy 36 w(Nucleic) j(A) l(cids) g(R) l(es.) p Fx(,) e(28\(1\):37{40,) d (2000.) 945 1744 y(R.) 40 b(Mro) n(wk) -5 b(a,) 38 b(W.) i(Lieb) r (erneister,) f(and) g(D.) h(Holste.) 72 b(Do) r(es) 39 b(mapping) h(rev) n(eal) e(correlation) 1028 1852 y(b) r(et) n(w) n (een) 33 b(gene) f(expression) f(and) i(protein-protein) f(in) n (teraction?) p Fy 51 w(Natur) l(e) i(Genetics) p Fx(,) f(33:) 1028 1960 y(15{16,) 25 b(2003.) 945 2093 y(Akihiro) 20 b(Nak) -5 b(a) n(y) n(a,) 19 b(Susum) n(u) h(Goto,) g(and) h(Minoru) f(Kanehisa.) k(Extraction) 19 b(of) h(correlated) f(gene) 1028 2201 y(clusters) i(b) n(y) g(m) n(ultiple) h(graph) e(comparison.) 25 b(In) c(H.) h(Matsuda,) f(S.) h(Miy) n(ano,) e(T.) h(T) -7 b(ak) i(agi,) 21 b(and) 1028 2309 y(L.) 35 b(W) -7 b(ong,) 35 b(editors,) p Fy 35 w(Genome) i(Informatics) h(2001) p Fx(,) f(pages) d(44{53.) f(Univ) n(ersal) h(Academ) n(y) 1028 2417 y(Press,) 26 b(2001.) 945 2549 y(Y.) 43 b(Nestero) n(v) f(and) g (A.) h(Nemiro) n(vsky) -7 b(.) p Fy 81 w(Interior) 44 b(p) l(oint) g(p) l(olynomial) i(metho) l(ds) f(in) f(c) l(onvex) 1028 2657 y(pr) l(o) l(gr) l(amming:) 31 b(The) l(ory) g(and) f(applic) l (ations) p Fx(.) 40 b(SIAM,) 28 b(Philadelphia,) f(P) -7 b(A,) 27 b(1994.) 945 2790 y(P) -7 b(.) 32 b(P) n(a) n(vlidis,) f(J.) h (W) -7 b(eston,) 33 b(J.) f(Cai,) g(and) h(W.) f(N.) h(Grundy) -7 b(.) 52 b(Gene) 32 b(functional) h(classi\014cation) 1028 2898 y(from) 27 b(heterogeneous) f(data.) 36 b(In) p Fy 28 w(RECOMB) p Fx(,) 28 b(pages) f(242{248,) d(2001.) 945 3031 y(B.) 30 b(Sc) n(h\177) -42 b(olk) n(opf) 30 b(and) g(A.) h (Smola.) p Fy 45 w(L) l(e) l(arning) i(with) g(Kernels) p Fx(.) 46 b(MIT) 31 b(Press,) e(Cam) n(bridge,) g(MA,) 1028 3139 y(2002.) 945 3272 y(T.) 18 b(F.) h(Smith) h(and) e(M.) h(S.) g(W) -7 b(aterman.) 22 b(Iden) n(ti\014cation) c(of) h(common) f(molecular) f (subsequences.) p Fy 1028 3380 a(Journal) 29 b(of) i(Mole) l(cular) g (Biolo) l(gy) p Fx(,) f(147\(1\):195{197,) 23 b(1981.) 945 3512 y(E.) 39 b(Sonnhammer,) f(S.) i(Eddy) -7 b(,) 39 b(and) g(R.) h(Durbin.) 72 b(Pfam:) 39 b(a) g(comprehensiv) n(e) f (database) g(of) 1028 3620 y(protein) 22 b(domain) g(families) g(based) g(on) g(seed) h(alignmen) n(ts.) p Fy 27 w(Pr) l(oteins) p Fx(,) g(28\(3\):405{420,) 18 b(1997.) 945 3753 y(P) -7 b(.) 34 b(T.) h(Sp) r(ellman,) g(G.) g(Sherlo) r(c) n(k,) f(M.) h(Q.) f (Zhang,) g(V.) h(R.) g(Iy) n(er,) f(K.) g(Anders,) h(M.) g(B.) f (Eisen,) 1028 3861 y(P) -7 b(.) 42 b(O.) h(Bro) n(wn,) f(D.) h (Botstein,) g(and) f(B.) h(F) -7 b(utc) n(her.) 82 b(Comprehensiv) n(e) 42 b(iden) n(ti\014cation) h(of) 1028 3969 y(cell) 36 b(cycle-regulated) d(genes) i(of) h(the) g(y) n(east) p Fy 35 w(Sac) l(char) l(omyc) l(es) j(c) l(er) l(evisiae) p Fx 37 w(b) n(y) d(microarra) n(y) 1028 4077 y(h) n(ybridization.) p Fy 36 w(Mol) 30 b(Biol) i(Cel) t(l) p Fx(,) d(9:3273{3297,) 23 b(1998.) 945 4210 y(J.) 45 b(F.) h(Sturm.) 90 b(Using) 45 b(SeDuMi) h(1.02,) e(a) h(MA) -7 b(TLAB) 46 b(to) r(olb) r(o) n(x) f (for) f(optimization) h(o) n(v) n(er) 1028 4318 y(symmetric) e(cones.) p Fy 83 w(Optimization) i(Metho) l(ds) h(and) e(Softwar) l(e) p Fx(,) h(11{12:625{653,) 37 b(1999.) 1028 4426 y(Sp) r(ecial) 27 b(issue) h(on) f(In) n(terior) f(P) n(oin) n(t) h(Metho) r(ds) h(\(CD) g (supplemen) n(t) g(with) g(soft) n(w) n(are\).) 945 4558 y(Amos) g(T) -7 b(ana) n(y) g(,) 27 b(Ro) r(ded) h(Sharan,) g(and) g (Ron) g(Shamir.) 38 b(Disco) n(v) n(ering) 27 b(statistically) g (signi\014can) n(t) 1028 4666 y(biclusters) g(in) h(gene) f(expression) f(data.) p Fy 36 w(Bioinformatics) p Fx(,) 31 b(18:S136{S144,) 24 b(2002.) 945 4799 y(K.) 44 b(Tsuda.) 87 b(Supp) r(ort) 44 b(v) n(ector) f(classi\014cation) h(with) h(asymmetric) e(k) n(ernel) h (function.) 87 b(In) 1028 4907 y(M.) 28 b(V) -7 b(erleysen,) 27 b(editor,) p Fy 27 w(Pr) l(o) l(c) l(e) l(e) l(dings) j(ESANN) p Fx(,) d(pages) g(183{188,) d(1999.) p 90 rotate dyy eop %%Page: 34 34 34 33 bop -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V FB -389 5700 a(2003/09/30) 35 b(13:09) p Fw -34 100 a(34) 670 b(R) l(efer) l(enc) l(es) p Fx 714 349 a(L.) 32 b(V) -7 b(anden) n(b) r(erghe) 32 b(and) h(S.) g(Bo) n(yd.) 51 b(Semide\014nite) 34 b(programming.) p Fy 50 w(SIAM) g(R) l(eview) p Fx(,) f(38\(1\):) 797 457 y(49{95,) 25 b(1996.) 714 590 y(C.) d(v) n(on) f(Mering,) g(R.) h(Krause,) e(B.) i(Snel,) h(M.) f (Cornell,) f(S.) h(G.) g(Olivier,) f(S.) h(Fields,) g(and) g(P) -7 b(.) 22 b(Bork.) 797 698 y(Comparativ) n(e) 29 b(assessmen) n(t) i(of) g (large-scale) e(data) i(sets) h(of) f(protein-protein) g(in) n (teractions.) p Fy 797 806 a(Natur) l(e) p Fx(,) c(417:399{403,) c (2002.) 714 939 y(A.) 33 b(Zien,) f(G.) h(R\177) -42 b(atc) n(h,) 33 b(S.) g(Mik) -5 b(a,) 32 b(B.) h(Sc) n(h\177) -42 b(olk) n(opf,) 32 b(T.) h(Lengauer,) e(and) h(K.-R.) h(M) r(\177) -44 b(uller.) 52 b(Engi-) 797 1047 y(neering) 21 b(supp) r(ort) i(v) n (ector) e(mac) n(hine) h(k) n(ernels) g(that) h(recognize) d (translation) i(initiation) h(sites.) p Fy 797 1155 a(Bioinformatics) p Fx(,) 30 b(16\(9\):799{807,) 24 b(2000.) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF