Fixed Points in Spectral Complexity (original) (raw)
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Calcolo, 1998
Any attempt to find connections between mathematical properties of functions and their computational complexity has strong relevance to theory of computation. Indeed, there is the hope that developing new mathematical techniques could lead to discovering properties that might be responsible for lower bounds. The current situation is that none of the known techniques has yet led to lower bounds in general models of computation.
The multiplicative complexity of discrete cosine transforms
Advances in Applied Mathematics, 1992
We obtain the multiplicative complexity of discrete cosine transforms in all cases. It is given as a function of the multiplicative complexity of discrete Fourier transforms. The latter have all been determined previously. o 19% Academic press, Inc.
Complexity of the Fourier transform on the Johnson graph
arXiv: Combinatorics, 2017
The set XXX of kkk-subsets of an nnn-set has a natural graph structure where two kkk-subsets are connected if and only if the size of their intersection is k−1k-1k−1. This is known as the Johnson graph. The symmetric group SnS_nSn acts on the space of complex functions on XXX and this space has a multiplicity-free decomposition as sum of irreducible representations of SnS_nSn, so it has a well-defined Gelfand-Tsetlin basis up to scalars. The Fourier transform on the Johnson graph is defined as the change of basis matrix from the delta function basis to the Gelfand-Tsetlin basis. The direct application of this matrix to a generic vector requires binomnk2\binom{n}{k}^2binomnk2 arithmetic operations. We show that --in analogy with the standard Fast Fourier Transform on the discrete circle-- this matrix can be factorized as a product of n−1n-1n−1 orthogonal matrices, each one with at most two nonzero elements in each column. This factorization shows that the number of arithmetic operations required to apply this...
On the representation of functions as Fourier transforms
Canadian Journal of Mathematics, 1959
If f ∈ Lp (— ∞, ∞), 1 < p ≤ 2, then f has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞) where p-1 + q-1 = 1. Also if ∣x∣1-2/qf(x) ∈ Lq (— ∞, ∞), q ≥ 2, then / has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞). These results can be found in (2, Theorems 74 and 79). In neither case, however, does the collection of transforms cover Lq , except when p = q = 2, and in neither case, with the same exception, has the collection of transforms been characterized. Further, if f ∈ Lp, (— ∞, ∞), 1 < p ≤ 2, then its transform F has the property |x|1-2/p F(x) ∈ Lp (— ∞, ∞) (see 2, Theorem 80) but, except when p = 2, the collection of transforms does not cover the set of functions with this property, and again, except when p = 2, the collection of transforms has not been characterized.
Complexity of symmetric functions in perceptron-like models
1992
We examine the size complexity of the symmetric boolean functions in two circuit models containing threshold gates: the d-perceptron model BRS, ABFR] (a single threshold function of constant-depth AND/OR circuits) and the parity-threshold model studied by Bruck Br] (a single threshold function of exclusive-ORs). These models are intermediate between the well-understood model of constant-depth AND/OR circuits and the still mysterious model of general constantdepth threshold circuits. In the d-perceptron model, we give an if and only if condition for a symmetric boolean function to be computable by a quasi-polynomial size d-perceptron: we show that a symmetric boolean function can be computed by a quasi-polynomial size dperceptron i it has only poly-log many sign changes, i.e. the number of times the function changes output value as the number of inputs on varies from zero through n (we call this parameter the degree of the symmetric function) is bounded above by log c n for some c. This extends the work of Fagin et al. FKPS] which gave a very nice characterization of symmetric functions computable by AC 0 circuits. An interesting consequence of our result is that a recent construction of Beigel Be] is optimal. In the parity-threshold model, we nd a similar parameter as a measure of size complexity, the odd-even degree, or number of output value changes as the number of inputs on varies through the odd numbers from 0 through n and then through the even numbers. We observe that poly-log odd-even degree implies quasipolynomial size, conjecture the converse, and prove the converse in the presence of a certain technical condition on the function's Fourier coe cients. In particular, we prove that the modulo-q function for any constant q > 2 has more than quasi-polynomial size. i Acknowledement This is a joint work with my advisor David A. Mix Barrington, I am grateful to him for many helpful discussions and patient guidance. I thank Neil Immerman for being the Second Reader. The proof of Theorem 10 in Section 3 was also independently discovered by Jun Tarui. I also thank him for many insightful comments.
Recursive analysis is the most classical approach to model and discuss computations over the reals. It is usually presented using Type 2 or higher order Turing machines. Recently, it has been shown that computability classes of functions computable in recursive analysis can also be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub-and sup-classes) over the reals and algebraically defined (sub-and sup-) classes of R-recursive functionsà la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of functions over the reals. In particular we provide the first algebraic characterization of polynomial time computable functions over the reals. This framework opens the field of implicit complexity of functions over the reals, and also provide a new reading of some of the existing characterizations at the computability level.
On the Solvability Complexity Index Hierarchy and Towers of Algorithms
2015
This paper establishes some of the fundamental barriers in the theory of computations and finally settles the long-standing computational spectral problem. That is to determine the existence of algorithms that can compute spectra mathrmsp(A)\mathrm{sp}(A)mathrmsp(A) of classes of bounded operators A=aiji,jinmathbbNinmathcalB(l2(mathbbN))A = \{a_{ij}\}_{i,j \in \mathbb{N}} \in \mathcal{B}(l^2(\mathbb{N}))A=aiji,jinmathbbNinmathcalB(l2(mathbbN)), given the matrix elements aiji,jinmathbbN\{a_{ij}\}_{i,j \in \mathbb{N}}aiji,jinmathbbN, that are sharp in the sense that they achieve the boundary of what a digital computer can achieve. Similarly, for a Schr\"odinger operator H=−triangle+VH = -\triangle+VH=−triangle+V, determine the existence of algorithms that can compute the spectrum mathrmsp(H)\mathrm{sp}(H)mathrmsp(H) given point samples of the potential function VVV. In order to solve these problems, we establish the Solvability Complexity Index (SCI) hierarchy and provide a collection of new algorithms that allow for problems that were previously out of reach. The SCI is the smallest number of limits needed in the computation, yielding a classifi...
Complexity Theoretic Aspects of Some Cryptographic Functions
Lecture Notes in Computer Science, 2003
In this work, we are interested in non-trivial upper bounds on the spectral norm of binary matrices M from {−1, 1} N×N . It is known that the distributed Boolean function represented by M is hard to compute in various restricted models of computation if the spectral norm is bounded from above by N 1−ε , where ε > 0 denotes a fixed constant. For instance, the size of a two-layer threshold circuit (with polynomially bounded weights for the gates in the hidden layer, but unbounded weights for the output gate) grows exponentially fast with n := log N . We prove sufficient conditions on M that imply small spectral norms (and thus high computational complexity in restricted models). Our general results cover specific cases, where the matrix M represents a bit (the least significant bit or other fixed bits) of a cryptographic decoding function. For instance, the decoding functions of the Pointcheval [9], the El Gamal , and the RSA-Paillier [2] cryptosystems can be addressed by our technique. In order to obtain our results, we make a detour on exponential sums and on spectral norms of matrices with complex entries. This method might be considered interesting in its own right.
Spectral Norm of Symmetric Functions
Lecture Notes in Computer Science, 2012
The spectral norm of a Boolean function f : {0, 1} n → {−1, 1} is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as r(f) log(n/r(f)) where r(f) = max{r 0 , r 1 }, and r 0 and r 1 are the smallest integers less than n/2 such that f (x) or f (x) • PARITY(x) is constant for all x with x i ∈ [r 0 , n − r 1 ]. We mention some applications to the decision tree and communication complexity of symmetric functions.