Luke Oeding | Auburn University (original) (raw)

Papers by Luke Oeding

Collectanea Mathematica, Mar 10, 2023

Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to... more Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to k-planes with the restriction that their intersection has a prescribed dimension. We study dimensions of restricted secant of Grassmannians and relate them to the analogous question for secants of Grassmannians via an incidence variety construction. We define a notion of expected dimension and give a formula for the dimension of all restricted secant varieties of Grassmannians that holds if the BDdG conjecture [4, Conjecture 4.1] on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay2, and point out ways to make these calculations more efficient. We also show a potential application to coding theory.

arXiv (Cornell University), May 19, 2022

arXiv (Cornell University), Jun 27, 2022

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra which acts on ... more We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra which acts on it and embed them into an auxiliary algebra. Viewed as endomorphisms of this algebra we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which endows the tensor with a Jordan decomposition. We utilize aspects of the Jordan decomposition to study orbit separation and classification in examples that are relevant for quantum information.

The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irr... more The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their multiplicities. The analogous decomposition for the \mu-discriminant of polynomial is found.

By using a result from the numerical algebraic geometry package Bertini we show that (up to high ... more By using a result from the numerical algebraic geometry package Bertini we show that (up to high numerical accuracy) a specific set of degree 6 and degree 9 polynomials cut out the secant variety σ 4 (P 2 × P 2 × P 3). This, combined with an argument provided by Landsberg and Manivel (whose proof was corrected by Friedland), implies set-theoretic defining equations in degrees 5, 6 and 9 for a much larger set of secant varieties, including σ 4 (P 3 × P 3 × P 3) which is of particular interest in light of the salmon prize offered by E. Allman for the ideal-theoretic defining equations.

We describe the defining ideal of the rth secant variety of P 2 × P n embedded by O(1, 2), for ar... more We describe the defining ideal of the rth secant variety of P 2 × P n embedded by O(1, 2), for arbitrary n and r ≤ 5. We also present the Schur module decomposition of the space of generators of each such ideal. Our main results are based on a more general construction for producing explicit matrix equations that vanish on secant varieties of products of projective spaces. This extends previous work of Strassen and Ottaviani. 2. The κ-invariant of a 3-tensor From a tensor in U * ⊗ V * ⊗ W * , we will construct a series of linear maps, whose ranks we define to be the κ-invariants of the tensor. The κ-invariants give inequalities on the rank of the tensor, and thus, determinantal equations which vanish on the secant variety. There is a natural map U * ⊗ j U * → j+1 U * defined by sending u ⊗ u ′ → u ∧ u ′ for any 0 ≤ j ≤ m − 1. This induces an inclusion U * ⊆ j U ⊗ j+1 U *. By tensoring on both sides by V * ⊗ W * , we get an inclusion U * ⊗ V * ⊗ W * ⊆ (V ⊗ j U *) * ⊗ (W * ⊗ j+1 U *). An element of the tensor product on the right-hand side may be interpreted as a linear homomorphism, meaning that for any x ∈ U * ⊗ V * ⊗ W * we have a homomorphism ψ j,x : V ⊗ j U * → W * ⊗ j+1 U * , and ψ j,x depends linearly on x. We call ψ j,x an exterior flattening of x, as it generalizes the flattening of a tensor, as discussed below. Definition 2.1. Following [7, Defn. 1.1], we define κ j (x) to be the rank of ψ j,x , and we let κ(x) denote the vector of κ-invariants (κ 0 (x),. .. , κ m−1 (x)). More concretely, by choosing bases for the vector spaces, we can represent ψ j,x as a matrix. If e 1 ,. .. , e m is a basis for U * , then a basis for j U * is given by the set of all e i 1 ∧. .. ∧ e i j for 1 ≤ i 1 < • • • < i j ≤ m, and analogously for j+1 U *. For a fixed u = m i=1 u i e i in U * , the map j U * → j+1 U * defined by u ′ → u ∧ u ′ will send e i 1 ∧. .. ∧ e i j to i u i e i ∧ e i 1 ∧. .. ∧ e i j. Thus, this map will be represented in the above bases by a matrix whose entries are either

The variety of principal minors of n × n symmetric matrices, denoted Z n , is invariant under the... more The variety of principal minors of n × n symmetric matrices, denoted Z n , is invariant under the action of a group G ⊂ GL(2 n) isomorphic to (SL(2) ×n) ⋉ S n. We describe an irreducible G-module of degree 4 polynomials constructed from Cayley's 2×2×2 hyperdeterminant and show that it cuts out Z n set-theoretically. This solves the set-theoretic version of a conjecture of Holtz and Sturmfels. Standard techniques from representation theory and geometry are explored and developed for the proof of the conjecture and may be of use for studying similar G-varieties.

We determine the minimal generators of the ideal of the tangential variety of a Segre-Veronese va... more We determine the minimal generators of the ideal of the tangential variety of a Segre-Veronese variety, as well as the decomposition into irreducible GL-representations of its homogeneous coordinate ring. In the special case of a Segre variety, our results confirm a conjecture of Landsberg and Weyman.

ArXiv, 2020

Stochastic Alternating Least Squares (SALS) is a method that approximates the canonical decomposi... more Stochastic Alternating Least Squares (SALS) is a method that approximates the canonical decomposition of averages of sampled random tensors. Its simplicity and efficient memory usage make SALS an ideal tool for decomposing tensors in an online setting. We show, under mild regularization and readily verifiable assumptions on the boundedness of the data, that the SALS algorithm is globally convergent. Numerical experiments validate our theoretical findings and demonstrate the algorithm's performance and complexity.

Algebraic Geometry, 2019

We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddi... more We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P 1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening laws.

This is a technical report on the proceedings of the workshop held July 21 to July 25, 2008 at th... more This is a technical report on the proceedings of the workshop held July 21 to July 25, 2008 at the American Institute of Mathematics, Palo Alto, California, organized by Joseph Landsberg, Lek-Heng Lim, Jason Morton, and Jerzy Weyman. We include a list of open problems coming from applications in 4 different areas: signal processing, the Mulmuley-Sohoni approach to P vs. NP, matchgates and holographic algorithms, and entanglement and quantum information theory. We emphasize the interactions between geometry and representation theory and these applied areas.

arXiv (Cornell University), Jan 25, 2023

arXiv: Algebraic Geometry, 2016

When present, the Cohen-Macaulay property can be useful for finding the minimal defining equation... more When present, the Cohen-Macaulay property can be useful for finding the minimal defining equations of an algebraic variety. It is conjectured that all secant varieties of Segre products of projective spaces are arithmetically Cohen-Macaulay. A summary of the known cases where the conjecture is true is given. An inductive procedure based on the work of Landsberg and Weyman (LW-lifting) is described and used to obtain resolutions of orbits of secant varieties from those of smaller secant varieties. A new computation of the minimal free resolution of the variety of border rank 4 tensors of format 3times3times43 \times 3 \times 43times3times4 is given together with its equivariant presentation. LW-lifting is used to prove several cases where secant varieties are arithmetically Cohen-Macaulay and arithmetically Gorenstein.

G-Varieties and the Principal Minors of Symmetric Matrices. (May 2009) Luke Aaron Oeding, B.A., F... more G-Varieties and the Principal Minors of Symmetric Matrices. (May 2009) Luke Aaron Oeding, B.A., Franklin & Marshall College Chair of Advisory Committee: Dr. J.M. Landsberg The variety of principal minors of n×n symmetric matrices, denoted Zn, can be described naturally as a projection from the Lagrangian Grassmannian. Moreover, Zn is invariant under the action of a group G ⊂ GL(2) isomorphic to (SL(2)) ⋉ Sn. One may use this symmetry to study the defining ideal of Zn as a G-module via a coupling of classical representation theory and geometry. The need for the equations in the defining ideal comes from applications in matrix theory, probability theory, spectral graph theory and statistical physics. I describe an irreducible G-module of degree 4 polynomials called the hyperdeterminantal module (which is constructed as the span of the G-orbit of Cayley’s hyperdeterminant of format 2 × 2 × 2) and show that it that cuts out Zn set theoretically. This result solves the set-theoretic vers...

Frontiers in Neuroscience

We explore the use of superconducting quantum phase slip junctions (QPSJs), an electromagnetic du... more We explore the use of superconducting quantum phase slip junctions (QPSJs), an electromagnetic dual to Josephson Junctions (JJs), in neuromorphic circuits. These small circuits could serve as the building blocks of neuromorphic circuits for machine learning applications because they exhibit desirable properties such as inherent ultra-low energy per operation, high speed, dense integration, negligible loss, and natural spiking responses. In addition, they have a relatively straight-forward micro/nano fabrication, which shows promise for implementation of an enormous number of lossless interconnections that are required to realize complex neuromorphic systems. We simulate QPSJ-only, as well as hybrid QPSJ + JJ circuits for application in neuromorphic circuits including artificial synapses and neurons, as well as fan-in and fan-out circuits. We also design and simulate learning circuits, where a simplified spike timing dependent plasticity rule is realized to provide potential learning...

Quantum Information Processing

Linear Algebra and its Applications

Journal für die reine und angewandte Mathematik (Crelles Journal)

Let T be a general complex tensor of format (n 1 , ..., n d). When the fraction i n i /[1+ i (n i... more Let T be a general complex tensor of format (n 1 , ..., n d). When the fraction i n i /[1+ i (n i −1)] is an integer, and a natural inequality (called balancedness) is satisfied, it is expected that T has finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions of T , starting from a given one. In particular this gives a guess, true with high probability, about the total number of these decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats (3, 4, 5) and (2, 2, 2, 3) which have a unique decomposition as the sum of 6 and 4 decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the only identifiable cases. Building on the computational experiments, we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable.