arXiv subjects

James Saunderson

Publications and source records attributed to James Saunderson.

At least 37 records · Page 2Linked to original sources

arXiv2022

Spectral Polyhedra

A "spectral convex set" is a collection of symmetric matrices whose range of eigenvalues form a symmetric convex set. Spectral convex sets generalize the Schur-Horn orbitopes studied by Sanyal-Sottile-Sturmfels (2011). We study this class of convex bodies, which is closed under intersections, polarity, and Minkowski sums. We describe orbits of faces and give a formula for their Steiner polynomials. We then focus on spectral polyhedra. We prove that spectral polyhedra are spectrahedra and give small representations as spectrahedral shadows. We close with observations and questions regarding hyperbolicity cones, polar convex bodies, and spectral zonotopes.

math.MG↗

arXiv2022

Amenable cones are particularly nice

Amenability is a geometric property of convex cones that is stronger than facial exposedness and assists in the study of error bounds for conic feasibility problems. In this paper we establish numerous properties of amenable cones, and investigate the relationships between amenability and other properties of convex cones, such as niceness and projectional exposure. We show that the amenability of a compact slice of a closed convex cone is equivalent to the amenability of the cone, and prove several results on the preservation of amenability under intersections and other convex operations. It then follows that homogeneous, doubly nonnegative and other cones that can be represented as slices of the cone of positive semidefinite matrices, are amenable. It is known that projectionally exposed cones are amenable and that amenable cones are nice, however the converse statements have been open questions. We construct an example of a four-dimensional cone that is nice but not amenable. We also show that amenable cones are projectionally exposed in dimensions up to and including four. We conclude with a discussion on open problems related to facial structure of convex sets that we came across in the course of this work, but were not able to fully resolve.

math.OC↗

arXiv2022

Spinsim: a GPU optimized python package for simulating spin-half and spin-one quantum systems

The Spinsim python package simulates spin-half and spin-one quantum mechanical systems following a time dependent Shroedinger equation. It makes use of numba.cuda, which is an LLVM (Low Level Virtual Machine) compiler for Nvidia Cuda compatible systems using GPU parallelization. Along with other optimizations, this allows for speed improvements from 3 to 4 orders of magnitude while staying just as accurate, compared to industry standard packages. It is available for installation on PyPI, and the source code is available on github. The initial use-case for the Spinsim will be to simulate quantum sensing-based ultracold atom experiments for the Monash University School of Physics \& Astronomy spinor Bose-Einstein condensate (spinor BEC) lab, but we anticipate it will be useful in simulating any range of spin-half or spin-one quantum systems with time dependent Hamiltonians that cannot be solved analytically. These appear in the fields of nuclear magnetic resonance (NMR), nuclear quadrupole resonance (NQR) and magnetic resonance imaging (MRI) experiments and quantum sensing, and with the spin-one systems of nitrogen vacancy centres (NVCs), ultracold atoms, and BECs.

quant-ph↗

arXiv2022

Terracini Convexity

We present a generalization of the notion of neighborliness to non-polyhedral convex cones. Although a definition of neighborliness is available in the non-polyhedral case in the literature, it is fairly restrictive as it requires all the low-dimensional faces to be polyhedral. Our approach is more flexible and includes, for example, the cone of positive-semidefinite matrices as a special case (this cone is not neighborly in general). We term our generalization Terracini convexity due to its conceptual similarity with the conclusion of Terracini's lemma from algebraic geometry. Polyhedral cones are Terracini convex if and only if they are neighborly. More broadly, we derive many families of non-polyhedral Terracini convex cones based on neighborly cones, linear images of cones of positive semidefinite matrices, and derivative relaxations of Terracini convex hyperbolicity cones. As a demonstration of the utility of our framework in the non-polyhedral case, we give a characterization based on Terracini convexity of the tightness of semidefinite relaxations for certain inverse problems.

math.OC↗

arXiv2021

Lifting for Simplicity: Concise Descriptions of Convex Sets

This paper presents a selected tour through the theory and applications of lifts of convex sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original set. Many convex sets have lifts that are dramatically simpler to describe than the original set. Finding such simple lifts has significant algorithmic implications, particularly for optimization problems. We consider both the classical case of polyhedral lifts, described by linear inequalities, as well as spectrahedral lifts, defined by linear matrix inequalities, with a focus on recent developments related to spectrahedral lifts. Given a convex set, ideally we would either like to find a (low-complexity) polyhedral or spectrahedral lift, or find an obstruction proving that no such lift is possible. To this end, we explain the connection between the existence of lifts of a convex set and certain structured factorizations of its associated slack operator. Based on this characterization, we describe a uniform approach, via sums of squares, to the construction of spectrahedral lifts of convex sets and illustrate the method on several families of examples. Finally, we discuss two flavors of obstruction to the existence of lifts: one related to facial structure, and the other related to algebraic properties of the set in question. Rather than being exhaustive, our aim is to illustrate the richness of the area. We touch on a range of different topics related to the existence of lifts, and present many examples of lifts from different areas of mathematics and its applications.

math.OC↗

arXiv2021

Memory-Efficient Approximation Algorithms for Max-k-Cut and Correlation Clustering

Max-k-Cut and correlation clustering are fundamental graph partitioning problems. For a graph with G=(V,E) with n vertices, the methods with the best approximation guarantees for Max-k-Cut and the Max-Agree variant of correlation clustering involve solving SDPs with $O(n^2)$ variables and constraints. Large-scale instances of SDPs, thus, present a memory bottleneck. In this paper, we develop simple polynomial-time Gaussian sampling-based algorithms for these two problems that use $O(n+|E|)$ memory and nearly achieve the best existing approximation guarantees. For dense graphs arriving in a stream, we eliminate the dependence on $|E|$ in the storage complexity at the cost of a slightly worse approximation ratio by combining our approach with sparsification.

math.OC↗

arXiv2021

Memory-efficient structured convex optimization via extreme point sampling

Memory is a key computational bottleneck when solving large-scale convex optimization problems such as semidefinite programs (SDPs). In this paper, we focus on the regime in which storing an $n\times n$ matrix decision variable is prohibitive. To solve SDPs in this regime, we develop a randomized algorithm that returns a random vector whose covariance matrix is near-feasible and near-optimal for the SDP. We show how to develop such an algorithm by modifying the Frank-Wolfe algorithm to systematically replace the matrix iterates with random vectors. As an application of this approach, we show how to implement the Goemans-Williamson approximation algorithm for \textsc{MaxCut} using $\mathcal{O}(n)$ memory in addition to the memory required to store the problem instance. We then extend our approach to deal with a broader range of structured convex optimization problems, replacing decision variables with random extreme points of the feasible region.

math.OC↗

arXiv2021

A convex form that is not a sum of squares

Every convex homogeneous polynomial (or form) is nonnegative. Blekherman has shown that there exist convex forms that are not sums of squares via a nonconstructive argument. We provide an explicit example of a convex form of degree four in 272 variables that is not a sum of squares. The form is related to the Cauchy-Schwarz inequality over the octonions. The proof uses symmetry reduction together with the fact (due to Blekherman) that forms of even degree, that are near-constant on the unit sphere, are convex. Using this same connection, we obtain improved bounds on the approximation quality achieved by the basic sum-of-squares relaxation for optimizing quaternary quartic forms on the sphere.

math.OC↗

arXiv2020

Lieb's concavity theorem, matrix geometric means, and semidefinite optimization

A famous result of Lieb establishes that the map $(A,B) \mapsto \text{tr}\left[K^* A^{1-t} K B^t\right]$ is jointly concave in the pair $(A,B)$ of positive definite matrices, where $K$ is a fixed matrix and $t \in [0,1]$. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational $t \in [0,1]$. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.

math.OC↗

arXiv2019

Certifying polynomial nonnegativity via hyperbolic optimization

We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems---a class of convex optimization problems that generalize semidefinite programs. We show how to produce families of nonnegative polynomials (which we call hyperbolic certificates of nonnegativity) from any hyperbolic polynomial. We investigate the pairs $(n,d)$ for which there is a hyperbolic polynomial of degree $d$ in $n$ variables such that an associated hyperbolic certificate of nonnegativity is not a sum of squares. If $d\geq 4$ we show that this occurs whenever $n\geq 4$. In the degree three case, we find an explicit hyperbolic cubic in $43$ variables that gives hyperbolic certificates that are not sums of squares. As a corollary, we obtain the first known hyperbolic cubic no power of which has a definite determinantal representation. Our approach also allows us to show that, given a cubic $p$, and a direction $e$, the decision problem "Is $p$ hyperbolic with respect to $e$?" is co-NP hard.

math.OC↗

arXiv2019

Limitations on the expressive power of convex cones without long chains of faces

A convex optimization problem in conic form involves minimizing a linear functional over the intersection of a convex cone and an affine subspace. In some cases, it is possible to replace a conic formulation using a certain cone, with a 'lifted' conic formulation using another cone that is higher-dimensional, but simpler, in some sense. One situation in which this can be computationally advantageous is when the higher-dimensional cone is a Cartesian product of many 'low-complexity' cones, such as second-order cones, or small positive semidefinite cones. This paper studies obstructions to a convex cone having a lifted representation with such a product structure. The main result says that whenever a convex cone has a certain neighborliness property, then it does not have a lifted representation using a finite product of cones, each of which has only short chains of faces. This is a generalization of recent work of Averkov ('Optimal size of linear matrix inequalities in semidefinite approaches to polynomial optimization', SIAM J. Appl. Alg. Geom., Vol. 3, No. 1, 128--151, 2019) which considers only lifted representations using products of positive semidefinite cones of bounded size. Among the consequences of the main result is that various cones related to nonnegative polynomials do not have lifted representations using products of 'low-complexity' cones, such as smooth cones, the exponential cone, and cones defined by hyperbolic polynomials of low degree.

math.OC↗

arXiv2018

Competitive Online Algorithms for Resource Allocation over the Positive Semidefinite Cone

We consider a new and general online resource allocation problem, where the goal is to maximize a function of a positive semidefinite (PSD) matrix with a scalar budget constraint. The problem data arrives online, and the algorithm needs to make an irrevocable decision at each step. Of particular interest are classic experiment design problems in the online setting, with the algorithm deciding whether to allocate budget to each experiment as new experiments become available sequentially. We analyze two greedy primal-dual algorithms and provide bounds on their competitive ratios. Our analysis relies on a smooth surrogate of the objective function that needs to satisfy a new diminishing returns (PSD-DR) property (that its gradient is order-reversing with respect to the PSD cone). Using the representation for monotone maps on the PSD cone given by Löwner's theorem, we obtain a convex parametrization of the family of functions satisfying PSD-DR. We then formulate a convex optimization problem to directly optimize our competitive ratio bound over this set. This design problem can be solved offline before the data start arriving. The online algorithm that uses the designed smoothing is tailored to the given cost function, and enjoys a competitive ratio at least as good as our optimized bound. We provide examples of computing the smooth surrogate for D-optimal and A-optimal experiment design, and demonstrate the performance of the custom-designed algorithm.

math.OC↗

arXiv2018

A Projection Method for Metric-Constrained Optimization

We outline a new approach for solving optimization problems which enforce triangle inequalities on output variables. We refer to this as metric-constrained optimization, and give several examples where problems of this form arise in machine learning applications and theoretical approximation algorithms for graph clustering. Although these problem are interesting from a theoretical perspective, they are challenging to solve in practice due to the high memory requirement of black-box solvers. In order to address this challenge we first prove that the metric-constrained linear program relaxation of correlation clustering is equivalent to a special case of the metric nearness problem. We then developed a general solver for metric-constrained linear and quadratic programs by generalizing and improving a simple projection algorithm originally developed for metric nearness. We give several novel approximation guarantees for using our framework to find lower bounds for optimal solutions to several challenging graph clustering problems. We also demonstrate the power of our framework by solving optimizing problems involving up to 10^{8} variables and 10^{11} constraints.

math.NA↗

arXiv2018

Semidefinite approximations of the matrix logarithm

The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this paper we show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size. Such approximations allow us to use off-the-shelf semidefinite optimization solvers for convex optimization problems involving the matrix logarithm and related functions, such as the quantum relative entropy. The basic ingredients of our approach apply, beyond the matrix logarithm, to functions that are operator concave and operator monotone. As such, we introduce strategies for constructing semidefinite approximations that we expect will be useful, more generally, for studying the approximation power of functions with small semidefinite representations.

math.OC↗

arXiv2018

Estimating the Spectral Density of Large Implicit Matrices

Many important problems are characterized by the eigenvalues of a large matrix. For example, the difficulty of many optimization problems, such as those arising from the fitting of large models in statistics and machine learning, can be investigated via the spectrum of the Hessian of the empirical loss function. Network data can be understood via the eigenstructure of a graph Laplacian matrix using spectral graph theory. Quantum simulations and other many-body problems are often characterized via the eigenvalues of the solution space, as are various dynamic systems. However, naive eigenvalue estimation is computationally expensive even when the matrix can be represented; in many of these situations the matrix is so large as to only be available implicitly via products with vectors. Even worse, one may only have noisy estimates of such matrix vector products. In this work, we combine several different techniques for randomized estimation and show that it is possible to construct unbiased estimators to answer a broad class of questions about the spectra of such implicit matrices, even in the presence of noise. We validate these methods on large-scale problems in which graph theory and random matrix theory provide ground truth.

stat.ML↗

arXiv2018

A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone

If $X$ is an $n\times n$ symmetric matrix, then the directional derivative of $X \mapsto \det(X)$ in the direction $I$ is the elementary symmetric polynomial of degree $n-1$ in the eigenvalues of $X$. This is a polynomial in the entries of $X$ with the property that it is hyperbolic with respect to the direction $I$. The corresponding hyperbolicity cone is a relaxation of the positive semidefinite (PSD) cone known as the first derivative relaxation (or Renegar derivative) of the PSD cone. A spectrahedal cone is a convex cone that has a representation as the intersection of a subspace with the cone of PSD matrices in some dimension. We show that the first derivative relaxation of the PSD cone is a spectrahedral cone, and give an explicit spectrahedral description of size $\binom{n+1}{2}-1$. The construction provides a new explicit example of a hyperbolicity cone that is also a spectrahedron. This is consistent with the generalized Lax conjecture, which conjectures that every hyperbolicity cone is a spectrahedron.

math.OC↗

arXiv2017

Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations

It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.

math.OC↗

arXiv2016

Symmetric Sums of Squares over $k$-Subset Hypercubes

We consider the problem of finding sum of squares (sos) expressions to establish the non-negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the $k$-element subsets of $[n]$. For simplicity, we focus on the case $k=2$, but our results extend naturally to all values of $k \geq 2$. We develop a variant of the Gatermann-Parrilo symmetry-reduction method tailored to our setting that allows for several simplifications and a connection to flag algebras. We show that every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sos expression whose size depends only on the degree and not on the number of variables. Our method bypasses much of the technical difficulties needed to apply the Gatermann-Parrilo method, and offers flexibility in obtaining succinct sos expressions that are combinatorially meaningful. As a byproduct of our results, we arrive at a natural representation-theoretic justification for the concept of flags as introduced by Razborov in his flag algebra calculus. Furthermore, this connection exposes a family of non-negative polynomials that cannot be certified with any fixed set of flags, answering a question of Razborov in the context of our finite setting.

math.CO↗