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arXiv:2004.07403

On the computability of continuous maximum entropy distributions with applications

Jonathan Leake, Nisheeth K. Vishnoi

Published Apr 16, 2020• 0 citations
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Abstract

We initiate a study of the following problem: Given a continuous domain $Ω$ along with its convex hull $\mathcal{K}$, a point $A \in \mathcal{K}$ and a prior measure $μ$ on $Ω$, find the probability density over $Ω$ whose marginal is $A$ and that minimizes the KL-divergence to $μ$. This framework gives rise to several extremal distributions that arise in mathematics, quantum mechanics, statistics, and theoretical computer science. Our technical contributions include a polynomial bound on the norm of the optimizer of the dual problem that holds in a very general setting and relies on a "balance" property of the measure $μ$ on $Ω$, and exact algorithms for evaluating the dual and its gradient for several interesting settings of $Ω$ and $μ$. Together, along with the ellipsoid method, these results imply polynomial-time algorithms to compute such KL-divergence minimizing distributions in several cases. Applications of our results include: 1) an optimization characterization of the Goemans-Williamson measure that is used to round a positive semidefinite matrix to a vector, 2) the computability of the entropic barrier for polytopes studied by Bubeck and Eldan, and 3) a polynomial-time algorithm to compute the barycentric quantum entropy of a density matrix that was proposed as an alternative to von Neumann entropy in the 1970s: this corresponds to the case when $Ω$ is the set of rank one projections matrices and $μ$ corresponds to the Haar measure on the unit sphere. Our techniques generalize to the setting of Hermitian rank $k$ projections using the Harish-Chandra-Itzykson-Zuber formula, and are applicable even beyond, to adjoint orbits of compact Lie groups.

DOI
10.48550/arxiv.2004.07403

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The MAXENT (maximum entropy) principle gives a natural correspondance between (classical and quantum) statistical mechanics and information theory. Its general formulation leverages on log-concavity (or Klein's inequality) and the celebrated information inequality, which is equivalent to Gibbs' inequality. The resulting convex optimization allows one to relate entropy to the log-canonical partition function by Legendre-Fenchel duality. This can be further generalized to dual variational characterization of relative entropy or Kullback-Leibler divergence, relating Gibbs variationnal principle to Donsker-Vardhan variational representations.From an information theoretical point of view, the Gibbs principle and its associated evidence lower bound serves as the basis of modern machine learning algorithms, such as the expectation-maximization (EM) algorithm and its restriction to parameters output by a neural network. Interestingly, a large class of alternating optimization algorithms including EM and Blahut-Arimoto for computing channel capacity and source rate-distortion functions can be cast in a generic information geometric alternating projections on convex sets of measures. On the other hand, Donsker-Vardhan representations, when restricted to the output of a neural network, allows one to efficiently estimate information measures, and when applied to parametric estimators with quadratic risk, provide HCR-like and CRB-like statistical lower bounds. An alternative derivation on such lower bounds in the Bayesian setting makes use of the Weyl-Heisenberg uncertainty principle, which opens up new perspectives.In this keynote, I will select and review several of the aforementioned topics, and discuss the interactions between classical and quantum interpretations.
We give a method for computing asymptotic formulas and approximations for the volumes of spectrahedra, based on the maximum-entropy principle from statistical physics. The method gives an approximate volume formula based on a single convex optimization problem of minimizing −log⁡det⁡P-\log \det P−logdetP over the spectrahedron. Spectrahedra can be described as affine slices of the convex cone of positive semi-definite (PSD) matrices, and the method yields efficient deterministic approximation algorithms and asymptotic formulas whenever the number of affine constraints is sufficiently dominated by the dimension of the PSD cone. Our approach is inspired by the work of Barvinok and Hartigan who used an analogous framework for approximately computing volumes of polytopes. Spectrahedra, however, possess a remarkable feature not shared by polytopes, a new fact that we also prove: central sections of the set of density matrices (the quantum version of the simplex) all have asymptotically the same volume. This allows for very general approximation algorithms, which apply to large classes of naturally occurring spectrahedra. We give two main applications of this method. First, we apply this method to what we call the "multi-way Birkhoff spectrahedron" and obtain an explicit asymptotic formula for its volume. This spectrahedron is the set of quantum states with maximal entanglement (i.e., the quantum states having univariant quantum marginals equal to the identity matrix) and is the quantum analog of the multi-way Birkhoff polytope. Second, we apply this method to explicitly compute the asymptotic volume of central sections of the set of density matrices.
Given two n×nn \times nn×n Hermitian matrices YYY and ΛΛΛ, the Harish-Chandra-Itzykson-Zuber (HCIZ) distribution on the unitary group U(n)\text{U}(n)U(n) is etr(UΛU∗Y)dμ(U)e^{\text{tr}(UΛU^*Y)}dμ(U)etr(UΛU∗Y)dμ(U), where μμμ is the Haar measure on U(n)\text{U}(n)U(n). The density etr(UΛU∗Y)e^{\text{tr}(UΛU^*Y)}etr(UΛU∗Y) is known as the HCIZ density. Random unitary matrices distributed according to the HCIZ density are important in various settings in physics and random matrix theory. However, the basic question of efficient sampling from the HCIZ distribution has remained open. We present two efficient algorithms to sample matrices from distributions that are close to the HCIZ distribution. The first algorithm outputs samples that are ξξξ-close in total variation distance and requires polynomially many arithmetic operations in log⁡1/ξ\log 1/ξlog1/ξ and the number of bits needed to encode YYY and ΛΛΛ. The second algorithm comes with a stronger guarantee that the samples are ξξξ-close in infinity divergence, but the number of arithmetic operations depends polynomially on 1/ξ1/ξ1/ξ, the number of bits needed to encode YYY and ΛΛΛ, and the differences of the largest and the smallest eigenvalues of YYY and ΛΛΛ. HCIZ densities can also be viewed as exponential densities on U(n)\text{U}(n)U(n)-orbits, and these densities have been studied in statistics, machine learning, and theoretical computer science. Thus our results have the following applications: 1) an efficient algorithm to sample from complex versions of matrix Langevin distributions studied in statistics, 2) an efficient algorithm to sample from continuous max-entropy distributions on unitary orbits, which implies an efficient algorithm to sample a pure quantum state from the entropy-maximizing ensemble representing a given density matrix, and 3) an efficient algorithm for differentially private rank-kkk approximation, with improved utility bounds for k>1k>1k>1.
An ensemble of quantum states can be described by a Hermitian, positive semidefinite and unit trace matrix called density matrix. Thus, the study of methods for optimizing a certain function (energy, entropy) over the set of density matrices has a direct application to important problems in quantum information and computation. We propose a projected gradient method for solving such problems. By exploiting the geometry of the feasible set, which is the intersection of the cone of Hermitian positive semidefinite matrices with the hyperplane defined by the unit trace constraint, we describe an efficient procedure to compute the projection onto this set using the Frobenius norm. Some important applications, such as quantum state tomography, are described and numerical experiments illustrate the effectiveness of the method when compared to previous methods based on fixed-point iterations or semidefinite programming.
This thesis focuses on the intersection of mathematical and computational optimization and quantum information. Main contributions are open-source software code: A hybrid approach mixing "traditional" nonconvex and convex methods can make difficult problems more accessible. A demonstration of how to efficiently implement such an algorithm, avoiding interfacial bottlenecks, is provided, finding optimal protocols to establish entanglement through a lossy channel. The central software package developed addresses polynomial optimization problems. Many problems naturally involve only a polynomial objective and constraint polynomials. Such problems can automatically be cast into semidefinite programs that provide a hierarchy of outer approximations. The resulting problems are often so large and scale so unfavorably with respect to the variable number and degree involved that the boundary of the doable is reached quickly. However, technical progress both in hardware and algorithms has pushed this boundary - but software frameworks for polynomial optimization have not followed in the same manner, often now making them the bottleneck that before was the solver. The package PolynomialOptimization.jl developed during this thesis aims to fill the gap and provide a very resource-efficient intermediate layer together with a wide number of algorithms to reduce the problem size, and naturally supporting complex numbers and semidefinite constraints ubiquitous in quantum information problems. Its application on an entanglement distribution problem is demonstrated, showing that even relaxations with semidefinite matrices of three- and four-digit size can be solved conveniently. Finally, a new way to calculate interior-point barriers for the cone of sums-of-squares matrices in a nearly time-optimal way is developed, whose efficient implementation has the potential of further reducing resource consumption.
Entropy is a fundamental property of both classical and quantum systems, spanning myriad theoretical and practical applications in physics and computer science. We study the problem of obtaining estimates to within a multiplicative factor γ>1γ>1γ>1 of the Shannon entropy of probability distributions and the von Neumann entropy of mixed quantum states. Our main results are: ∙\quad\bullet∙ an O~(n1+η2γ2)\widetilde{\mathcal{O}}\left( n^{\frac{1+η}{2γ^2}}\right)O(n2γ21+η​)-query quantum algorithm that outputs a γγγ-multiplicative approximation of the Shannon entropy H(p)H(\mathbf{p})H(p) of a classical probability distribution p=(p1,…,pn)\mathbf{p} = (p_1,\ldots,p_n)p=(p1​,…,pn​); ∙\quad\bullet∙ an O~(n12+1+η2γ2)\widetilde{\mathcal{O}}\left( n^{\frac12+\frac{1+η}{2γ^2}}\right)O(n21​+2γ21+η​)-query quantum algorithm that outputs a γγγ-multiplicative approximation of the von Neumann entropy S(ρ)S(ρ)S(ρ) of a density matrix ρ∈Cn×nρ\in\mathbb{C}^{n\times n}ρ∈Cn×n. In both cases, the input is assumed to have entropy bounded away from zero by a quantity determined by the parameter η>0η>0η>0, since, as we prove, no polynomial query algorithm can multiplicatively approximate the entropy of distributions with arbitrarily low entropy. In addition, we provide Ω(n13γ2)Ω\left(n^{\frac{1}{3γ^2}}\right)Ω(n3γ21​) lower bounds on the query complexity of γγγ-multiplicative estimation of Shannon and von Neumann entropies. We work with the quantum purified query access model, which can handle both classical probability distributions and mixed quantum states, and is the most general input model considered in the literature.