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Additive estimates of the permanent using Gaussian fields

Tantrik Mukerji, Wei-Shih Yang

TL;DR

This work addresses the problem of additively estimating the permanent $\mathrm{perm}(A)$ of a real matrix by embedding $A$ into a Gaussian field and exploiting Wick calculus. The authors show that $\mathrm{perm}(A)$ equals the expectation of a product of $2M$ Gaussian variables under a carefully chosen covariance $C$, and they estimate it via the empirical mean of products across $N$ samples, achieving a polynomial-time bound $P(|S_N-\mathrm{perm}(A)|>t)$ with a complexity of $O(M^{3}+M^{2}N+MN)$. A central contribution is the universal $2M$-dimensional Gaussian embedding for any $A$ and a semidefinite-programming framework to choose (and possibly modify) the covariance to minimize variance, yielding additive FPRAS in favorable regimes and connections to Max-Cut and cut-norms. The paper thus provides a novel, geometry- and graph-theory-inspired approach to permanent approximation that can offer improvements over Gurvits for certain matrix classes and opens avenues for further exploration of Gaussian embeddings in combinatorial counting problems.

Abstract

We present a randomized algorithm for estimating the permanent of an $M \times M$ real matrix $A$ up to an additive error. We do this by viewing the permanent $\mathrm{perm}(A)$ of $A$ as the expectation of a product of centered joint Gaussian random variables with a particular covariance matrix $C$. The algorithm outputs the empirical mean $S_{N}$ of this product after sampling $N$ times. Our algorithm runs in total time $O(M^{3} + M^{2}N + MN)$ with failure probability \begin{equation*} P(|S_{N}-\text{perm}(A)| > t) \leq \frac{3^{M}}{t^{2}N} \prod^{2M}_{i=1} C_{ii}. \end{equation*} In particular, we can estimate $\mathrm{perm}(A)$ to an additive error of $ε\bigg(\sqrt{3^{2M}\prod^{2M}_{i=1} C_{ii}}\bigg)$ in polynomial time. We compare to a previous procedure due to Gurvits. We discuss how to find a particular $C$ using a semidefinite program and a relation to the Max-Cut problem and cut-norms.

Additive estimates of the permanent using Gaussian fields

TL;DR

This work addresses the problem of additively estimating the permanent of a real matrix by embedding into a Gaussian field and exploiting Wick calculus. The authors show that equals the expectation of a product of Gaussian variables under a carefully chosen covariance , and they estimate it via the empirical mean of products across samples, achieving a polynomial-time bound with a complexity of . A central contribution is the universal -dimensional Gaussian embedding for any and a semidefinite-programming framework to choose (and possibly modify) the covariance to minimize variance, yielding additive FPRAS in favorable regimes and connections to Max-Cut and cut-norms. The paper thus provides a novel, geometry- and graph-theory-inspired approach to permanent approximation that can offer improvements over Gurvits for certain matrix classes and opens avenues for further exploration of Gaussian embeddings in combinatorial counting problems.

Abstract

We present a randomized algorithm for estimating the permanent of an real matrix up to an additive error. We do this by viewing the permanent of as the expectation of a product of centered joint Gaussian random variables with a particular covariance matrix . The algorithm outputs the empirical mean of this product after sampling times. Our algorithm runs in total time with failure probability \begin{equation*} P(|S_{N}-\text{perm}(A)| > t) \leq \frac{3^{M}}{t^{2}N} \prod^{2M}_{i=1} C_{ii}. \end{equation*} In particular, we can estimate to an additive error of in polynomial time. We compare to a previous procedure due to Gurvits. We discuss how to find a particular using a semidefinite program and a relation to the Max-Cut problem and cut-norms.
Paper Structure (10 sections, 30 theorems, 99 equations)

This paper contains 10 sections, 30 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The Algorithm and Analysis
  4. Gaussian embedding of an arbitrary matrix $A$
  5. Adjusting the Covariance
  6. Additive FPRAS
  7. Diagonally-dominant matrices
  8. A semidefinite program to reduce variance
  9. Relationship to Max-Cut and cut-norms
  10. Discussion

Key Result

Theorem 1.1

There is a randomized algorithm such that for any $M \times M$ complex matrix $A$ returns an empirical sample $S_{N}$ on $N$ samples from a uniformly distributed bit string $\{-1, 1\}$, which can be computed in $O(M^{2}N)$ time, whose additive estimate for $\text{perm}(A)$ has the upper bound on the In particular, if we take $\frac{K}{\epsilon^{2}}$ samples then this algorithm in time $O(M^{2}/\ep

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Corollary 2.1
  • Lemma 2.1
  • proof
  • proof
  • ...and 38 more