[Paper Review] New Permanent Estimators via Non-Commutative Determinants

Alexander Barvinok|ArXiv.org|Jul 25, 2000

Markov Chains and Monte Carlo Methods8 references20 citations

TL;DR

This paper introduces a new non-commutative determinant, the symmetrized determinant, for matrices over finite-dimensional associative algebras, enabling polynomial-time computation. It constructs randomized permanent estimators using this determinant, with a conjecture that for matrix algebras $\operatorname{Mat}(d,\mathbb{R})$ endowed with Gaussian measure, the estimator approximates the permanent within an $O(\gamma_d^n)$ factor, where $\gamma_d \to 1$ as $d \to \infty$, suggesting improved concentration with increasing algebra dimension.

ABSTRACT

We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $Å$. The monomial expansion of the symmetrized determinant is obtained from the standard expansion of the commutative determinant by averaging the products of entries of the matrix in all possible orders. We show that for any fixed finite-dimensional associative algebra $Å$, the symmetrized determinant of an $n imes n$ matrix with the entries in $Å$ can be computed in polynomial in $n$ time (the degree of the polynomial is linear in the dimension of $Å$). Then, for every associative algebra $Å$ endowed with a scalar product and unbiased probability measure, we construct a randomized polynomial time algorithm to estimate the permanent of non-negative matrices. We conjecture that if $Å=\Mat(d, {\Bbb R})$ is the algebra of $d imes d$ real matrices endowed with the standard scalar product and Gaussian measure, the algorithm approximates the permanent of a non-negative $n imes n$ matrix within $O(γ_d^n)$ factor, where $\lim_{d \longrightarrow +\infty} γ_d=1$. Finally, we provide some informal arguments why the conjecture might be true.

Motivation & Objective

• To develop a new class of permanent estimators based on non-commutative determinants that outperform existing methods in concentration and computational efficiency.
• To define and analyze the symmetrized determinant for matrices over finite-dimensional associative algebras, ensuring polynomial-time computability.
• To construct randomized polynomial-time algorithms for estimating the permanent of non-negative matrices using scalar products and unbiased measures on associative algebras.
• To conjecture that estimators based on $\operatorname{Mat}(d,\mathbb{R})$ with Gaussian measure achieve exponentially shrinking relative error as $d$ increases.
• To provide theoretical justification for the improved concentration behavior of the symmetrized determinant compared to Cayley determinants.

Proposed method

• Introduces the symmetrized determinant as the average of all monomial products over permutations, generalizing the commutative determinant to non-commutative algebras.
• Proves that for any fixed finite-dimensional associative algebra $\mathcal{A}$, the symmetrized determinant of an $n \times n$ matrix over $\mathcal{A}$ can be computed in polynomial time in $n$, with degree linear in $\dim \mathcal{A}$.
• Constructs a permanent estimator by sampling $n^2$ matrices $u_{ij}$ from a Gaussian measure on $\mathcal{A} = \operatorname{Mat}(d,\mathbb{R})$, forming a matrix $B$ with entries $b_{ij} = u_{ij}\sqrt{a_{ij}}$, and computing $\alpha = \|\operatorname{sdet} B\|^2 / \|\operatorname{sdet} E\|^2$, where $E$ is a diagonal matrix of the $u_{ii}$.
• Uses the scalar product $\langle a,b \rangle = \operatorname{Tr}(ab^t)$ on $\mathcal{A}$ to define the inner product space for the measure and estimator construction.
• Applies Chebyshev's inequality and measure concentration heuristics to argue for tight concentration of the estimator around the permanent.
• Leverages the fact that the symmetrized determinant is the average of $n!$ Cayley determinants, suggesting better concentration than the individual Cayley determinant.

Experimental results

Research questions

• RQ1 Can a new non-commutative determinant be defined such that its monomial expansion averages over all orderings of matrix entries, enabling efficient computation and permanent estimation?
• RQ2 Does the symmetrized determinant over $\operatorname{Mat}(d,\mathbb{R})$ with Gaussian measure yield a permanent estimator with exponentially shrinking relative error as $d$ increases?
• RQ3 Is the concentration of the symmetrized determinant estimator at least as sharp as that of the Cayley determinant, given that it is an average over all row permutations?
• RQ4 Can the expected value of the squared norm of the symmetrized determinant be related to the permanent via a computable normalization factor?
• RQ5 What is the asymptotic behavior of the approximation factor $\gamma_d^n$ as $d \to \infty$, and does it converge to 1?

Key findings

• The symmetrized determinant of an $n \times n$ matrix over a fixed finite-dimensional associative algebra $\mathcal{A}$ can be computed in polynomial time in $n$, with degree linear in $\dim \mathcal{A}$.
• For any associative algebra $\mathcal{A}$ with a scalar product and unbiased measure, a randomized polynomial-time algorithm estimates the permanent of non-negative matrices, satisfying properties (1.2.1)–(1.2.4).
• The estimator based on $\mathcal{A} = \operatorname{Mat}(d,\mathbb{R})$ with standard Gaussian measure is conjectured to approximate $\operatorname{per} A$ within a factor of $\gamma_d^n$, where $\lim_{d \to \infty} \gamma_d = 1$.
• The algorithm's complexity is $O(n^{d^2 + 3})$, scaling with the dimension of the matrix algebra.
• Theoretical arguments suggest that the symmetrized determinant's concentration is at least as strong as that of the Cayley determinant, due to its averaging nature over all permutations.
• The expected value of the squared norm of the symmetrized determinant, normalized by the diagonal case, equals the permanent, ensuring unbiased estimation.