New Permanent Estimators via Non-Commutative Determinants

TL;DR

This paper introduces a new symmetrized determinant for matrices over associative algebras, enabling polynomial-time computation and randomized approximation of permanents for non-negative matrices, with conjectured near-accurate estimates in high-dimensional cases.

Contribution

It defines a novel symmetrized determinant for associative algebra matrices and develops a polynomial-time algorithm to estimate permanents of non-negative matrices using this concept.

Findings

Symmetrized determinant can be computed in polynomial time.

A randomized polynomial-time algorithm estimates permanents with conjectured near-accuracy.

The approach generalizes permanent estimation to matrices over associative algebras.

Abstract

We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $\AA$. 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 $\AA$, the symmetrized determinant of an $n\times n$ matrix with the entries in $\AA$ can be computed in polynomial in $n$ time (the degree of the polynomial is linear in the dimension of $\AA$). Then, for every associative algebra $\AA$ 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 $\AA=\Mat(d, {\Bbb R})$ is the algebra of $d\times d$ real matrices endowed with the standard scalar product and Gaussian measure, the algorithm approximates the permanent of a non-negative $n \times n$ matrix within $O(\gamma_d^n)$ factor, where $\lim_{d \longrightarrow +\infty} \gamma_d=1$. Finally, we provide some informal arguments why the conjecture might be true.