Complex-Valued-Matrix Permanents: SPA-based Approximations and Double-Cover Analysis

TL;DR

This paper extends factor-graph-based sum-product algorithm methods to approximate the permanent of complex matrices, analyzing the behavior of SPA fixed points and Bethe approximations in complex-valued settings.

Contribution

It introduces a novel SPA-based approach for complex matrices and analyzes the Bethe approximation using double-edge normal factor graphs, providing new insights into complex-valued permanent approximations.

Findings

SPA fixed points change with complex matrices

Bethe approximation remains meaningful in complex settings

Graph cover analysis clarifies approximation behavior

Abstract

Approximating the permanent of a complex-valued matrix is a fundamental problem with applications in Boson sampling and probabilistic inference. In this paper, we extend factor-graph-based methods for approximating the permanent of non-negative-real-valued matrices that are based on running the sum-product algorithm (SPA) on standard normal factor graphs, to factor-graph-based methods for approximating the permanent of complex-valued matrices that are based on running the SPA on double-edge normal factor graphs. On the algorithmic side, we investigate the behavior of the SPA, in particular how the SPA fixed points change when transitioning from real-valued to complex-valued matrix ensembles. On the analytical side, we use graph covers to analyze the Bethe approximation of the permanent, i.e., the approximation of the permanent that is obtained with the help of the SPA. This combined algorithmic and analytical perspective provides new insight into the structure of Bethe approximations in complex-valued problems and clarifies when such approximations remain meaningful beyond the non-negative-real-valued settings.