On Convex Programming Relaxations for the Permanent

TL;DR

This paper provides a unified conceptual framework for understanding convex relaxations used to estimate the permanent of non-negative matrices, linking them to max-entropy programs and capacity-based approaches.

Contribution

It reveals the origins and relationships of various convex relaxations for the permanent, connecting them through a systematic max-entropy perspective.

Findings

Establishes equivalence of different relaxations via convex duality

Links relaxations to capacity-like quantities in the literature

Provides a unified framework for understanding permanent approximations

Abstract

In recent years, several convex programming relaxations have been proposed to estimate the permanent of a non-negative matrix, notably in the works of Gurvits and Samorodnitsky. However, the origins of these relaxations and their relationships to each other have remained somewhat mysterious. We present a conceptual framework, implicit in the belief propagation literature, to systematically arrive at these convex programming relaxations for estimating the permanent -- as approximations to an exponential-sized max-entropy convex program for computing the permanent. Further, using standard convex programming techniques such as duality, we establish equivalence of these aforementioned relaxations to those based on capacity-like quantities studied by Gurvits and Anari et al.