On Convex Programming Relaxations for the Permanent

In recent years, several convex programming relaxations have been proposed to estimate the permanent of a non-negative matrix, notably in the works of [GS02, Gur11, GS14]. 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 in [Gur06, Gur09, AMOV16].