Gauges, loops, and polynomials for partition functions of graphical models

Abstract

Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and optimization applications. The problem is #P-hard that is of an exponential complexity with respect to the number of variables. In this manuscript, aimed at approximating the partition function, we consider multi-graph models where binary variables and multivariable factors are associated with edges and nodes, respectively, of an undirected multi-graph. We suggest a new methodology for analysis and computations that combines the Gauge function technique from Chertkov and Chernyak (2006 Phys. Rev. E 73 065102; 2006 J. Stat. Mech. 2006 P06009) with the technique developed in Anari and Oveis Gharan 2017 arXiv:1702.02937; Gurvits 2011 arXiv:1106.2844; Straszak and Vishnoi 2017 55th Annual Allerton Conf. on Communication, Control, and Computing, based on the recent progress in the field of real stable polynomials. We show that the Gauge function, representing a single-out term in a finite sum expression for the partition function which achieves extremum at the so-called belief-propagation gauge, has a natural polynomial representation in terms of gauges/variables associated with edges of the multi-graph. Moreover, Gauge function can be used to recover the partition function through a sequence of transformations allowing appealing algebraic and graphical interpretations. Algebraically, one step in the sequence consists of the application of a differential operator over gauges associated with an edge. Graphically, the sequence is interpreted as a repetitive elimination/contraction of edges resulting in multi-graph models on decreasing in size (number of edges) graphs with the same partition function as in the original multi-graph model. Even though the complexity of computing factors in the sequence of the derived multi-graph models and respective Gauge functions grow exponentially with the number of eliminated edges, polynomials associated with the new factors remain bi-stable if the original factors have this property. Moreover, we show that BP estimations in the sequence do not decrease, each low-bounding the partition function.