Loop Calculus and Belief Propagation for q-ary Alphabet: Loop Tower

TL;DR

This paper extends Loop Calculus, a theoretical framework connecting MAP solutions and Belief Propagation, to q-ary alphabets, enhancing its applicability in statistical inference.

Contribution

It introduces an invariant formulation that generalizes Loop Calculus to handle q-ary alphabets, broadening the scope of the original approach.

Findings

Generalization of Loop Calculus to q-ary alphabets

Invariant formulation for the generalized approach

Enhanced understanding of BP in complex inference problems

Abstract

Loop Calculus introduced in [Chertkov, Chernyak '06] constitutes a new theoretical tool that explicitly expresses the symbol Maximum-A-Posteriori (MAP) solution of a general statistical inference problem via a solution of the Belief Propagation (BP) equations. This finding brought a new significance to the BP concept, which in the past was thought of as just a loop-free approximation. In this paper we continue a discussion of the Loop Calculus. We introduce an invariant formulation which allows to generalize the Loop Calculus approach to a q-are alphabet.