Concavity of reweighted Kikuchi approximation

TL;DR

This paper investigates a reweighted Kikuchi approximation for the log partition function, establishing conditions for its concavity, and demonstrating that a reweighted sum product algorithm can find global optima, with practical benefits shown through simulations.

Contribution

The paper provides a comprehensive analysis of the concavity conditions for the reweighted Kikuchi approximation and links these conditions to the convergence of a reweighted sum product algorithm.

Findings

Sufficient conditions for concavity are established.

Reweighted sum product algorithm converges to global optima under certain conditions.

Simulations show practical advantages of the reweighted Kikuchi approach.

Abstract

We analyze a reweighted version of the Kikuchi approximation for estimating the log partition function of a product distribution defined over a region graph. We establish sufficient conditions for the concavity of our reweighted objective function in terms of weight assignments in the Kikuchi expansion, and show that a reweighted version of the sum product algorithm applied to the Kikuchi region graph will produce global optima of the Kikuchi approximation whenever the algorithm converges. When the region graph has two layers, corresponding to a Bethe approximation, we show that our sufficient conditions for concavity are also necessary. Finally, we provide an explicit characterization of the polytope of concavity in terms of the cycle structure of the region graph. We conclude with simulations that demonstrate the advantages of the reweighted Kikuchi approach.