Variational Inference on the Boolean Hypercube with the Quantum Entropy

TL;DR

This paper introduces quantum relaxation-based variational inference bounds for pairwise Markov random fields on the Boolean hypercube, along with efficient algorithms and hierarchical improvements, validated through extensive experiments.

Contribution

It presents novel quantum relaxation bounds for variational inference on the Boolean hypercube and proposes algorithms for their efficient computation and hierarchy-based improvements.

Findings

Quantum relaxations improve inference bounds.

Hierarchical relaxations enhance bound tightness.

Algorithms outperform state-of-the-art methods in experiments.

Abstract

In this paper, we derive variational inference upper-bounds on the log-partition function of pairwise Markov random fields on the Boolean hypercube, based on quantum relaxations of the Kullback-Leibler divergence. We then propose an efficient algorithm to compute these bounds based on primal-dual optimization. An improvement of these bounds through the use of ''hierarchies,'' similar to sum-of-squares (SoS) hierarchies is proposed, and we present a greedy algorithm to select among these relaxations. We carry extensive numerical experiments and compare with state-of-the-art methods for this inference problem.