A Generalization of Spatial Monte Carlo Integration

TL;DR

This paper introduces a generalized version of Spatial Monte Carlo Integration (GSMCI) that overcomes previous limitations and provides a statistical accuracy bound, along with a new PBM learning method that enhances learning accuracy.

Contribution

The paper proposes GSMCI to relax previous limitations and proves its statistical accuracy bound, and introduces a new PBM learning method combining SMCI with persistent contrastive divergence.

Findings

GSMCI overcomes limitations of previous SMCI in dense systems.

GSMCI has a proven statistical accuracy bound.

The new PBM learning method significantly improves learning accuracy.

Abstract

Spatial Monte Carlo integration (SMCI) is an extension of standard Monte Carlo integration and can approximate expectations on Markov random fields with high accuracy. SMCI was applied to pairwise Boltzmann machine (PBM) learning, with superior results to those from some existing methods. The approximation level of SMCI can be changed, and it was proved that a higher-order approximation of SMCI is statistically more accurate than a lower-order approximation. However, SMCI as proposed in the previous studies suffers from a limitation that prevents the application of a higher-order method to dense systems. This study makes two different contributions as follows. A generalization of SMCI (called generalized SMCI (GSMCI)) is proposed, which allows relaxation of the above-mentioned limitation; moreover, a statistical accuracy bound of GSMCI is proved. This is the first contribution of this study. A new PBM learning method based on SMCI is proposed, which is obtained by combining SMCI and the persistent contrastive divergence. The proposed learning method greatly improves the accuracy of learning. This is the second contribution of this study.