Efficient MCMC Sampling for Bayesian Matrix Factorization by Breaking Posterior Symmetries

TL;DR

This paper introduces a modification to Gaussian priors in Bayesian matrix factorization that breaks posterior symmetries, leading to more efficient MCMC sampling and improved accuracy in relational data analysis.

Contribution

It proposes a simple prior adjustment that provably eliminates symmetries in the posterior, enhancing MCMC efficiency and accuracy in Bayesian matrix factorization.

Findings

Non-zero prior means reduce MCMC autocorrelation

Breaking symmetries improves sampling efficiency

Lower reconstruction errors achieved

Abstract

Bayesian low-rank matrix factorization techniques have become an essential tool for relational data analysis and matrix completion. A standard approach is to assign zero-mean Gaussian priors on the columns or rows of factor matrices to create a conjugate system. This choice of prior leads to simple implementations; however it also causes symmetries in the posterior distribution that can severely reduce the efficiency of Markov-chain Monte-Carlo (MCMC) sampling approaches. In this paper, we propose a simple modification to the prior choice that provably breaks these symmetries and maintains/improves accuracy. Specifically, we provide conditions that the Gaussian prior mean and covariance must satisfy so the posterior does not exhibit invariances that yield sampling difficulties. For example, we show that using non-zero linearly independent prior means significantly lowers the autocorrelation of MCMC samples, and can also lead to lower reconstruction errors.