Efficient MCMC Sampling for Bayesian Matrix Factorization by Breaking Posterior Symmetries
This addresses sampling inefficiencies for researchers using Bayesian methods in relational data analysis and matrix completion, though it is an incremental improvement over existing prior choices.
The paper tackles the problem of inefficient MCMC sampling in Bayesian matrix factorization due to posterior symmetries from zero-mean Gaussian priors, and shows that using non-zero linearly independent prior means reduces autocorrelation and can lower reconstruction errors.
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.