Posterior contraction in sparse Bayesian factor models for massive covariance matrices

TL;DR

This paper provides theoretical analysis of Bayesian methods for estimating high-dimensional covariance matrices using sparse factor models, demonstrating their consistency and optimal convergence rates.

Contribution

It introduces new continuous shrinkage priors for factor loadings and establishes their effectiveness in high-dimensional covariance estimation.

Findings

Bayesian estimators achieve consistent covariance matrix estimation when p > n.

New continuous shrinkage priors perform comparably to traditional point mass mixture priors.

Posterior convergence rates match minimax rates up to a logarithmic factor.

Abstract

Sparse Bayesian factor models are routinely implemented for parsimonious dependence modeling and dimensionality reduction in high-dimensional applications. We provide theoretical understanding of such Bayesian procedures in terms of posterior convergence rates in inferring high-dimensional covariance matrices where the dimension can be larger than the sample size. Under relevant sparsity assumptions on the true covariance matrix, we show that commonly-used point mass mixture priors on the factor loadings lead to consistent estimation in the operator norm even when $p\gg n$. One of our major contributions is to develop a new class of continuous shrinkage priors and provide insights into their concentration around sparse vectors. Using such priors for the factor loadings, we obtain similar rate of convergence as obtained with point mass mixture priors. To obtain the convergence rates, we construct test functions to separate points in the space of high-dimensional covariance matrices using insights from random matrix theory; the tools developed may be of independent interest. We also derive minimax rates and show that the Bayesian posterior rates of convergence coincide with the minimax rates upto a $\sqrt{\log n}$ term.