PAC-Bayesian Matrix Completion with a Spectral Scaled Student Prior

TL;DR

This paper introduces a PAC-Bayesian matrix completion method using a spectral scaled Student prior, providing theoretical guarantees, efficient sampling algorithms, and demonstrating superior performance in image inpainting tasks.

Contribution

The paper proposes a novel PAC-Bayesian approach with a spectral scaled Student prior for matrix completion, including theoretical bounds and efficient Langevin Monte Carlo sampling.

Findings

Achieves minimax-optimal oracle inequalities.

Langevin Monte Carlo sampling is faster than Gibbs sampler.

Effective in image inpainting applications.

Abstract

We study the problem of matrix completion in this paper. A spectral scaled Student prior is exploited to favour the underlying low-rank structure of the data matrix. We provide a thorough theoretical investigation for our approach through PAC-Bayesian bounds. More precisely, our PAC-Bayesian approach enjoys a minimax-optimal oracle inequality which guarantees that our method works well under model misspecification and under general sampling distribution. Interestingly, we also provide efficient gradient-based sampling implementations for our approach by using Langevin Monte Carlo. More specifically, we show that our algorithms are significantly faster than Gibbs sampler in this problem. To illustrate the attractive features of our inference strategy, some numerical simulations are conducted and an application to image inpainting is demonstrated.