PAC-Bayesian Matrix Completion with a Spectral Scaled Student Prior
This work addresses matrix completion for applications like image inpainting, offering a robust and efficient method, though it appears incremental as it builds on prior PAC-Bayesian and spectral techniques.
The paper tackles matrix completion by using a spectral scaled Student prior to promote low-rank structure, achieving a minimax-optimal oracle inequality that ensures robustness under model misspecification and general sampling, with efficient gradient-based sampling via Langevin Monte Carlo that is significantly faster than Gibbs sampler.
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.