# PAC-Bayesian Matrix Completion with a Spectral Scaled Student Prior

arXiv: https://arxiv.org/abs/2104.08191 · HTML: https://mlsanity.com/paper/2104.08191/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 52 references that are in the map.
- First posted on arXiv 16 Apr 2021.

## Builds on (most cited first, 13 of 13)

- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [On the properties of variational approximations of Gibbs posteriors](https://mlsanity.com/paper/1506.04091/) (2015)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares](https://mlsanity.com/paper/1410.2596/) (2014)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Inference and Uncertainty Quantification for Noisy Matrix Completion](https://mlsanity.com/paper/1906.04159/) (2019)
- [1-bit Matrix Completion: PAC-Bayesian Analysis of a Variational Approximation](https://mlsanity.com/paper/1604.04191/) (2016)
- [Fast Low-Rank Bayesian Matrix Completion with Hierarchical Gaussian Prior Models](https://mlsanity.com/paper/1708.02455/) (2017)
- [Bayesian methods for low-rank matrix estimation: short survey and theoretical study](https://mlsanity.com/paper/1306.3862/) (2013)
- [PAC-Bayesian High Dimensional Bipartite Ranking](https://mlsanity.com/paper/1511.02729/) (2015)
- [Bayesian matrix completion: prior specification](https://mlsanity.com/paper/1406.1440/) (2014)

## Abstract (arXiv)

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.