From bilinear regression to inductive matrix completion: a quasi-Bayesian analysis

TL;DR

This paper introduces a quasi-Bayesian framework for bilinear regression and inductive matrix completion, providing statistical guarantees and a practical Langevin Monte Carlo algorithm, supported by numerical experiments.

Contribution

It develops a novel quasi-Bayesian approach for bilinear regression and inductive matrix completion, with theoretical analysis and an efficient sampling method.

Findings

The proposed estimators have favorable statistical properties under low-rank assumptions.

The Langevin Monte Carlo method effectively approximates the estimators.

Numerical studies demonstrate the practical performance of the methods.

Abstract

In this paper we study the problem of bilinear regression and we further address the case when the response matrix contains missing data that referred as the problem of inductive matrix completion. We propose a quasi-Bayesian approach first to the problem of bilinear regression where a quasi-likelihood is employed. Then, we adapt this approach to the context of inductive matrix completion. Under a low-rankness assumption and leveraging PAC-Bayes bound technique, we provide statistical properties for our proposed estimators and for the quasi-posteriors. We propose a Langevin Monte Carlo method to approximately compute the proposed estimators. Some numerical studies are conducted to demonstrated our methods.