Optimal quasi-Bayesian reduced rank regression with incomplete response

TL;DR

This paper introduces a quasi-Bayesian method for reduced rank regression with incomplete responses, providing theoretical guarantees and an efficient algorithm, demonstrating competitive performance on synthetic and real datasets.

Contribution

It proposes a novel quasi-Bayesian approach for incomplete reduced rank regression with theoretical oracle inequalities and a Langevin Monte Carlo algorithm for computation.

Findings

Method is adaptive to the rank of the coefficient matrix.

Competitively performs against state-of-the-art methods.

Sometimes improves over existing approaches.

Abstract

The aim of reduced rank regression is to connect multiple response variables to multiple predictors. This model is very popular, especially in biostatistics where multiple measurements on individuals can be re-used to predict multiple outputs. Unfortunately, there are often missing data in such datasets, making it difficult to use standard estimation tools. In this paper, we study the problem of reduced rank regression where the response matrix is incomplete. We propose a quasi-Bayesian approach to this problem, in the sense that the likelihood is replaced by a quasi-likelihood. We provide a tight oracle inequality, proving that our method is adaptive to the rank of the coefficient matrix. We describe a Langevin Monte Carlo algorithm for the computation of the posterior mean. Numerical comparison on synthetic and real data show that our method are competitive to the state-of-the-art where the rank is chosen by cross validation, and sometimes lead to an improvement.