# Capturing Between-Tasks Covariance and Similarities Using Multivariate Linear Mixed Models

## TL;DR

This paper introduces MrRCE, a multivariate linear mixed model approach that captures within-group coefficient similarities for multi-response prediction, outperforming existing methods in synthetic and real data scenarios.

## Contribution

The paper proposes a novel multivariate linear mixed model estimator that directly models and estimates within-group coefficient similarities, improving prediction accuracy.

## Key findings

- Outperforms competitors in synthetic data experiments.
- Effective in real-world multi-response prediction tasks.
- Encourages coefficients for the same variable to share signs and magnitudes.

## Abstract

We consider the problem of predicting several response variables using the same set of explanatory variables. This setting naturally induces a group structure over the coefficient matrix, in which every explanatory variable corresponds to a set of related coefficients. Most of the existing methods that utilize this group formation assume that the similarities between related coefficients arise solely through a joint sparsity structure. In this paper, we propose a procedure for constructing an estimator of a multivariate regression coefficient matrix that directly models and captures the within-group similarities, by employing a multivariate linear mixed model formulation, with a joint estimation of covariance matrices for coefficients and errors via penalized likelihood. Our approach, which we term Multivariate random Regression with Covariance Estimation (MrRCE) encourages structured similarity in parameters, in which coefficients for the same variable in related tasks sharing the same sign and similar magnitude. We illustrate the benefits of our approach in synthetic and real examples, and show that the proposed method outperforms natural competitors and alternative estimators under several model settings.