Convex Optimization Methods for Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression

TL;DR

This paper compares convex optimization techniques for estimating multivariate linear regression models with trace norm regularization, highlighting the efficiency of a Nesterov-based method over interior point approaches.

Contribution

It introduces a variant of Nesterov's smooth method for this problem and demonstrates its superior performance and memory efficiency compared to traditional interior point methods.

Findings

Nesterov's method outperforms interior point methods in speed

The proposed method is more memory efficient

Experimental results on random instances validate the approach

Abstract

In this paper, we study convex optimization methods for computing the trace norm regularized least squares estimate in multivariate linear regression. The so-called factor estimation and selection (FES) method, recently proposed by Yuan et al. [22], conducts parameter estimation and factor selection simultaneously and have been shown to enjoy nice properties in both large and finite samples. To compute the estimates, however, can be very challenging in practice because of the high dimensionality and the trace norm constraint. In this paper, we explore a variant of Nesterov's smooth method [20] and interior point methods for computing the penalized least squares estimate. The performance of these methods is then compared using a set of randomly generated instances. We show that the variant of Nesterov's smooth method [20] generally outperforms the interior point method implemented in SDPT3 version 4.0 (beta) [19] substantially . Moreover, the former method is much more memory efficient.