Convex optimization methods for dimension reduction and coefficient estimation in multivariate linear regression

In this paper, we study convex optimization methods for computing the nuclear (or, trace) norm regularized least squares estimate in multivariate linear regression. The so-called factor estimation and selection method, recently proposed by Yuan et al. (J Royal Stat Soc Ser B (Statistical Methodology) 69(3):329-346, 2007) 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 nuclear norm constraint. In this paper, we explore a variant due to Tseng of Nesterov's smooth method 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 generally outperforms the interior point method implemented in SDPT3 version 4.0 (beta) (Toh et al. On the implementation and usage of sdpt3-a matlab software package for semidefinite-quadratic-linear programming, version 4.0. Manuscript, Department of Mathematics, National University of Singapore (2006)) substantially. Moreover, the former method is much more memory efficient.