Alternating DCA for reduced-rank multitask linear regression with covariance matrix estimation

We study a challenging problem in machine learning that is the reduced-rank multitask linear regression with covariance matrix estimation. The objective is to build a linear relationship between multiple output variables and input variables of a multitask learning process, taking into account the general covariance structure for the errors of the regression model in one hand, and reduced-rank regression model in another hand. The problem is formulated as minimizing a nonconvex function in two joint matrix variables (X,Theta) under the low-rank constraint on X and positive definiteness constraint on Theta. It has a double difficulty due to the non-convexity of the objective function as well as the low-rank constraint. We investigate a nonconvex, nonsmooth optimization approach based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) for this hard problem. A penalty reformulation is considered which takes the form of a partial DC program. An alternating DCA and its inexact version are developed, both algorithms converge to a weak critical point of the considered problem. Numerical experiments are performed on several synthetic and benchmark real multitask linear regression datasets. The numerical results show the performance of the proposed algorithms and their superiority compared with three classical alternating/joint methods.