MixLasso: Generalized Mixed Regression via Convex Atomic-Norm Regularization

We consider a generalization of mixed regression where the response is an additive combination of several mixture components. Standard mixed regression is a special case where each response is generated from exactly one component. Typical approaches to the mixture regression problem employ local search methods such as Expectation Maximization (EM) that are prone to spurious local optima. On the other hand, a number of recent theoretically-motivated \emph{Tensor-based methods} either have high sample complexity, or require the knowledge of the input distribution, which is not available in most of practical situations. In this work, we study a novel convex estimator \emph{MixLasso} for the estimation of generalized mixed regression, based on an atomic norm specifically constructed to regularize the number of mixture components. Our algorithm gives a risk bound that trades off between prediction accuracy and model sparsity without imposing stringent assumptions on the input/output distribution, and can be easily adapted to the case of non-linear functions. In our numerical experiments on mixtures of linear as well as nonlinear regressions, the proposed method yields high-quality solutions in a wider range of settings than existing approaches.