Estimation of Simultaneously Sparse and Low Rank Matrices

TL;DR

This paper proposes a convex penalized estimation method for simultaneously recovering sparse and low-rank matrices, with applications in social networks and protein interactions, supported by theoretical bounds and efficient algorithms.

Contribution

It introduces a novel mixed penalty combining  and trace norms for joint sparsity and low-rankness, along with theoretical guarantees and practical algorithms.

Findings

Oracle inequality characterizes the interaction of sparsity and low-rankness.

Bounded generalization error in link prediction.

Efficient proximal algorithms demonstrated on synthetic and real data.

Abstract

The paper introduces a penalized matrix estimation procedure aiming at solutions which are sparse and low-rank at the same time. Such structures arise in the context of social networks or protein interactions where underlying graphs have adjacency matrices which are block-diagonal in the appropriate basis. We introduce a convex mixed penalty which involves $\ell_1$-norm and trace norm simultaneously. We obtain an oracle inequality which indicates how the two effects interact according to the nature of the target matrix. We bound generalization error in the link prediction problem. We also develop proximal descent strategies to solve the optimization problem efficiently and evaluate performance on synthetic and real data sets.