Simultaneously Sparse and Low-Rank Matrix Reconstruction via Nonconvex and Nonseparable Regularization

Many real-world problems involve the recovery of a matrix from linear measurements, where the matrix lies close to some low-dimensional structure. This paper considers the problem of reconstructing a matrix with a simultaneously sparse and low-rank model. As surrogate functions of the sparsity and the matrix rank that are non-convex and discontinuous, the $\ell _1$ norm and the nuclear norm are often used instead to derive efficient algorithms to promote sparse and low-rank characteristics, respectively. However, the $\ell _1$ norm and the nuclear norm are loose approximations, and furthermore, recent study reveals using convex regularizations for joint structures cannot do better, orderwise, than exploiting only one of the structures. Motivated by the construction of non-convex and nonseparable regularization in sparse Bayesian learning, a new optimization problem is formulated in the latent space for recovering a simultaneously sparse and low-rank matrix. The newly proposed non-convex cost function is proved to have the ability to recover a simultaneously sparse and low-rank matrix with a sufficient number of noiseless linear measurements. In addition, an algorithm is derived to solve the resulting non-convex optimization problem, and convergence analysis of the proposed algorithm is provided in this paper. The performance of the proposed approach is demonstrated by experiments using both synthetic data and real hyperspectral images for compressive sensing applications.