We address the problem of designing a convex nonsmooth regularizer encouraging multiple structural effects simultaneously. Focus- ing on the inference of sparse and low-rank matrices we suggest a new complexity index and a convex penalty approximating it. The new penalty term can be written as the trace norm of a linear function of the matrix. By analyzing theoretical properties of this family of regularizers we come up with oracle in- equalities and compressed sensing results ensuring the quality of our regularized estimator. We also provide algorithms and support- ing numerical experiments.