| Learning by using the structures of signals is crucial in data understanding. This talk focuses the study on three interesting structure. First, we use nonconvex functions instead of nuclear norm to better approximate the matrix rank. We propose a general solver to compute the generalized singular value of thresholding operator, which is a key subproblem in nonconvex low-rank minimization. Second, we study the subspace clustering problem. Many existing methods obtain block diagonal solutions which lead to correct clustering, but with their proofs given case by case. We provide a unified theoretical guarantee of the block diagonal property. We propose the block diagonal regularizer for directly pursuing the block diagonal matrix, and use it to solve the subspace clustering problem. For low-rank tensor recovery, we propose a new tensor nuclear norm induced by the tensor-tensor product, solve the tensor completion and tensor robust PCA by convex optimization and provide the sharp bound for the exact recovery. Finally, we study the widely used solver Alternating Direction Method of Multipliers. Many variants have been proposed but with their convergence proofs given case by case. We propose a general framework to unify existing methods and several new ADMMs and strategies to improve the convergence. |
