Tensor completion and low-n-rank tensor recovery via convex optimization

In this paper we consider sparsity on a tensor level, as given by the n-rank of a tensor. In the important sparse-vector approximation problem (compressed sensing) and the low-rank matrix recovery problem, using a convex relaxation technique proved to be a valuable solution strategy. Here, we will adapt these techniques to the tensor setting. We use the n-rank of a tensor as sparsity measure and consider the low-n-rank tensor recovery problem, i.e., the problem of finding the tensor of lowest n-rank that fulfills some linear constraints. We introduce a tractable convex relaxation of the n-rank and propose efficient algorithms to solve the low-n-rank tensor recovery problem numerically. The algorithms are based on the Douglas-Rachford splitting technique and its dual variant, the alternating direction method of multipliers (ADM).