Convex optimization and tensors

It is shown that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space

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

The low-n-rank tensor recovery problem (1) (also its variant (2)) is a difficult non-convex problem

Therefore we will relax it to the following convex problem: Problem setting 3