Tensor Ranks and Norms

Abstract

Tensor decompositions have been studied for nearly a century, but the well-known notion of tensor rank does not capture all the properties that one may desire of a rank function on tensors. For that reason, in recent years many alternative notions of tensor rank have been developed and studied. Some have also studied related notions of norms on tensors, especially the nuclear norm, which can be viewed as a convex relaxation of tensor rank.

In this thesis, we calculate the value of the recently introduced G-stable rank for all weights on 2x2x2 and 2x2x3 complex-valued tensors, and introduce X-rank, which can be viewed as a refinement of G-stable rank.

We then investigate the nuclear norm, and how it and some other norms on tensor product spaces behave with respect to the vertical, or Kronecker, tensor product.

Finally, we introduce some notions of stable ranks on tensors, built from common norms on tensor products, and discuss how these stable ranks relate to other notions of tensor rank.