Dissertation Defense Seminar

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 $2 \times 2 \times 2$ and $2 \times 2 \times 3$ 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.
Speaker(s): Robert Cochrane (UM)