Tensor-Tensor Products, Group Representations, and Semidefinite Programming

Abstract

The ${\star }_M$-family of tensor-tensor products is a framework which generalizes many properties from linear algebra to third order tensors. Here, we investigate positive semidefiniteness and semidefinite programming under the ${\star }_M$-product. Critical to our investigation is a connection between the choice of matrix M in the ${\star }_M$-product and the representation theory of an underlying group action. Using this framework, third order tensors equipped with the ${\star }_M$-product are a natural setting for the study of invariant semidefinite programs. As applications of the M-SDP framework, we provide a characterization of certain nonnegative quadratic forms and solve low-rank tensor completion problems.

34 Pages, 7 figures