Tensor-Tensor Products, Group Representations, and Semidefinite Programming

TL;DR

This paper introduces a general tensor-tensor product framework that extends linear algebra concepts to third order tensors, linking group representation theory to semidefinite programming and enabling new applications like tensor completion.

Contribution

It develops the $ ext{star}_M$-product framework, connecting tensor algebra with group representations, and applies it to invariant semidefinite programs and tensor completion problems.

Findings

Characterization of nonnegative quadratic forms

Solution methods for low-rank tensor completion

Framework unifies tensor algebra with 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.