Tensor-Tensor Products, Group Representations, and Semidefinite Programming
This work addresses foundational challenges in tensor algebra and optimization for researchers in applied mathematics and machine learning, though it appears incremental by building on existing tensor product frameworks.
The paper tackles the problem of extending positive semidefiniteness and semidefinite programming to third-order tensors using the $\star_M$-product framework, resulting in applications like characterizing nonnegative quadratic forms and solving low-rank tensor completion problems.
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.