Tensor-Tensor Products, Group Representations, and Semidefinite Programming

• URL: http://arxiv.org/abs/2507.12729v1
• Date: Thu, 17 Jul 2025 02:08:14 GMT
• Title: Tensor-Tensor Products, Group Representations, and Semidefinite Programming
• Authors: Alex Dunbar, Elizabeth Newman,
• Abstract summary: We investigate positive semidefiniteness and semidefinite programming under the $star_M$-product.<n>Using this framework, third order tensors equipped with the $star_M$-product are a natural setting for the study of invariant semidefinite programs.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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