Tensor-Tensor Products, Group Representations, and Semidefinite Programming

Alex Dunbar; Elizabeth Newman

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arXiv · 2507.12729

Tensor-Tensor Products, Group Representations, and Semidefinite Programming

Alex Dunbar·Elizabeth Newman

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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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Alex Dunbar, Elizabeth Newman. 2025-07-17. Tensor-Tensor Products, Group Representations, and Semidefinite Programming. https://arxiv.org/abs/2507.12729

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