{"ID":2893300,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.12729","arxiv_id":"2507.12729","title":"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.","short_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...","url_abs":"https://arxiv.org/abs/2507.12729","url_pdf":"https://arxiv.org/pdf/2507.12729v1","authors":"[\"Alex Dunbar\",\"Elizabeth Newman\"]","published":"2025-07-17T02:08:14Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"cs.CV\",\"math.NA\",\"math.RT\"]","methods":"[]","has_code":false}