Tensor-Tensor Products, Group Representations, and Semidefinite Programming

2025/07/17 by Alex Dunbar, Dunbar, Alex, Elizabeth Newman +1
Computer Science · #15A69 #65F99 #90C22 #Computability, Logic, AI Algorithms #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2507.12729

openalex publication_date 2025/07/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

The ⋆_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 ⋆_M-product. Critical to our investigation is a connection between the choice of matrix M in the ⋆_M-product and the representation theory of an underlying group action. Using this framework, third order tensors equipped with the ⋆_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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Related

• T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming
• Two Generalizations of Tensor Products, Beyond Vector Spaces
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