Tensor-Tensor Products, Group Representations, and Semidefinite Programming

2025-07-16

AI Summary

This work addresses the problem of characterizing semidefiniteness and formulating semidefinite programs (SDPs) for third-order tensors under the $star_M$-product. Methodologically, it embeds group representation theory into tensor algebra, leveraging the choice of matrix $M$ to encode invariance under a base group action, thereby constructing symmetry-adapted tensor SDP models; it further integrates the $star_M$-product, nonnegative quadratic form analysis, and low-rank tensor completion to derive higher-order semidefiniteness criteria. The main contributions are threefold: (i) the first algebraic framework for tensor SDPs driven by group representations, generalizing classical linear SDP theory to higher-order settings; (ii) a theoretical characterization of the structure of group-invariant nonnegative quadratic forms; and (iii) empirical validation of the framework’s optimization efficacy and data recovery accuracy on low-rank tensor completion tasks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• Investigates positive semidefiniteness under $star_M$-product
• Connects matrix M choice to group representation theory
• Solves low-rank tensor completion problems via M-SDP

Innovation

• Generalizes linear algebra to third order tensors
• Connects matrix M with group representation theory
• Solves invariant semidefinite programs using M-SDP