# Tensor-Tensor Products, Group Representations, and Semidefinite Programming

arXiv: https://arxiv.org/abs/2507.12729 · HTML: https://mlsanity.com/paper/2507.12729/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 40 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 11 of 11)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [TTHRESH: Tensor Compression for Multidimensional Visual Data](https://mlsanity.com/paper/1806.05952/) (2018)
- [Equivariant neural networks for inverse problems](https://mlsanity.com/paper/2102.11504/) (2021)
- [Tensor Methods and Recommender Systems](https://mlsanity.com/paper/1603.06038/) (2016)
- [Tensor-based formulation and nuclear norm regularization for multi-energy computed tomography](https://mlsanity.com/paper/1307.5348/) (2013)
- [A Tensor-Based Dictionary Learning Approach to Tomographic Image Reconstruction](https://mlsanity.com/paper/1506.04954/) (2015)
- [T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming](https://mlsanity.com/paper/2006.11151/) (2020)
- [Symmetry Adapted Gram Spectrahedra](https://mlsanity.com/paper/2004.09641/) (2020)
- [Group-invariant tensor train networks for supervised learning](https://mlsanity.com/paper/2206.15051/) (2022)

## Built on by (most cited first, 1 of 1)

- [Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules](https://mlsanity.com/paper/2605.20440/) (2026)

## Abstract (arXiv)

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