# Optimal Matrix-Mimetic Tensor Algebras via Variable Projection

arXiv: https://arxiv.org/abs/2406.06942 · HTML: https://mlsanity.com/paper/2406.06942/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 59 references that are in the map.
- First posted on arXiv 11 Jun 2024.

## Builds on (most cited first, 9 of 9)

- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Randomized Numerical Linear Algebra : A Perspective on the Field With an Eye to Software](https://mlsanity.com/paper/2302.11474/) (2023)
- [Train Like a (Var)Pro: Efficient Training of Neural Networks with Variable Projection](https://mlsanity.com/paper/2007.13171/) (2020)
- [Tensor Q-Rank: New Data Dependent Definition of Tensor Rank](https://mlsanity.com/paper/1910.12016/) (2019)
- [Stable Tensor Neural Networks for Rapid Deep Learning](https://mlsanity.com/paper/1811.06569/) (2018)
- [A Tensor SVD-based Classification Algorithm Applied to fMRI Data](https://mlsanity.com/paper/2111.00587/) (2021)
- [Automatic Differentiation for Complex Valued SVD](https://mlsanity.com/paper/1909.02659/) (2019)
- [Non-negative Tensor Patch Dictionary Approaches for Image Compression and Deblurring Applications](https://mlsanity.com/paper/1910.00993/) (2019)

## Built on by (most cited first, 2 of 2)

- [Projected Tensor-Tensor Products for Efficient Computation of Optimal Multiway Data Representations](https://mlsanity.com/paper/2409.19402/) (2024)
- [Robust Low-Tubal-Rank Tensor Completion under Cross-Concentrated Sampling](https://mlsanity.com/paper/2608.03928/) (2026)

## Abstract (arXiv)

Recent advances in {matrix-mimetic} tensor frameworks have made it possible to preserve linear algebraic properties for multilinear data analysis and, as a result, to obtain optimal representations of multiway data. Matrix mimeticity arises from interpreting tensors as operators that can be multiplied, factorized, and analyzed analogous to matrices. Underlying the tensor operation is an algebraic framework parameterized by an invertible linear transformation. The choice of linear mapping is crucial to representation quality and, in practice, is made heuristically based on expected correlations in the data. However, in many cases, these correlations are unknown and common heuristics lead to suboptimal performance. In this work, we simultaneously learn optimal linear mappings and corresponding tensor representations without relying on prior knowledge of the data. Our new framework explicitly captures the coupling between the transformation and representation using variable projection. We preserve the invertibility of the linear mapping by learning orthogonal transformations with Riemannian optimization. We provide original theory of uniqueness of the transformation and convergence analysis of our variable-projection-based algorithm. We demonstrate the generality of our framework through numerical experiments on a wide range of applications, including financial index tracking, image compression, and reduced order modeling. We have published all the code related to this work at https://github.com/elizabethnewman/star-M-opt.