# Projected Tensor-Tensor Products for Efficient Computation of Optimal Multiway Data Representations

arXiv: https://arxiv.org/abs/2409.19402 · HTML: https://mlsanity.com/paper/2409.19402/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 38 references that are in the map.
- First posted on arXiv 28 Sep 2024.

## Builds on (most cited first, 14 of 14)

- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion](https://mlsanity.com/paper/1909.06982/) (2019)
- [Dictionary Learning with Low-rank Coding Coefficients for Tensor Completion](https://mlsanity.com/paper/2009.12507/) (2020)
- [SDRBench: Scientific Data Reduction Benchmark for Lossy Compressors](https://mlsanity.com/paper/2101.03201/) (2021)
- [Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recovery](https://mlsanity.com/paper/2105.14320/) (2021)
- [Subspace power method for symmetric tensor decomposition](https://mlsanity.com/paper/1912.04007/) (2019)
- [Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion](https://mlsanity.com/paper/2110.08774/) (2021)
- [Tensor Q-Rank: New Data Dependent Definition of Tensor Rank](https://mlsanity.com/paper/1910.12016/) (2019)
- [A Tensor SVD-based Classification Algorithm Applied to fMRI Data](https://mlsanity.com/paper/2111.00587/) (2021)
- [Cost-efficient Gaussian Tensor Network Embeddings for Tensor-structured Inputs](https://mlsanity.com/paper/2205.13163/) (2022)
- [Optimal Matrix-Mimetic Tensor Algebras via Variable Projection](https://mlsanity.com/paper/2406.06942/) (2024)
- [Dimensionality Reduction of Longitudinal 'Omics Data using Modern Tensor Factorization](https://mlsanity.com/paper/2111.14159/) (2021)

## Abstract (arXiv)

Tensor decompositions have become essential tools for feature extraction and compression of multiway data. Recent advances in tensor operators have enabled desirable properties of standard matrix algebra to be retained for multilinear factorizations. Behind this matrix-mimetic tensor operation is an invertible matrix whose size depends quadratically on certain dimensions of the data. As a result, for large-scale multiway data, the invertible matrix can be computationally demanding to apply and invert and can lead to inefficient tensor representations in terms of construction and storage costs. In this work, we propose a new projected tensor-tensor product that relaxes the invertibility restriction to reduce computational overhead and still preserves fundamental linear algebraic properties. The transformation behind the projected product is a tall-and-skinny matrix with unitary columns, which depends only linearly on certain dimensions of the data, thereby reducing computational complexity by an order of magnitude. We provide extensive theory to prove the matrix mimeticity and the optimality of compressed representations within the projected product framework. We further prove that projected-product-based approximations outperform a comparable, non-matrix-mimetic tensor factorization. We support the theoretical findings and demonstrate the practical benefits of projected products through numerical experiments on video and hyperspectral imaging data.