Algebraic Methods for Tensor Data

TL;DR

This paper introduces algebraic techniques for tensor data analysis, including invariant feature extraction, spectral norm approximation, and tensor amplification, supported by new diagrammatic tools and numerical experiments.

Contribution

It presents novel algebraic methods and colored Brauer diagrams for tensor computations, improving low rank tensor approximation performance.

Findings

Tensor amplification enhances low rank approximation accuracy.

Colored Brauer diagrams facilitate algebraic tensor computations.

Numerical experiments demonstrate improved ALS algorithm performance.

Abstract

We develop algebraic methods for computations with tensor data. We give 3 applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and amplification of low rank tensor structure. We introduce colored Brauer diagrams, which are used for algebraic computations and in analyzing their computational complexity. We present numerical experiments whose results show that the performance of the alternating least square algorithm for the low rank approximation of tensors can be improved using tensor amplification.