Randomized Interpolative Decomposition of Separated Representations

TL;DR

This paper presents a new tensor Interpolative Decomposition method for reducing the separation rank of Canonical Tensor Decompositions, utilizing randomized techniques and offering an alternative to traditional algorithms.

Contribution

It introduces tensor ID as a novel, efficient approach for tensor rank reduction, combining it with Q-factorization and randomized methods for improved performance.

Findings

Tensor ID effectively reduces tensor separation rank with controlled accuracy.

Randomized sampling and projection enable efficient computation of tensor IDs.

The approach offers cost estimates and practical examples demonstrating its effectiveness.

Abstract

We introduce tensor Interpolative Decomposition (tensor ID) for the reduction of the separation rank of Canonical Tensor Decompositions (CTDs). Tensor ID selects, for a user-defined accuracy \epsilon, a near optimal subset of terms of a CTD to represent the remaining terms via a linear combination of the selected terms. Tensor ID can be used as an alternative to or a step of the Alternating Least Squares (ALS) algorithm. In addition, we briefly discuss Q-factorization to reduce the size of components within an ALS iteration. Combined, tensor ID and Q-factorization lead to a new paradigm for the reduction of the separation rank of CTDs. In this context, we also discuss the spectral norm as a computational alternative to the Frobenius norm. We reduce the problem of finding tensor IDs to that of constructing Interpolative Decompositions of certain matrices. These matrices are generated via either randomized projection or randomized sampling of the given tensor. We provide cost estimates and several examples of the new approach to the reduction of separation rank.