Fast truncation of mode ranks for bilinear tensor operations

TL;DR

This paper introduces a rapid algorithm for mode rank truncation in bilinear tensor operations, significantly reducing computational costs while maintaining high approximation accuracy, suitable for tensors in Tucker or canonical form.

Contribution

The paper presents a novel, efficient algorithm based on cross approximation of Gram matrices for mode rank truncation in 3-tensors, with computational complexity of O(nr^3 + r^4).

Findings

Computational cost is reduced to O(nr^3 + r^4).

Approximation accuracy is limited by square root of machine precision.

Applicable to tensors in Tucker or canonical form.

Abstract

We propose a fast algorithm for mode rank truncation of the result of a bilinear operation on 3-tensors given in the Tucker or canonical form. If the arguments and the result have mode sizes n and mode ranks r, the computation costs $O(nr^3 + r^4)$. The algorithm is based on the cross approximation of Gram matrices, and the accuracy of the resulted Tucker approximation is limited by square root of machine precision.