A multilinear Nystr\"om algorithm for low-rank approximation of tensors in Tucker format

TL;DR

This paper introduces a multilinear Nyström algorithm for low-rank tensor approximation in Tucker format, extending the Nyström method to higher-order tensors with stability and efficiency advantages.

Contribution

It develops a higher-order Nyström-based method for tensor approximation, offering a stable, efficient alternative to randomized higher-order SVD.

Findings

The method achieves strong stability properties.

It is a cost-effective, streamable alternative to existing tensor decompositions.

The algorithm maintains key attributes of the generalized Nyström method.

Abstract

The Nystr\"om method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the generalized Nystr\"om method). It is a randomized, single-pass, streamable, cost-effective, and accurate alternative to the randomized SVD, and it facilitates the computation of several matrix low-rank factorizations. In this paper, we take these advancements a step further by introducing a higher-order variant of Nystr\"om's methodology tailored to approximating low-rank tensors in the Tucker format: the multilinear Nystr\"om technique. We show that, by introducing appropriate small modifications in the formulation of the higher-order method, strong stability properties can be obtained. This algorithm retains the key attributes of the generalized Nystr\"om method, positioning it as a viable substitute for the randomized higher-order SVD algorithm.