Effective Tensor Sketching via Sparsification

TL;DR

This paper introduces a new tensor sparsification method that efficiently approximates high-dimensional tensors with fewer samples, maintaining accuracy and enabling faster tensor decompositions, especially for large, high-order tensors.

Contribution

A novel tensor sparsification algorithm that reduces sample complexity for accurate spectral norm approximation, independent of tensor order, and facilitates efficient HOSVD.

Findings

Sample complexity is significantly reduced compared to existing methods.

Achieves near-optimal accuracy with fewer samples, especially for small error levels.

Enables efficient approximation of HOSVD for large tensors.

Abstract

In this paper, we investigate effective sketching schemes via sparsification for high dimensional multilinear arrays or tensors. More specifically, we propose a novel tensor sparsification algorithm that retains a subset of the entries of a tensor in a judicious way, and prove that it can attain a given level of approximation accuracy in terms of tensor spectral norm with a much smaller sample complexity when compared with existing approaches. In particular, we show that for a $k$th order $d\times\cdots\times d$ cubic tensor of {\it stable rank} $r_s$, the sample size requirement for achieving a relative error $\varepsilon$ is, up to a logarithmic factor, of the order $r_s^{1/2} d^{k/2} /\varepsilon$ when $\varepsilon$ is relatively large, and $r_s d /\varepsilon^2$ and essentially optimal when $\varepsilon$ is sufficiently small. It is especially noteworthy that the sample size requirement for achieving a high accuracy is of an order independent of $k$. To further demonstrate the utility of our techniques, we also study how higher order singular value decomposition (HOSVD) of large tensors can be efficiently approximated via sparsification.