Multi-resolution Low-rank Tensor Formats

TL;DR

This paper introduces a multiresolution tensor compression method that captures low-rank structures across scales, outperforming traditional formats in compression efficiency and providing algorithms with convergence guarantees.

Contribution

It presents a novel multiresolution tensor format, an alternating algorithm for its computation, and demonstrates superior compression compared to existing methods on real datasets.

Findings

Can outperform Eckart-Young theorem in 2D

Achieves higher compression than tensor-train format in higher dimensions

Provides stable and closed tensor representations

Abstract

We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale, and we show how this leads to an increase in compression for a fixed accuracy. We devise an alternating algorithm to represent a given tensor in the multiresolution format and prove local convergence guarantees. In two dimensions, we provide examples that show that this approach can beat the Eckart-Young theorem, and for dimensions higher than two, we achieve higher compression than the tensor-train format on six real-world datasets. We also provide results on the closedness and stability of the tensor format and discuss how to perform common linear algebra operations on the level of the compressed tensors.