On Algorithms for and Computing with the Tensor Ring Decomposition

TL;DR

This paper explores advanced algorithms for the tensor ring decomposition, enabling lower storage costs and higher compression ratios for high-dimensional data through novel conversion, rounding, and graph transformation techniques.

Contribution

It introduces new algorithms for converting tensors to tensor ring format, a rounding operation requiring new linear algebra definitions, and efficient graph structure transformations.

Findings

Algorithms achieve higher compression ratios than previous methods.

Demonstrated significant storage cost reductions in numerical examples.

Efficient transformations with lower complexity than existing approaches.

Abstract

Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high-dimensional data, achieving linear scaling with the input dimension instead of exponential scaling. In this paper, we investigate even lower storage-cost representations in the tensor ring format, which is an extension of the tensor train format with variable end-ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage-cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph-based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format.