Efficient construction of tensor ring representations from sampling

TL;DR

This paper introduces an efficient ALS-based method for compressing high-dimensional functions into tensor ring format using a sampling scheme that requires only O(d) samples, resulting in fewer parameters and better structure preservation.

Contribution

The paper presents a novel sampling scheme and initialization method for tensor ring approximation, enabling efficient compression of high-dimensional functions with fewer samples and faster convergence.

Findings

Tensor ring format uses fewer parameters than tensor-train for similar accuracy.

Sampling scheme requires only O(d) samples, reducing computational cost.

Numerical examples demonstrate improved structure preservation and efficiency.

Abstract

In this paper we propose an efficient method to compress a high dimensional function into a tensor ring format, based on alternating least-squares (ALS). Since the function has size exponential in $d$ where $d$ is the number of dimensions, we propose efficient sampling scheme to obtain $O(d)$ important samples in order to learn the tensor ring. Furthermore, we devise an initialization method for ALS that allows fast convergence in practice. Numerical examples show that to approximate a function with similar accuracy, the tensor ring format provided by the proposed method has less parameters than tensor-train format and also better respects the structure of the original function.