Efficient Leverage Score Sampling for Tensor Train Decomposition

TL;DR

This paper introduces an efficient leverage score sampling algorithm for Tensor Train decomposition that significantly speeds up the ALS method by leveraging a specialized data structure and the canonical form of TT.

Contribution

The paper presents a novel leverage score sampling technique combined with a data structure that enables logarithmic time sampling, accelerating TT decomposition with ALS.

Findings

Outperforms SVD-based algorithms in experiments.

Achieves significant speed-up in ALS-based TT decomposition.

Effective on both synthetic and real tensor data.

Abstract

Tensor Train~(TT) decomposition is widely used in the machine learning and quantum physics communities as a popular tool to efficiently compress high-dimensional tensor data. In this paper, we propose an efficient algorithm to accelerate computing the TT decomposition with the Alternating Least Squares (ALS) algorithm relying on exact leverage scores sampling. For this purpose, we propose a data structure that allows us to efficiently sample from the tensor with time complexity logarithmic in the tensor size. Our contribution specifically leverages the canonical form of the TT decomposition. By maintaining the canonical form through each iteration of ALS, we can efficiently compute (and sample from) the leverage scores, thus achieving significant speed-up in solving each sketched least-square problem. Experiments on synthetic and real data on dense and sparse tensors demonstrate that our method outperforms SVD-based and ALS-based algorithms.