# Efficient Leverage Score Sampling for Tensor Train Decomposition

arXiv: https://arxiv.org/abs/2406.02749 · HTML: https://mlsanity.com/paper/2406.02749/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2024.

## Built on by (most cited first, 2 of 2)

- [Interaction Tensor SHAP](https://mlsanity.com/paper/2512.05338/) (2025)
- [Fast Tensor Completion via Approximate Richardson Iteration](https://mlsanity.com/paper/2502.09534/) (2025)

## Abstract (arXiv)

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.