Efficient Algorithms for Low Tubal Rank Tensor Approximation with Applications

TL;DR

This paper introduces efficient randomized algorithms for low tubal rank tensor approximation, demonstrating improved speed, robustness, and applicability to real-world data like images and videos, with successful applications in image compression, super-resolution, and deep learning.

Contribution

The paper develops faster, more robust randomized fixed-precision algorithms for low tubal rank tensor approximation and extends single-pass randomized methods to tensors based on the T-product.

Findings

Proposed algorithms outperform existing methods in speed and accuracy.

Single-pass algorithms with equal-sized sketches are often ill-conditioned, but the new methods are robust.

Numerical experiments confirm the effectiveness in image compression, super-resolution, and deep learning applications.

Abstract

In this paper we propose efficient randomized fixed-precision techniques for low tubal rank approximation of tensors. The proposed methods are faster and more efficient than the existing fixed-precision algorithms for approximating the truncated tensor SVD (T-SVD). Besides, there are a few works on randomized single-pass algorithms for computing low tubal rank approximation of tensors, none of them experimentally reports the robustness of such algorithms for low-rank approximation of real-world data tensors e.g., images and videos. The current single-pass algorithms for tensors are generalizations of those developed for matrices to tensors. However, the single-pass randomized algorithms for matrices have been recently improved and stabilized. Motivated by this progress, in this paper, we also generalize them to the tensor case based on the tubal product (T-product). We conduct extensive simulations to study the robustness of them compared with the existing single-pass randomized algorithms. In particular, we experimentally found that single-pass algorithms with the sketching parameters of equal sizes usually lead to ill-conditioned tensor least-squares problems and inaccurate results. It is experimentally shown that our proposed single-pass algorithms are robust in this sense. Numerical results demonstrate that under the same conditions (setting the same hyper-parameters), our proposed algorithms provide better performance. Three applications to image compression, super-resolution problem and deep learning are also presented.