Efficient Techniques for Low-Rank Tensor Approximation and Applications in Robust Object Detection

Abstract

This paper introduces efficient randomized fixed-precision and single-pass algorithms for low-tubal-rank approximation of third-order tensors. The proposed fixed-precision algorithms are faster and more efficient than the existing algorithms for approximating the truncated tensor SVD (T-SVD). Furthermore, unlike existing single-pass methods, which directly extend early, unstable matrix algorithms, the proposed approach adapts enhanced and stabilized matrix techniques to the tensor setting. Through extensive numerical experiments, we identify a critical flaw in current single-pass algorithms: using sketching parameters of equal size often produces ill-conditioned tensor least-squares problems, leading to inaccurate approximations. The proposed algorithms are demonstrably robust to this issue, achieving superior performance under identical conditions. We also evaluate the robustness of existing single-pass methods on real-world data tensors, including images and videos, a topic that has not been thoroughly examined before. Numerical results confirm the effectiveness of the proposed methods. Three applications are presented: image compression, video super-resolution, and deep learning.