Efficient randomized algorithms for the fixed Tucker-rank problem of Tucker decomposition with adaptive shifts

TL;DR

This paper presents fast randomized algorithms with adaptive shifts for tensor Tucker decomposition, significantly improving efficiency and accuracy over existing methods through dynamic iteration strategies.

Contribution

It introduces novel adaptive shifted power iteration algorithms for Tucker decomposition, enhancing convergence speed and accuracy in randomized tensor approximation.

Findings

Reduced runtime compared to state-of-the-art methods

Maintained or improved approximation accuracy

Validated effectiveness on synthetic and real datasets

Abstract

Randomized numerical linear algebra is proved to bridge theoretical advancements to offer scalable solutions for approximating tensor decomposition. This paper introduces fast randomized algorithms for solving the fixed Tucker-rank problem of Tucker decomposition, through the integration of adaptive shifted power iterations. The proposed algorithms enhance randomized variants of truncated high-order singular value decomposition (T-HOSVD) and sequentially T-HOSVD (ST-HOSVD) by incorporating dynamic shift strategies, which accelerate convergence by refining the singular value gap and reduce the number of required power iterations while maintaining accuracy. Theoretical analyses provide probabilistic error bounds, demonstrating that the proposed methods achieve comparable or superior accuracy compared to deterministic approaches. Numerical experiments on synthetic and real-world datasets validate the efficiency and robustness of the proposed algorithms, showing a significant decline in runtime and approximation error over state-of-the-art techniques.