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

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.