A fast Lanczos-based hierarchical algorithm for tensor ring decomposition

Tensor ring (TR) decomposition has made remarkable achievements in numerous high-order data processing tasks. However, the current alternating least squares (ALS)- and singular value decomposition (SVD)-based algorithms for TR decomposition, i.e., TR-ALS and TR-SVD, especially the former, are computationally expensive, making them unfriendly for large-scale data processing. This paper adopts three strategies to propose a novel fast TR decomposition algorithm: (1) Use a more efficient Lanczos bidiagonalization algorithm than SVD to generate the TR core tensors. (2) Exploit the hierarchical strategy to generate the TR core tensors in parallel. (3) Employ new reshaping and unfolding operations to reduce the dimensionality of the data used to generate TR core tensors. By incorporating these three strategies, we propose the TR-HLanczos algorithm for fast TR decomposition. This algorithm seamlessly produces the TR core tensors through the Lanczos bidiagonalization algorithm in a hierarchical manner. The effectiveness of the proposed TR-HLanczos algorithm is demonstrated through experimental results on both highly oscillatory functions and real-world datasets. For instance, when dealing with data of size $50^5$, TR-HLanczos is nearly 561 times and 18 times faster than algorithms based on ALS and SVD, respectively.