Efficient Alternating Least Squares Algorithms for Truncated HOSVD of Higher-Order Tensors

The truncated Tucker decomposition, also known as the truncated higher-order singular value decomposition (HOSVD), has been extensively utilized as an efficient tool in many applications. Popular direct methods for truncated HOSVD often suffer from the notorious intermediate data explosion issue, especially for large-scale tensors, and are usually not easy to parallelize. In this paper, we propose a class of new truncated HOSVD algorithms based on alternating least squares (ALS). The proposed ALS-based approaches are able to eliminate the redundant computations of the singular vectors of intermediate matrices and are therefore free of data explosion. Also, the new methods are more flexible with adjustable convergence tolerance and are intrinsically parallelizable on high-performance computers. Theoretical analysis reveals that the ALS iteration in the proposed algorithms is q-linear convergent with a relatively wide convergence region. Numerical experiments with large-scale tensors from both synthetic and real-world applications demonstrate that ALS-based methods can substantially reduce the total cost of the original ones and are highly scalable for parallel computing.