A fast alternating least squares method for third-order tensors based on a compression procedure

The alternating least squares (ALS) method is frequently used for the computation of the canonical polyadic decomposition (CPD) of tensors. It generally gives accurate solutions, but demands much time. A strong alternative to this is the alternating slice-wise diagonalization (ASD) method. It limits its targets only to third-order tensors, and in exchange for this restriction, it fully utilizes a compression technique based on matrix singular value decomposition and consequently achieves high efficiency. In this paper, we propose a new simple algorithm, Reduced ALS, which lies somewhere between ALS and ASD; it employs a similar compression procedure to ASD, but applies it more directly to ALS. Numerical experiments show that Reduced ALS runs as fast as ASD, avoiding instability ASD sometimes exhibits.