On Fast Computation of Gradients for CANDECOMP/PARAFAC Algorithms

Product between mode-n unfolding Y(n) of an N-D tensor Y and Khatri-Rao products of (N − 1) factor matrices A(m), m = 1, . . . , n − 1, n + 1, . . . , N exists in algorithms for CANDECOMP/PARAFAC (CP). If Y is an error tensor of a tensor approximation, this product is the gradient of a cost function with respect to factors, and has the largest workload in most CP algorithms. In this paper, a fast method to compute this product is proposed. Experimental verification shows that the fast CP gradient can accelerate the CP ALS algorithm 2 times and 8 times faster for factorizations of 3-D and 4-D tensors, and the speed-up ratios can be 20-30 times for higher dimensional tensors.