Tensor Deflation for CANDECOMP/PARAFAC. Part 3: Rank Splitting

CANDECOMP/PARAFAC (CPD) approximates multiway data by sum of rank-1 tensors. Our recent study has presented a method to rank-1 tensor deflation, i.e. sequential extraction of the rank-1 components. In this paper, we extend the method to block deflation problem. When at least two factor matrices have full column rank, one can extract two rank-1 tensors simultaneously, and rank of the data tensor is reduced by 2. For decomposition of order-3 tensors of size R x R x R and rank-R, the block deflation has a complexity of O(R^3) per iteration which is lower than the cost O(R^4) of the ALS algorithm for the overall CPD.