Canonical Polyadic Decomposition: From 3-way to N-Way

Abstract

Canonical Polyadic (or CANDECOMP/PARAFAC, CP) decompositions are widely applied to analyze high order data, i.e. N-way tensors. Existing CP decomposition methods use alternating least square (ALS) iterations and hence need to compute the inverse of matrices and unfold tensors frequently, which are very time consuming for large-scale data and when N is large. Fortunately, once at least one factor has been correctly estimated, all the remaining factors can be computed efficiently and uniquely by using a series of rank-one approximations. Motivated by this fact, to perform a full N-way CP decomposition, we run 3-way CP decompositions on a sub-tensor to estimate two factors first. Then the remaining factors are estimated via an efficient Khatri-Rao product recovering procedure. In this way the whole ALS iterations with respect to each mode are avoided and the efficiency can be significantly improved. Simulations show that, compared with ALS based CP decomposition methods, the proposed method is more efficient and is easier to escape from local solutions for high order tensors.