Rank-one tensor injection: A novel method for canonical polyadic tensor decomposition

Abstract

Canonical polyadic decomposition of tensor is to approximate or express the tensor by sum of rank-1 tensors. When all or almost all components of factor matrices of the tensor are highly collinear, the decomposition becomes difficult. Algorithms, e.g., the alternating algorithms, require plenty of iterations, and may get stuck in false local minima. This paper proposes a novel method for such decompositions. The method injects one or a few rank-1 tensors into the data tensor in order to control the decompositions of the rank-expanded data, while still preserving the estimation accuracy of the original tensor. To achieve this, we develop a method to automatically generate the injected tensor which satisfies a specific estimation accuracy such that this tensor should not dominate rank-1 tensors of the data tensor, but is still able to be retrieved with a sufficient accuracy. Simulations on tensors with highly collinear factor matrices will illustrate efficiency of the proposed injecting method.