A Generating Polynomial Based Two-Stage Optimization Method for Tensor Rank Decomposition

The rank decomposition, also known as canonical polyadic (CP) or simply tensor decomposition, has a long-standing history in multilinear algebra. However, computing a CP decomposition becomes particularly challenging when the tensor's rank lies between its largest and second-largest dimensions. Moreover, a common approach to address high-order tensor decompositions is to first flatten them into an order-3 tensor, where a significant gap often exists between the largest and the second-largest dimension, also making this case crucial in practice. In such a case, traditional optimization methods, such as nonlinear least squares or alternative least squares methods, often fail to produce accurate tensor decompositions. There are also direct methods, such as the normal form algorithm and the method by Domanov and De Lathauwer, that solve tensor decompositions algebraically. However, these methods can be computationally expensive and demand substantial memory, especially when the tensor rank is high. This paper introduces a novel generating polynomial (GP) based two-stage algorithm for the order-3 nonsymmetric tensor decomposition problem, assuming the rank does not exceed the largest dimension. The proposed method reformulates the tensor decomposition problem into two sequential optimization problems. Notably, if the first-stage optimization yields only a partial solution, it will be effectively utilized in the second stage. We establish the theoretical equivalence between the CP decomposition and the global minimizers of those two-stage optimization problems. Numerical experiments demonstrate that our approach is both efficient and robust, capable of finding tensor decompositions in scenarios where the current state-of-the-art methods often fail.