Rank-1 Tensor Approximation Methods and Application to Deflation
For researchers in tensor decomposition, this work offers improved algorithms and theoretical guarantees for deflation-based CP decomposition, though the improvements are incremental.
The paper presents a new algebraic rank-1 approximation method for three-way tensors that outperforms HOSVD, and a novel iterative algorithm that improves any rank-1 approximation method. It also provides a probabilistic convergence analysis for deflation-based CP decomposition, with experiments showing improved efficiency over existing algorithms.
Because of the attractiveness of the canonical polyadic (CP) tensor decomposition in various applications, several algorithms have been designed to compute it, but efficient ones are still lacking. Iterative deflation algorithms based on successive rank-1 approximations can be used to perform this task, since the latter are rather easy to compute. We first present an algebraic rank-1 approximation method that performs better than the standard higher-order singular value decomposition (HOSVD) for three-way tensors. Second, we propose a new iterative rank-1 approximation algorithm that improves any other rank-1 approximation method. Third, we describe a probabilistic framework allowing to study the convergence of deflation CP decomposition (DCPD) algorithms based on successive rank-1 approximations. A set of computer experiments then validates theoretical results and demonstrates the efficiency of DCPD algorithms compared to other ones.