Group Sparse-based Tensor CP Decomposition: Model, Algorithms, and Applications in Chemometrics

Abstract

The CANDECOMP/PARAFAC (or Canonical polyadic, CP) decomposition of tensors has numerous applications in various fields, such as chemometrics, signal processing, machine learning, etc. Tensor CP decomposition assumes the knowledge of the exact CP rank, i.e., the total number of rank-one components of a tensor. However, accurately estimating the CP rank is very challenging. In this work, to address this issue, we prove that the CP rank can be exactly estimated by minimizing the group sparsity of any one of the factor matrices under the unit length constraints on the columns of the other factor matrices. Based on this result, we propose a CP decomposition model with group sparse regularization, which integrates the rank estimation and the tensor decomposition as an optimization problem, whose set of optimal solutions is proved to be nonempty. To solve the proposed model, we propose a double-loop block-coordinate proximal gradient descent algorithm with extrapolation and prove that each accumulation point of the sequence generated by the algorithm is a stationary point of the proposed model. Furthermore, we incorporate a rank reduction strategy into the algorithm to reduce the computational complexity. Finally, we apply the proposed model and algorithms to the component separation problem in chemometrics using real data. Numerical experiments demonstrate the robustness and effectiveness of the proposed methods.