Joint Factors and Rank Estimation for the Canonical Polyadic Decomposition Based on Convex Optimization

Abstract

Estimating the minimal number of rank-1 tensors in the Canonical Polyadic Decomposition (CPD), known as the canonical rank, is a challenging area of research. To address this problem, we propose a method based on convex optimization to jointly estimate the CP factors and the canonical rank called FARAC for joint FActors and RAnk canonical estimation for the PolyadiC Decomposition. We formulate the FARAC method as a convex optimization problem in which a sparse promoting constraint is added to the superdiagonal of the core tensor of the CPD, whereas the Frobenius norm of the offdiagonal terms is constrained to be bounded. We propose an alternated minimization strategy for the Lagrangien to solve the optimization problem. The FARAC method has been validated on synthetic data with varying levels of noise, as well as on three real data sets. Compared to state-of-the-art methods, FARAC exhibits very good performance in terms of rank estimation accuracy for a large range of SNR values. Additionally, FARAC can handle the case in which the canonical rank exceeds one of the dimensions of the input tensor.

Keywords

• Rank and factors estimation
• canonical polyadic decomposition
• convex optimization