Efficient Nonnegative Tensor Decomposition Using Alternating Direction Proximal Method of Multipliers

Nonnegative CANDECOMP/PARAFAC (NCP) tensor decomposition is a powerful tool for multiway signal processing. The alternating direction method of multipliers (ADMM) optimization algorithm has become increasingly popular for solving tensor decomposition problems in the block coordinate descent framework. However, the ADMM-based NCP algorithm suffers from rank deficiency and slow convergence for some large-scale and highly sparse tensor data. The proximal algorithm is preferred to enhance optimization algorithms and improve convergence properties. In this study, we propose a novel NCP algorithm using the alternating direction proximal method of multipliers (ADPMM) that consists of the proximal algorithm. The proposed NCP algorithm can guarantee convergence and overcome the rank deficiency. Moreover, we implement the proposed NCP using an inexact scheme that alternatively optimizes the subproblems. Each subproblem is optimized by a finite number of inner iterations yielding fast computation speed. Our NCP algorithm is a hybrid of alternating optimization and ADPMM and is named <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{A}^{2}\text{DPMM}$</tex>. The experimental results on synthetic and real-world tensors demonstrate the effectiveness and efficiency of our proposed algorithm.