Primal-Dual Algorithms for Non-negative Matrix Factorization with the Kullback-Leibler Divergence
This work addresses performance bottlenecks in NMF for applications like data analysis and signal processing, offering incremental improvements over existing methods.
The authors tackled the slow convergence and local minima issues in non-negative matrix factorization (NMF) with Kullback-Leibler divergence by proposing a primal-dual algorithm based on the Chambolle-Pock method. The result showed that on synthetic, face recognition, and music source separation datasets, the algorithm was either faster than existing methods, led to improved local optima, or both.
Non-negative matrix factorization (NMF) approximates a given matrix as a product of two non-negative matrices. Multiplicative algorithms deliver reliable results, but they show slow convergence for high-dimensional data and may be stuck away from local minima. Gradient descent methods have better behavior, but only apply to smooth losses such as the least-squares loss. In this article, we propose a first-order primal-dual algorithm for non-negative decomposition problems (where one factor is fixed) with the KL divergence, based on the Chambolle-Pock algorithm. All required computations may be obtained in closed form and we provide an efficient heuristic way to select step-sizes. By using alternating optimization, our algorithm readily extends to NMF and, on synthetic examples, face recognition or music source separation datasets, it is either faster than existing algorithms, or leads to improved local optima, or both.