2024/05/18 by Shota Takahashi, Mirai Tanaka, Takahashi, Shota +3 · 3 citations
Engineering · Mathematics · #Advanced Optimization Algorithms Research #Bregman divergence #Convex function #Function (biology) #Non-negative matrix factorization #Numerical methods in inverse problems #Separable space #Sequence (biology) #Sparse and Compressive Sensing Techniques #Stationary point #Theory of computation #Variable (mathematics)
paper · pdf · doi:10.1007/s10957-025-02833-y
published in Journal of Optimization Theory and Applications 208(1) (Springer Science+Business Media)
openalex publication_date 2025/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
Abstract Nonnegative matrix factorization (NMF) is a popular method in machine learning and signal processing to decompose a given nonnegative matrix into two nonnegative matrices. In this paper, we propose new algorithms, called majorization-minimization Bregman proximal gradient algorithm (MMBPG) and MMBPG with extrapolation (MMBPGe) to solve NMF. These iterative algorithms minimize the objective function and its potential function monotonically. Assuming the Kurdyka–Łojasiewicz property, we establish that a sequence generated by MMBPG(e) globally converges to a stationary point. We apply MMBPG and MMBPGe to the Kullback–Leibler (KL) divergence-based NMF. While most existing KL-based NMF methods update two blocks or each variable alternately, our algorithms update all variables simultaneously. MMBPG and MMBPGe for KL-based NMF are equipped with a separable Bregman distance that satisfies the smooth adaptable property and that makes its subproblem solvable in closed form. Using this fact, we guarantee that a sequence generated by MMBPG(e) globally converges to a Karush–Kuhn–Tucker (KKT) point of KL-based NMF. In numerical experiments, we compare proposed algorithms with existing algorithms on synthetic data and real-world data.