Convergence Analysis of Single Latent Factor-Dependent, Nonnegative, and Multiplicative Update-Based Nonnegative Latent Factor Models

A single latent factor (LF)-dependent, nonnegative, and multiplicative update (SLF-NMU) learning algorithm is highly efficient in building a nonnegative LF (NLF) model defined on a high-dimensional and sparse (HiDS) matrix. However, convergence characteristics of such NLF models are never justified in theory. To address this issue, this study conducts rigorous convergence analysis for an SLF-NMU-based NLF model. The main idea is twofold: 1) proving that its learning objective keeps nonincreasing with its SLF-NMU-based learning rules via constructing specific auxiliary functions; and 2) proving that it converges to a stable equilibrium point with its SLF-NMU-based learning rules via analyzing the Karush-Kuhn-Tucker (KKT) conditions of its learning objective. Experimental results on ten HiDS matrices from real applications provide numerical evidence that indicates the correctness of the achieved proof.