Abstract: Supervised constrained matrix factorization (SCMF) is a classical machine learning method that seeks low-dimensional feature extraction and classification tasks at the same time. Training an SCMF model involves solving a non-convex and constrained optimization problem with at least three blocks of parameters. Due to the high non-convexity and constraints, theoretical understanding of the optimization landscape of SCMF has been limited. In this paper, we provide an extensive local landscape analysis for SCMF and derive several theoretical and practical applications. Analyzing diagonal blocks of the Hessian naturally leads to a block coordinate descent (BCD) algorithm with adaptive step sizes. We provide global convergence and iteration complexity guarantees for this algorithm. Full Hessian analysis gives minimum $L_{2}$-regularization to guarantee local strong convexity and robustness of parameters. We establish a local estimation guarantee under a statistical SCMF model. We also propose a novel GPU-friendly neural implementation of the BCD algorithm and validate our theoretical findings through numerical experiments. Our work contributes to a deeper understanding of SCMF optimization, offering insights into the optimization landscape and providing practical solutions to enhance its performance.