Graph-Guided Fused Regularization for Single- and Multi-Task Regression on Spatiotemporal Data

Abstract: Spatiotemporal matrix-valued data arise frequently in modern applications, yet performing effective regression analysis remains challenging due to complex, dimension-specific dependencies. In this work, we propose a regularized framework for spatiotemporal matrix regression that characterizes temporal and spatial dependencies through tailored penalties. Specifically, the model incorporates a fused penalty to capture smooth temporal evolution and a graph-guided penalty to promote spatial similarity. The framework also extends to the multi-task setting, enabling joint estimation across related tasks. We provide a comprehensive analysis of the framework from both theoretical and computational perspectives. Theoretically, we establish the statistical consistency of the proposed estimators. Computationally, we develop an efficient solver based on the Halpern Peaceman-Rachford method for the resulting composite convex optimization problem. The proposed algorithm achieves a fast global non-ergodic $\mathcal{O}(1/k)$ convergence rate with low per-iteration complexity. Extensive numerical experiments demonstrate that our method significantly outperforms state-of-the-art approaches in terms of predictive accuracy and estimation error, while also exhibiting superior computational efficiency and scalability.