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

Meixia Lin, Ziyang Zeng, Yangjing Zhang

TL;DR

A regularized framework for spatiotemporal matrix regression that characterizes temporal and spatial dependencies through tailored penalties is proposed, which incorporates a fused penalty to capture smooth temporal evolution and a graph-guided penalty to promote spatial similarity.

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.

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

TL;DR

A regularized framework for spatiotemporal matrix regression that characterizes temporal and spatial dependencies through tailored penalties is proposed, which incorporates a fused penalty to capture smooth temporal evolution and a graph-guided penalty to promote spatial similarity.

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 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.
Paper Structure (17 sections, 3 theorems, 59 equations, 8 figures, 2 tables, 2 algorithms)

This paper contains 17 sections, 3 theorems, 59 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model Formulation
  3. GGFL: Single-Task Model
  4. MultiGGFL: Multi-Task Model
  5. Consistency Analysis
  6. A Halpern Peaceman-Rachford Method
  7. An HPR Method for Solving MultiGGFL
  8. Per-iteration Updates and Costs
  9. Practical Acceleration
  10. Numerical Experiments
  11. Synthetic Data
  12. Data Generation
  13. Evaluation Performance
  14. Computational Performance
  15. SWUS Precipitation Forecasting: Single-Task
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose Assumptions assu: iid and assu: regularity hold. If $\lambda_{i,n}/\sqrt{n}\rightarrow\lambda_{i,0}\geq 0$ for $i=1,2,t,g.$ Then in distribution as $n\rightarrow \infty$, where $\Theta$ is the true coefficient tensor from model eq: model, multi, and with for any vectors $v$ and $h$ of the same length. In particular, if $\lambda_{i,n}=o(\sqrt{n})$ for $i=1,2,t,g$, then we have with $\ba

Figures (8)

  • Figure 1: Illustrative spatiotemporal structure in the coefficient matrix. Left: Spatial graph on a $4\times4$ grid, where blue and red groups highlight sets of adjacent locations. Right: Estimated coefficient matrix $\theta$, exhibiting smooth evolution over time lags, and similarity among adjacent spatial locations.
  • Figure 2: Illustration of multi-task model \ref{['eq: model, multi']} and its notations. Each cube represents a coefficient. The highlighted blue cuboid corresponds to the grouped coefficient vector $\Theta_{[ij]}$ across tasks.
  • Figure 3: Illustration of a $10 \times 10$ spatial grid for $S$ (left) and $\widehat{S}$ (middle). Right: true coefficient matrix $\theta\in\mathbb R^{60\times100}$.
  • Figure 4: True coefficient matrix $\theta$ (left), and GGFL estimated coefficient matrices with sample sizes $n=200$ (middle) and $n=1000$ (right).
  • Figure 5: Runtime of HPR for GGFL under varying regularization strengths $\lambda_0$. Upper: Scaling with temporal length $t$ (fixed $n=500, s=100$). Lower: Scaling with spatial dimension $s$ (fixed $n=500, t=60$).
  • ...and 3 more figures

Theorems & Definitions (7)

  • Remark 1: Relation to the graph total variation (GTV) estimator
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • Theorem 3