Learning Time-Varying Graphs from Incomplete Graph Signals

2025-10-19

AI Summary

This paper addresses the joint inference of dynamic network topology and imputation of missing data from partially observed time-varying graph signals. We propose a unified nonconvex optimization framework that simultaneously recovers a sequence of graph Laplacians and unobserved signal values, enabling bidirectional structural–signal co-modeling. The formulation incorporates a Lasso-type temporal regularization to capture gradual topological evolution while ensuring robustness to noise and spurious edge suppression. An efficient ADMM-based algorithm is developed, with closed-form solutions for both the graph structure and signal subproblems. Theoretically, we derive non-asymptotic statistical error bounds and establish convergence of the algorithm to a stationary point. Experiments demonstrate that our method significantly outperforms existing baselines under high missingness rates, achieving superior joint estimation accuracy and faster convergence.

Technology Category

• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper tackles the challenging problem of jointly inferring time-varying network topologies and imputing missing data from partially observed graph signals. We propose a unified non-convex optimization framework to simultaneously recover a sequence of graph Laplacian matrices while reconstructing the unobserved signal entries. Unlike conventional decoupled methods, our integrated approach facilitates a bidirectional flow of information between the graph and signal domains, yielding superior robustness, particularly in high missing-data regimes. To capture realistic network dynamics, we introduce a fused-lasso type regularizer on the sequence of Laplacians. This penalty promotes temporal smoothness by penalizing large successive changes, thereby preventing spurious variations induced by noise while still permitting gradual topological evolution. For solving the joint optimization problem, we develop an efficient Alternating Direction Method of Multipliers (ADMM) algorithm, which leverages the problem's structure to yield closed-form solutions for both the graph and signal subproblems. This design ensures scalability to large-scale networks and long time horizons. On the theoretical front, despite the inherent non-convexity, we establish a convergence guarantee, proving that the proposed ADMM scheme converges to a stationary point. Furthermore, we derive non-asymptotic statistical guarantees, providing high-probability error bounds for the graph estimator as a function of sample size, signal smoothness, and the intrinsic temporal variability of the graph. Extensive numerical experiments validate the approach, demonstrating that it significantly outperforms state-of-the-art baselines in both convergence speed and the joint accuracy of graph learning and signal recovery.

Problem

• Jointly infer time-varying network topologies and impute missing data
• Develop a unified optimization framework for graph and signal recovery
• Establish convergence guarantees and error bounds for the proposed method

Innovation

• Unified non-convex optimization for joint graph and signal recovery
• Fused-lasso regularizer enabling smooth time-varying topologies
• ADMM algorithm ensuring convergence and scalability guarantees