Learning Time-Varying Graphs from Incomplete Graph Signals
Chuansen Peng, Xiaojing Shen
TL;DR
This work tackles the problem of learning time-varying graph topologies from incomplete graph signals by formulating a single non-convex optimization that jointly recovers a sequence of Laplacians and imputes missing signals. A structural fusion regularizer enforces shared topology patterns across graphs, while an ADMM algorithm with closed-form updates yields scalable, provably convergent solutions to the joint problem. The authors establish convergence to stationary points and derive non-asymptotic error bounds that quantify how sampling, sparsity, signal smoothness, and cross-graph similarity affect estimation accuracy. Empirical results on synthetic and real spatio-temporal data demonstrate faster convergence and superior joint graph learning and signal recovery compared to state-of-the-art baselines, highlighting strong practical impact for dynamic networks under partial observability.
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.