Time-Varying Graph Signal Recovery Using High-Order Smoothness and Adaptive Low-rankness

TL;DR

This work tackles time-varying graph signal recovery from partial, noisy observations on a fixed graph by integrating high-order Sobolev smoothness with an adaptive low-rank regularization via an ERF-weighted nuclear norm. Two ADMM-based algorithms are developed: Algorithm 1 uses an ERF regularizer with Sobolev temporal-spatial smoothness, while Algorithm 2 adds an L1 data-fidelity term for robustness to non-Gaussian noise. A convergence analysis guarantees that Algorithm 1 converges to a stationary point when the penalty parameter satisfies $\rho > L$, and numerical experiments on synthetic data and real datasets (PM$_{2.5}$ and sea surface temperature) show substantial RMSE improvements over baselines. The results demonstrate the effectiveness of combining high-order temporal smoothness with adaptive low-rankness in unsupervised graph signal recovery, with potential applications in climate, environmental monitoring, and epidemiology. Future work includes handling dynamic graph topology and exploring broader noise models.

Abstract

Time-varying graph signal recovery has been widely used in many applications, including climate change, environmental hazard monitoring, and epidemic studies. It is crucial to choose appropriate regularizations to describe the characteristics of the underlying signals, such as the smoothness of the signal over the graph domain and the low-rank structure of the spatial-temporal signal modeled in a matrix form. As one of the most popular options, the graph Laplacian is commonly adopted in designing graph regularizations for reconstructing signals defined on a graph from partially observed data. In this work, we propose a time-varying graph signal recovery method based on the high-order Sobolev smoothness and an error-function weighted nuclear norm regularization to enforce the low-rankness. Two efficient algorithms based on the alternating direction method of multipliers and iterative reweighting are proposed, and convergence of one algorithm is shown in detail. We conduct various numerical experiments on synthetic and real-world data sets to demonstrate the proposed method's effectiveness compared to the state-of-the-art in graph signal recovery.

Figures (9)

• Figure 1: The graph is constructed by kNN with $k=5$. The weight between any two nodes is inversely proportional to the square of their Euclidean distance.
• Figure 2: RMSE vs sampling rates. Averaged over 50 trials.
• Figure 3: RMSE vs noise level: $\eta^2 = \{0.01, 0.1, 0.2, 0.4, 0.6, 0.8, 1\}.$ Averaged over 50 trials.
• Figure 4: Graph with the places in California for the PM 2.5 concentration data. The graph was constructed with kNN for $k=5$.
• Figure 5: Average recovery accuracy comparison on the PM2.5 data.

Theorems & Definitions (2)

• theorem 1
• proof