Low-rank Updates in Slowly Time-varying Graphs for Spatial-Temporal Signal Interpolation

2026-06-22

AI Summary

Static graphs struggle to capture the gradually evolving similarity between nodes over time, which limits the performance of spatiotemporal signal interpolation. To address this limitation, this work proposes a dynamic graph learning method that models the difference between adjacency matrices at consecutive time steps as a low-rank matrix and jointly optimizes graph structure refinement and signal interpolation by incorporating a graph signal smoothness prior. A fast orthogonal matching pursuit (OMP) algorithm with linear time complexity is designed to approximate low-rank projection, and this algorithm is unrolled into a lightweight neural network to enable data-driven fine-tuning. Experimental results demonstrate that the proposed approach significantly outperforms existing dynamic graph models on spatiotemporal signal interpolation tasks.

Abstract

A crucial assumption in graph signal processing (GSP) is the existence of an underlying graph that captures the pairwise similarities between nodes, allowing filters to be designed based on this graph for tasks such as denoising. For spatial-temporal data in which node-to-node similarities evolve over time, a static spatial graph is insufficient. In this paper, to represent slowly time-varying pairwise relationships, we model the graph changes in two consecutive adjacency matrices $P = W^{(2)} - W^{(1)}$ across time as a low-rank matrix. % Specifically, given an initial adjacency matrix $W^{(1)}$ at time $t=1$, we jointly interpolate a signal $x_2$ and estimate $W^{(2)}$ at $t=2$ using both a graph signal smoothness prior for $x_2$ and a low-rank prior on $¶$. We alternate optimization steps. With $W^{(2)}$ fixed, $x_2$ is interpolated by solving a linear system. Alternatively, holding $x_2$ fixed, $W^{(2)}$ is updated via proximal gradient descent (PGD). The proximal mapping of the rank term $Gamma(W^{(2)} - W^{(1)})$ is approximated in linear time using a fast orthogonal matching pursuit (OMP) algorithm that selects a sparse combination of atoms from a dictionary $cR$ formed by the outer products of $W^{(1)}$'s eigenvectors. We unroll iterations of our algorithm into layers to build a lightweight neural network for limited data-driven parameter tuning. Experiments show that our joint optimization achieves better signal interpolation compared to existing time-varying graph models.

Problem

• time-varying graphs
• spatial-temporal signal interpolation
• low-rank updates
• graph signal processing
• adjacency matrix

Innovation

• low-rank update
• time-varying graph
• graph signal interpolation
• proximal gradient descent
• unrolled neural network