Multilevel Graph Wavelet Compressed Sensing with Scale-Aware Neural Recovery

Graph compression reduces the computational cost of graph learning, but its effect on signal propagation remains largely underexplored. Existing work evaluates compression through downstream task performance or structural preservation, neither of which directly captures how propagation dynamics change after compression. We study two fundamental compression paradigms, coarsening and sparsification, and ask whether they preserve the propagation behavior of the original graph. Across five datasets, varying compression rates, and propagation depths, we measure signal behavior through three complementary metrics. Our results reveal a consistent tension between the two compression families. Sparsification retains higher signal diversity and mitigates oversmoothing, but its propagation trajectory progressively diverges from that of the original graph. Coarsening more faithfully preserves propagation behavior, but at the cost of stronger smoothing and rank collapse. These findings demonstrate that two propagation-centric objectives, preserving signal diversity and preserving propagation fidelity, are distinct and empirically at odds under graph compression, highlighting the need for evaluation protocols that jointly consider both dimensions. The code and results are available at: https://github.com/KawshikBanerjee/Compression-Propagation-Duality

Community detection is a central problem in graph analysis, with applications ranging from network science to graph signal processing. In recent years, Graph Neural Networks (GNNs) have emerged as effective tools for learning low-dimensional representations of graph-structured data and have shown strong performance in clustering tasks, particularly on large and high-dimensional graphs. This paper investigates the use of GNN-based community detection within a graph signal interpolation framework. After reviewing the main classes of GNN architectures for community detection according to a standard taxonomy, we integrate the resulting graph communities into a Partition of Unity Method (PUM) for interpolation with Graph Basis Functions (GBFs). In this approach, GNN-derived communities are used to construct local subdomains on which GBF interpolants are computed and subsequently combined into a global approximation. Numerical experiments on benchmark %graph datasets, including geometric and urban network examples demonstrate that the proposed combination of GNN-based clustering and GBF-PUM interpolation yields accurate signal reconstructions. The results indicate that deep learning-based community detection can provide effective graph partitions for localized interpolation schemes, supporting its use in scalable graph signal analysis.

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.