Bespoke multiresolution analysis of graph signals

2025-07-25

AI Summary

To address the limitations of traditional Haar wavelets on graphs—including poor locality, absence of vanishing moments, and limited compression capability—this paper introduces the samplet transform to graph signal processing for the first time. We propose a novel orthogonal multi-resolution framework based on graph partitioning, Euclidean embedding, and pullback mapping. Specifically, the graph is partitioned via reweighted edge clustering; each subgraph is embedded into a low-dimensional Euclidean manifold using landmark Isomap; samplets with polynomial vanishing moments and strong spatial localization are constructed in the embedded space and then pulled back onto the original graph to yield a sparse, multi-scale representation. The framework rigorously guarantees orthogonality and controlled approximation error, significantly broadening the class of graph signals amenable to efficient compression. Experiments demonstrate superior performance over Haar wavelets in compression efficiency, multi-resolution fidelity, robustness to noise, and scalability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present a novel framework for discrete multiresolution analysis of graph signals. The main analytical tool is the samplet transform, originally defined in the Euclidean framework as a discrete wavelet-like construction, tailored to the analysis of scattered data. The first contribution of this work is defining samplets on graphs. To this end, we subdivide the graph into a fixed number of patches, embed each patch into a Euclidean space, where we construct samplets, and eventually pull the construction back to the graph. This ensures orthogonality, locality, and the vanishing moments property with respect to properly defined polynomial spaces on graphs. Compared to classical Haar wavelets, this framework broadens the class of graph signals that can efficiently be compressed and analyzed. Along this line, we provide a definition of a class of signals that can be compressed using our construction. We support our findings with different examples of signals defined on graphs whose vertices lie on smooth manifolds. For efficient numerical implementation, we combine heavy edge clustering, to partition the graph into meaningful patches, with landmark exttt{Isomap}, which provides low-dimensional embeddings for each patch. Our results demonstrate the method's robustness, scalability, and ability to yield sparse representations with controllable approximation error, significantly outperforming traditional Haar wavelet approaches in terms of compression efficiency and multiresolution fidelity.

Problem

• Defining samplets on graphs for signal analysis
• Enhancing graph signal compression beyond Haar wavelets
• Ensuring orthogonality and locality in graph signal processing

Innovation

• Samplet transform adapted for graph signals
• Heavy edge clustering for graph partitioning
• Landmark Isomap for low-dimensional patch embedding