Bespoke multiresolution analysis of graph signals

TL;DR

This paper introduces a new multiresolution analysis framework for graph signals using samplet transforms, enabling efficient compression and analysis of complex graph data by leveraging graph subdivision, embedding, and wavelet-like constructions.

Contribution

It defines samplets on graphs through patch subdivision and embedding, extending wavelet analysis to graph signals with properties like orthogonality and locality.

Findings

Outperforms Haar wavelets in compression and fidelity

Demonstrates robustness and scalability on manifold-based graph signals

Provides sparse representations with controllable approximation error

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 \texttt{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.