[Paper Review] A Framework for Multiscale Transforms on Graphs.

David I Shuman, Mohammad Javad Faraji+1 more|arXiv (Cornell University)|Aug 22, 2013

Advanced Graph Neural Networks12 citations

TL;DR

This paper proposes a novel multiscale transform framework for signals on weighted graphs by adapting the Laplacian pyramid from Euclidean signal processing. It introduces graph-specific operations—downsampling, reduction, filtering, and interpolation—to enable multiresolution analysis of both the graph structure and associated signals, yielding a hierarchical decomposition that captures intrinsic geometric structure.

ABSTRACT

Multiscale transforms designed to process analog and discrete-time signals and images cannot be directly applied to analyze high-dimensional data residing on the vertices of a weighted graph, as they do not capture the intrinsic geometric structure of the underlying graph data domain. In this paper, we adapt the Laplacian pyramid transform for signals on Euclidean domains so that it can be used to analyze high-dimensional data residing on the vertices of a weighted graph. Our approach is to study existing methods and develop new methods for the four fundamental operations of graph downsampling, graph reduction, and filtering and interpolation of signals on graphs. Equipped with appropriate notions of these operations, we leverage the basic multiscale constructs and intuitions from classical signal processing to generate a transform that yields both a multiresolution of graphs and an associated multiresolution of a graph signal on the underlying sequence of graphs.

Motivation & Objective

• To address the limitation of classical multiscale transforms in handling high-dimensional data on weighted graphs.
• To develop graph-specific analogs of downsampling, reduction, filtering, and interpolation for signal processing on graphs.
• To enable multiresolution analysis of both the graph topology and signals defined on its vertices.
• To preserve the intrinsic geometric structure of graph data during multiscale decomposition.

Proposed method

• Adapt the Laplacian pyramid transform from Euclidean domains to weighted graphs by redefining core operations for graph data.
• Define graph downsampling as a process that reduces vertex count while preserving signal and structural fidelity.
• Introduce graph reduction techniques to generate coarser approximations of the original graph at each scale.
• Develop graph filtering and interpolation methods that operate on signals defined on vertices, enabling signal reconstruction across scales.
• Use hierarchical decomposition to produce a sequence of increasingly coarser graphs and associated signals.
• Leverage classical multiscale intuition to ensure stability and interpretability in the resulting transform.

Experimental results

Research questions

• RQ1 How can classical multiscale transforms be adapted to operate on signals defined on weighted graphs?
• RQ2 What are appropriate graph-specific analogs of downsampling, filtering, and interpolation for multiscale analysis?
• RQ3 How can graph reduction be formalized to maintain structural and signal fidelity across scales?
• RQ4 Can a multiresolution representation of both graph and signal be constructed that captures intrinsic geometric structure?
• RQ5 What are the key properties of such a transform that ensure stability and utility in signal analysis?

Key findings

• The proposed framework successfully extends multiscale signal processing to graph-structured data by redefining fundamental operations for graphs.
• Graph downsampling and reduction preserve essential topological and signal characteristics across scales.
• The transform produces a coherent multiresolution hierarchy of both the graph and the signal on it.
• Filtering and interpolation operations are defined to maintain signal consistency during scale transitions.
• The approach enables analysis of high-dimensional data by capturing the intrinsic geometry of the underlying graph domain.
• The framework provides a systematic way to decompose graph signals across multiple resolutions while respecting the graph's structural complexity.