[Paper Review] Decimated Framelet System on Graphs and Fast G-Framelet Transforms

Xuebin Zheng, Bingxin Zhou+2 more|arXiv (Cornell University)|Dec 12, 2020

Advanced Graph Neural Networks109 references21 citations

TL;DR

This paper proposes a decimated framelet system on graphs that enables multiscale, localized representation of graph data through a hierarchical, coarse-grained chain structure. It introduces a fast $oldsymbol{ ext{G}}$-framelet transform (F$oldsymbol{ ext{G}}$T) with linear $oldsymbol{ ext{O}}(N)$ computational complexity, leveraging graph Laplacian-based orthonormal bases and a data-driven filter bank for efficient decomposition and reconstruction.

ABSTRACT

Graph representation learning has many real-world applications, from super-resolution imaging, 3D computer vision to drug repurposing, protein classification, social networks analysis. An adequate representation of graph data is vital to the learning performance of a statistical or machine learning model for graph-structured data. In this paper, we propose a novel multiscale representation system for graph data, called decimated framelets, which form a localized tight frame on the graph. The decimated framelet system allows storage of the graph data representation on a coarse-grained chain and processes the graph data at multi scales where at each scale, the data is stored at a subgraph. Based on this, we then establish decimated G-framelet transforms for the decomposition and reconstruction of the graph data at multi resolutions via a constructive data-driven filter bank. The graph framelets are built on a chain-based orthonormal basis that supports fast graph Fourier transforms. From this, we give a fast algorithm for the decimated G-framelet transforms, or FGT, that has linear computational complexity O(N) for a graph of size N. The theory of decimated framelets and FGT is verified with numerical examples for random graphs. The effectiveness is demonstrated by real-world applications, including multiresolution analysis for traffic network, and graph neural networks for graph classification tasks.

Motivation & Objective

• To develop a localized, tight framelet system for graph-structured data that supports multiscale analysis.
• To address the computational inefficiency of existing graph framelet transforms by enabling linear-time processing.
• To construct a hierarchical, coarse-grained chain of subgraphs that preserves geometric and spectral properties of the original graph.
• To enable efficient graph representation learning and multiresolution analysis for real-world applications such as traffic networks and graph neural networks.
• To establish a data-driven filter bank that ensures perfect reconstruction and stability in the transform process.

Proposed method

• Constructs a decimated framelet system on graphs using a chain-based orthonormal basis derived from the graph Laplacian, enabling multiscale decomposition.
• Employs graph clustering and coarse-graining to build a hierarchical subgraph chain, where each level represents a coarser approximation of the original graph.
• Utilizes the Haar basis on graphs as a low-SPOC (sparsity of the coefficient) orthonormal basis to ensure fast computation and localization.
• Designs a filter bank with low-pass and high-pass filters at each scale, ensuring the framelet system forms a tight frame for stable signal representation.
• Develops a fast algorithm, F$oldsymbol{ ext{G}}$T, by combining discrete Fourier transforms on each coarse-grained graph level with efficient downsampling and upsampling operations.
• Applies the discrete Fourier transform (DFT) and its adjoint (ADFT) on each level of the hierarchical graph structure to accelerate convolution and transform operations.

Experimental results

Research questions

• RQ1 Can a decimated framelet system be constructed on graphs to enable localized, multiscale representation of graph data with guaranteed stability and perfect reconstruction?
• RQ2 How can the computational complexity of graph framelet transforms be reduced to linear time while preserving accuracy and sparsity?
• RQ3 What is the role of the graph Laplacian and its eigenvectors in constructing orthonormal bases that support fast transforms on graphs?
• RQ4 How does the hierarchical coarse-graining of graphs affect the quality and interpretability of the resulting framelet coefficients?
• RQ5 To what extent can the proposed F$oldsymbol{ ext{G}}$T be leveraged to improve graph representation learning in graph neural networks and multiresolution analysis tasks?

Key findings

• The proposed decimated framelet system forms a localized tight frame on graphs, ensuring stable and invertible signal representation across multiple scales.
• The F$oldsymbol{ ext{G}}$T algorithm achieves linear computational complexity $oldsymbol{ ext{O}}(N)$ for a graph of size $N$, significantly improving efficiency over prior methods.
• Numerical experiments on random graphs confirm the linear scaling of computational time, validating the theoretical complexity analysis.
• The method successfully enables multiresolution analysis of a real-world road network, revealing structural patterns at different levels of granularity.
• When applied to graph neural networks, the F$oldsymbol{ ext{G}}$T-based graph convolution improves classification accuracy on benchmark datasets by leveraging multiscale features.
• The Haar basis on graphs achieves low SPOC (sparsity of coefficient), which contributes to the efficiency and localization of the framelet representation.