Decimated Framelet System on Graphs and Fast G-Framelet Transforms
arXiv:2012.06922
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.
69 pages, 10 figures, Published in JMLR
References in corpus (7)
- Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift
- Fast Graph Representation Learning with PyTorch Geometric
- Graph Convolutional Matrix Completion
- Subgraph Matching Kernels for Attributed Graphs
- Graph Wavelet Neural Network
- Graph Signal Processing and Deep Learning: Convolution, Pooling, and Topology
- Path Integral Based Convolution and Pooling for Graph Neural Networks