Graph Convolution with Low-rank Learnable Local Filters

TL;DR

This paper introduces a novel graph convolution method with learnable low-rank local filters that enhances expressiveness and robustness for non-Euclidean data, demonstrated through various experimental datasets.

Contribution

It proposes a new graph convolution approach with learnable low-rank local filters, unifying spectral and spatial methods, and provides theoretical stability guarantees.

Findings

Proven increased expressiveness over previous spectral methods

Demonstrated robustness to graph data perturbations

Empirical improvements on diverse datasets

Abstract

Geometric variations like rotation, scaling, and viewpoint changes pose a significant challenge to visual understanding. One common solution is to directly model certain intrinsic structures, e.g., using landmarks. However, it then becomes non-trivial to build effective deep models, especially when the underlying non-Euclidean grid is irregular and coarse. Recent deep models using graph convolutions provide an appropriate framework to handle such non-Euclidean data, but many of them, particularly those based on global graph Laplacians, lack expressiveness to capture local features required for representation of signals lying on the non-Euclidean grid. The current paper introduces a new type of graph convolution with learnable low-rank local filters, which is provably more expressive than previous spectral graph convolution methods. The model also provides a unified framework for both spectral and spatial graph convolutions. To improve model robustness, regularization by local graph Laplacians is introduced. The representation stability against input graph data perturbation is theoretically proved, making use of the graph filter locality and the local graph regularization. Experiments on spherical mesh data, real-world facial expression recognition/skeleton-based action recognition data, and data with simulated graph noise show the empirical advantage of the proposed model.