  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.
