Accurate and smooth shape matching is very hard to achieve. This is because for accuracy, one needs unique descriptors (signatures) on shapes that distinguish different vertices on a mesh accurately while at the same time being invariant to deformations. However, most existing unique shape descriptors are generally not smooth on the shape and are not noise-robust thus leading to non-smooth matches. On the other hand, for smoothness, one needs descriptors that are smooth and continuous on the shape. However, existing smooth descriptors are generally not unique and as such lose accuracy as they match neighborhoods (for smoothness) rather than exact vertices (for accuracy). In this work, we propose to use different k-hop neighborhoods of vertices as pairwise descriptors for shape matching. We use these descriptors in conjunction with local map distortion (LMD) to refine an initialized map for shape matching. We validate the effectiveness of our pipeline on benchmark datasets such as SCAPE, TOSCA, TOPKIDS, and others.

Attributed graph clustering (AGC) is an important problem in graph mining as more and more complex data in real-world have been represented in graphs with attributed nodes. While it is a common practice to leverage both attribute and structure information for improved clustering performance, most existing AGC algorithms consider only a specific type of relations, which hinders their applicability to integrate various complex relations into node attributes for AGC. In this article, we propose GRACE, an extended graph convolution framework for AGC tasks. Our framework provides a general and interpretative solution for clustering many different types of attributed graphs, including undirected, directed, heterogeneous and hyper attributed graphs. By building suitable graph Laplacians for each of the aforementioned graph types, GRACE can seamlessly perform graph convolution on node attributes to fuse all available information for clustering. We conduct extensive experiments on 14 real-world datasets of four different graph types. The experimental results show that GRACE outperforms the state-of-the-art AGC methods on the different graph types in terms of clustering quality, time, and memory usage.

For exact 3D shape correspondence (matching or alignment), i.e., the task of matching each point on a shape to its exact corresponding point on the other shape (or to be more specific, matching at geodesic error 0), most existing methods do not perform well due to two main problems. First, on nearly-isometric shapes (i.e., low noise levels), most existing methods use the eigen-vectors (eigen-functions) of the Laplace Beltrami Operator (LBO) or other shape descriptors to update an initialized correspondence which is not exact, leading to an accumulation of update errors. Thus, though the final correspondence may generally be smooth, it is generally inexact. Second, on non-isometric shapes (noisy shapes), existing methods are generally not robust to noise as they usually assume near-isometry. In addition, existing methods that attempt to address the non-isometric shape problem (e.g., GRAMPA) are generally computationally expensive and do not generalise to nearly-isometric shapes. To address these two problems, we propose a 2D graph convolution-based framework called 2D-GEM. 2D-GEM is robust to noise on non-isometric shapes and with a few additional constraints, it also addresses the errors in the update on nearly-isometric shapes. We demonstrate the effectiveness of 2D-GEM by achieving a high accuracy of 90.5$\%$ at geodesic error 0 on the non-isometric benchmark SHREC16, i.e., TOPKIDS (while being much faster than GRAMPA), and on nearly-isometric benchmarks by achieving a high accuracy of 92.5$\%$ on TOSCA and 84.9$\%$ on SCAPE at geodesic error 0.

Attributed Graph Clustering (AGC) and Attributed Hypergraph Clustering (AHC) are important topics in graph mining with many applications. For AGC, amongst the unsupervised methods that combine the graph structure with node attributes, graph convolution has been shown to achieve impressive results. However, the effects of graph convolution on AGC have not yet been adequately studied. In this paper, we show that graph convolution attempts to find the best trade-off between node attribute distance and the number of inter-cluster edges. On the one hand, we show that compared to clustering node attributes directly, graph convolution produces a greater distance between node attributes in the same cluster and a smaller distance between node attributes in different clusters (which is detrimental for clustering). On the other hand, we show that graph convolution benefits clustering by considerably reducing the number of edges among different clusters. We then extend our result on AGC to AHC and leverage the hypergraph convolution to propose an unsupervised, fast, and memory-efficient algorithm (GRAC) for AHC, which achieves excellent performance on popular supervised clustering measures.