Self-smoothing Effect In Graph Convolutional Neural Networks And Its Essence In Riemannian Manifolds

Graph convolutional network is now an effective tool to deal with non-Euclidean data,such as social behavior analysis,molecular structure analysis,and skeleton-based action recognition.Graph convolutional kernel is one of the most significant factors in graph convolutional networks to extract nodes’ feature,and some variants of it have achieved highly satisfactory performance theoretically and experimentally.However,there is limited research about how exactly different graph structures influence the performance of these kernels.Some existing methods still not explores the internal reasons.In our thesis,we firstly start from theoretical analysis of the spectral graph and study the properties of existing graph convolutional kernels,revealing the self-smoothing phenomenon and its effect in specific structured graphs,such as Small-Gap,Large-Gap and Fully connected graphs.On this basis,we propose the Poisson kernel that can avoid self-smoothing without training any adaptive kernel.Experimental results demonstrate that our Poisson kernel not only works well on the benchmark datasets where state-of-the-art methods work fine,but also is evidently superior to them in synthetic datasets.On the other hand,as a tool to describe non-Euclidean geometry in mathematical fields,Riemannian geometry has been applied in manifold learning,which is an earlier algorithm in machine learning.The algorithm of manifold learning points out that there is tight correlation between Riemannian manifolds and topological graphs.So if we can apply the method of Riemannian geometry into graph convolutional network,it is of great significance to the theoretical development of graph convolutional network.From the point of view of Riemannian geometry,by studying the isoperimetric problem and the minimum cut problem of Riemannian manifolds,our thesis demonstrates the isoperimetric inequality on the sphere and the minimum cut problem of some specific Riemannian manifolds in detail,and gives the definition of the self-smoothing effect on Riemannian manifolds step by step.By means of Stokes theorem in graphs,our thesis points out that there is a high degree of similarity between graph structures and Riemannian manifolds.In addition,from the perspective of minimum cut problem,our thesis verifies the consistency of the self-smoothing effect respectively on Riemannian manifold and graphs,and its form is re-examined from the perspective of geometry in our paper,which is also discussed using graph theory on the specific graphs,our thesis further reveals the correlation of the Riemannian manifolds and graphs,and points out the essence of the self-smoothing effect on Riemannian manifolds.