In a graph convolutional network, we assume that the graph @math is generated with respect to some observation noise. We make small random perturbations @math of the graph and try to improve generalization. Based on quantum information geometry, we can have quantitative measurements on the scale of @math . We try to maximize the intrinsic scale of the permutation with a small budget while minimizing the loss based on the perturbed @math . Our proposed model can consistently improve graph convolutional networks on semi-supervised node classification tasks with reasonable computational overhead. We present two different types of geometry on the manifold of graphs: one is for measuring the intrinsic change of a graph; the other is for measuring how such changes can affect externally a graph neural network. These new analytical tools will be useful in developing a good understanding of graph neural networks and fostering new techniques.
Bethe Hessian @cite_31 , or deformed Laplacian, was shown to improve the performance of spectral clustering on a par with non-symmetric and higher dimensional operators, yet, drawing advantages of symmetric positive-definite representation. Our graph Laplacian parameterisation also draws on this view.
@cite_31
2964029874 2148070710 2511280622 857387926
Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the non-backtracking walk on the graph, leads to improved performance in detecting clusters, and even to optimal performance for the stochastic block model. Here, we propose to use instead a simpler object, a symmetric real matrix known as the Bethe Hessian operator, or deformed Laplacian. We show that this approach combines the performances of the non-backtracking operator, thus detecting clusters all the way down to the theoretical limit in the stochastic block model, with the computational, theoretical and memory advantages of real symmetric matrices. This paper presents a new spectral partitioning formulation which directly incorporates vertex size information by modifying the Laplacian of the graph. Modifying the Laplacian produces a generalized eigenvalue problem, which is reduced to the standard eigenvalue problem. Experiments show that the scaled ratio-cut costs of results on benchmarks with arbitrary vertex size improve by 22 when the eigenvectors of the Laplacian in the spectral partitioner KP are replaced by the eigenvectors of our modified Laplacian. The inability to handle vertex sizes in the spectral partitioning formulation has been a limitation in applying spectral partitioning in a multilevel setting. We investigate whether our new formulation effectively removes this limitation by combining it with a simple multilevel bottom-up clustering algorithm and an iterative improvement algorithm for partition refinement. Experiments show that in a multilevel setting where the spectral partitioner KP provides the initial partitions of the most contracted graph, using the modified Laplacian in place of the standard Laplacian is more efficient and more effective in the partitioning of graphs with arbitrary-size and unit-size vertices; average improvements of 17 and 18 are observed for graphs with arbitrary-size and unit-size vertices, respectively. Comparisons with other ratio-cut based partitioners on hypergraphs with unit-size as well as arbitrary-size vertices, show that the multilevel spectral partitioner produces either better results or almost identical results more efficiently. The cost of computing the spectrum of Laplacian matrices hinders the application of spectral clustering to large data sets. While approximations recover computational tractability, they can potentially affect clustering performance. This paper proposes a practical approach to learn spectral clustering based on adaptive stochastic gradient optimization. Crucially, the proposed approach recovers the exact spectrum of Laplacian matrices in the limit of the iterations, and the cost of each iteration is linear in the number of samples. Extensive experimental validation on data sets with up to half a million samples demonstrate its scalability and its ability to outperform state-of-the-art approximate methods to learn spectral clustering for a given computational budget. In this paper, we examine a spectral clustering algorithm for similarity graphs drawn from a simple random graph model, where nodes are allowed to have varying degrees, and we provide theoretical bounds on its performance. The random graph model we study is the Extended Planted Partition (EPP) model, a variant of the classical planted partition model. The standard approach to spectral clustering of graphs is to compute the bottom k singular vectors or eigenvectors of a suitable graph Laplacian, project the nodes of the graph onto these vectors, and then use an iterative clustering algorithm on the projected nodes. However a challenge with applying this approach to graphs generated from the EPP model is that unnormalized Laplacians do not work, and normalized Laplacians do not concentrate well when the graph has a number of low degree nodes. We resolve this issue by introducing the notion of a degree-corrected graph Laplacian. For graphs with many low degree nodes, degree correction has a regularizing eect on the Laplacian. Our spectral clustering algorithm projects the nodes in the graph onto the bottom k right singular vectors of the degree-corrected random-walk Laplacian, and clusters the nodes in this subspace. We show guarantees on the performance of this algorithm, demonstrating that it outputs the correct partition under a wide range of parameter values. Unlike some previous work, our algorithm does not require access to any generative parameters of the model.