Global graph kernels using geometric embeddings (original) (raw)
Related papers
arXiv (Cornell University), 2021
The convolution operator at the core of many modern neural architectures can effectively be seen as performing a dot product between an input matrix and a filter. While this is readily applicable to data such as images, which can be represented as regular grids in the Euclidean space, extending the convolution operator to work on graphs proves more challenging, due to their irregular structure. In this paper we propose to use graph kernels, i.e., kernel functions that compute an inner product on graphs, to extend the standard convolution operator to the graph domain. This allows us to define an entirely structural model that does not require computing the embedding of the input graph. Our architecture allows to plug-in any type and number of graph kernels and has the added benefit of providing some interpretability in terms of the structural masks that are learned during the training process, similarly to what happens for convolutional masks in traditional convolutional neural networks. We perform an extensive ablation study to investigate the impact of the model hyper-parameters and we show that our model achieves competitive performance on standard graph classification datasets.
Unfolding Kernel embeddings of graphs: Enhancing class separation through manifold learning
Pattern Recognition, 2015
In this paper, we investigate the use of manifold learning techniques to enhance the separation properties of standard graph kernels. The idea stems from the observation that when we perform multidimensional scaling on the distance matrices extracted from the kernels, the resulting data tends to be clustered along a curve that wraps around the embedding space, a behaviour that suggests that long range distances are not estimated accurately, resulting in an increased curvature of the embedding space. Hence, we propose to use a number of manifold learning techniques to compute a low-dimensional embedding of the graphs in an attempt to unfold the embedding manifold, and increase the class separation.
Rethinking Kernel Methods for Node Representation Learning on Graphs
2019
Graph kernels are kernel methods measuring graph similarity and serve as a standard tool for graph classification. However, the use of kernel methods for node classification, which is a related problem to graph representation learning, is still ill-posed and the state-of-the-art methods are heavily based on heuristics. Here, we present a novel theoretical kernel-based framework for node classification that can bridge the gap between these two representation learning problems on graphs. Our approach is motivated by graph kernel methodology but extended to learn the node representations capturing the structural information in a graph. We theoretically show that our formulation is as powerful as any positive semidefinite kernels. To efficiently learn the kernel, we propose a novel mechanism for node feature aggregation and a data-driven similarity metric employed during the training phase. More importantly, our framework is flexible and complementary to other graph-based deep learning ...
Kernels on Graphs as Proximity Measures
Lecture Notes in Computer Science, 2017
Kernels and, broadly speaking, similarity measures on graphs are extensively used in graph-based unsupervised and semi-supervised learning algorithms as well as in the link prediction problem. We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This can potentially be useful for recommending the adoption of one or another similarity measure in a machine learning method. Also, we numerically compare various similarity measures in the context of spectral clustering and observe that normalized heat-type similarity measures with log modification generally perform the best.
On Graph Kernels: Hardness Results and Efficient Alternatives
2003
As most 'real-world' data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. An interesting and important challenge is thus to investigate kernels on instances that are represented by graphs. So far, only very specific graphs such as trees and strings have been considered. This paper investigates kernels on labeled directed graphs with general structure. It is shown that computing a strictly positive definite graph kernel is at least as hard as solving the graph isomorphism problem. It is also shown that computing an inner product in a feature space indexed by all possible graphs, where each feature counts the number of subgraphs isomorphic to that graph, is NP-hard. On the other hand, inner products in an alternative feature space, based on walks in the graph, can be computed in polynomial time. Such kernels are defined in this paper.
Generalized Shortest Path Kernel on Graphs
Lecture Notes in Computer Science, 2015
We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the number and length of shortest paths between nodes. For our example classification problem, we consider the task of classifying random graphs from two well-known families, by the number of clusters they contain. We verify empirically that the generalized shortest path kernel outperforms the original shortest path kernel on a number of datasets. We give a theoretical analysis for explaining our experimental results. In particular, we estimate distributions of the expected feature vectors for the shortest path kernel and the generalized shortest path kernel, and we show some evidence explaining why our graph kernel outperforms the shortest path kernel for our graph classification problem.
Graph Neural Networks with Composite Kernels
ArXiv, 2020
Learning on graph structured data has drawn increasing interest in recent years. Frameworks like Graph Convolutional Networks (GCNs) have demonstrated their ability to capture structural information and obtain good performance in various tasks. In these frameworks, node aggregation schemes are typically used to capture structural information: a node's feature vector is recursively computed by aggregating features of its neighboring nodes. However, most of aggregation schemes treat all connections in a graph equally, ignoring node feature similarities. In this paper, we re-interpret node aggregation from the perspective of kernel weighting, and present a framework to consider feature similarity in an aggregation scheme. Specifically, we show that normalized adjacency matrix is equivalent to a neighbor-based kernel matrix in a Krein Space. We then propose feature aggregation as the composition of the original neighbor-based kernel and a learnable kernel to encode feature similarit...
RetGK: Graph Kernels based on Return Probabilities of Random Walks
arXiv (Cornell University), 2018
Graph-structured data arise in wide applications, such as computer vision, bioinformatics, and social networks. Quantifying similarities among graphs is a fundamental problem. In this paper, we develop a framework for computing graph kernels, based on return probabilities of random walks. The advantages of our proposed kernels are that they can effectively exploit various node attributes, while being scalable to large datasets. We conduct extensive graph classification experiments to evaluate our graph kernels. The experimental results show that our graph kernels significantly outperform existing state-of-the-art approaches in both accuracy and computational efficiency.
Generating the Graph Gestalt: Kernel-Regularized Graph Representation Learning
ArXiv, 2021
Recent work on graph generative models has made remarkable progress towards generating increasingly realistic graphs, as measured by global graph features such as degree distribution, density, and clustering coefficients. Deep generative models have also made significant advances through better modelling of the local correlations in the graph topology, which have been very useful for predicting unobserved graph components, such as the existence of a link or the class of a node, from nearby observed graph components. A complete scientific understanding of graph data should address both global and local structure. In this paper, we propose a joint model for both as complementary objectives in a graph VAE framework. Global structure is captured by incorporating graph kernels in a probabilistic model whose loss function is closely related to the maximum mean discrepancy (MMD) between the global structures of the reconstructed and the input graphs. The ELBO objective derived from the mod...
Graph Classification via Neural Networks
2016
For a long time, the preferred machine learning algorithms for doing graph classification have been kernel based. The reasoning has been that kernels represent an elegant way to handle structured data that cannot be easily represented using numerical vectors or matrices. An important reason for the success of kernel methods, is the ’kernel trick’, which essentially replaces computing the feature representation, with a call to a kernel function, thus saving computation and memory cost. For some of the most successful kernels in the graph domain however, such as graphlets, this is not feasible, and one must compute the entire feature distribution in order to obtain the kernel. We present experimental evidence that using graphlet features presented to different neural networks gives comparable accuracy results to kernelized SVMs. As neural networks are parametric models that scale well with data size and can yield faster predictions than SVMs, our results suggest that they are attracti...