This document provides the foundations behind the functionality provided by the @math G library (this https URL), focusing on the basic operations the library provides: subsumption, refinement of directed labeled graphs, and distance similarity assessment between directed labeled graphs. @math G development was initially supported by the National Science Foundation, by the EAGER grant IIS-1551338.
Another related area is that of kernels for structured representations, which allow the application of techniques such as Support Vector Machines to structured data. Typically, kernels for graphs are based on the idea of finding common substructures between two graphs. For example, @cite_37 present a kernel for graphs based on random walks. @cite_5 also studied how to encapsulate their similarity measure for Description Logics into a kernel. For a survey on kernels for structured data the reader is referred to @cite_31 .
@cite_5 @cite_37 @cite_31
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Recently, kernel methods have become a popular tool for machine learning and data mining. 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. In this paper we study the trade-off between expressivity and efficiency of graph kernels. First, we motivate the need for this discussion by showing that fully general graph kernels can not even be approximated efficiently. We also discuss generalizations of graph kernels defined in literature and show that they are either not positive definite or not very useful. Finally, we propose a new graph kernel based on subtree patterns. We argue that while a little more computationally expensive, this kernel is more expressive than kernels based on walks. We introduce a novel kernel that upgrades the Weisfeiler-Lehman and other graph kernels to effectively exploit high-dimensional and continuous vertex attributes. Graphs are first decomposed into subgraphs. Vertices of the subgraphs are then compared by a kernel that combines the similarity of their labels and the similarity of their structural role, using a suitable vertex invariant. By changing this invariant we obtain a family of graph kernels which includes generalizations of Weisfeiler-Lehman, NSPDK, and propagation kernels. We demonstrate empirically that these kernels obtain state-of-the-art results on relational data sets. Positive definite kernels between labeled graphs have recently been proposed. They enable the application of kernel methods, such as support vector machines, to the analysis and classification of graphs, for example, chemical compounds. These graph kernels are obtained by marginalizing a kernel between paths with respect to a random walk model on the graph vertices along the edges. We propose two extensions of these graph kernels, with the double goal to reduce their computation time and increase their relevance as measure of similarity between graphs. First, we propose to modify the label of each vertex by automatically adding information about its environment with the use of the Morgan algorithm. Second, we suggest a modification of the random walk model to prevent the walk from coming back to a vertex that was just visited. These extensions are then tested on benchmark experiments of chemical compounds classification, with promising results. In this paper, we present Deep Graph Kernels, a unified framework to learn latent representations of sub-structures for graphs, inspired by latest advancements in language modeling and deep learning. Our framework leverages the dependency information between sub-structures by learning their latent representations. We demonstrate instances of our framework on three popular graph kernels, namely Graphlet kernels, Weisfeiler-Lehman subtree kernels, and Shortest-Path graph kernels. Our experiments on several benchmark datasets show that Deep Graph Kernels achieve significant improvements in classification accuracy over state-of-the-art graph kernels. With applications in biology, the world-wide web, and several other areas, mining of graph-structured objects has received significant interest recently. One of the major research directions in this field is concerned with predictive data mining in graph databases where each instance is represented by a graph. Some of the proposed approaches for this task rely on the excellent classification performance of support vector machines. To control the computational cost of these approaches, the underlying kernel functions are based on frequent patterns. In contrast to these approaches, we propose a kernel function based on a natural set of cyclic and tree patterns independent of their frequency, and discuss its computational aspects. To practically demonstrate the effectiveness of our approach, we use the popular NCI-HIV molecule dataset. Our experimental results show that cyclic pattern kernels can be computed quickly and offer predictive performance superior to recent graph kernels based on frequent patterns. Many real world graphs, such as the graphs of molecules, exhibit structure at multiple different scales, but most existing kernels between graphs are either purely local or purely global in character. In contrast, by building a hierarchy of nested subgraphs, the Multiscale Laplacian Graph kernels (MLG kernels) that we define in this paper can account for structure at a range of different scales. At the heart of the MLG construction is another new graph kernel, called the Feature Space Laplacian Graph kernel (FLG kernel), which has the property that it can lift a base kernel defined on the vertices of two graphs to a kernel between the graphs. The MLG kernel applies such FLG kernels to subgraphs recursively. To make the MLG kernel computationally feasible, we also introduce a randomized projection procedure, similar to the Nystr "om method, but for RKHS operators. The paper presents a kernel for learning from ordered hypergraphs, a formalization that captures relational data as used in Inductive Logic Programming (ILP). The kernel generalizes previous approaches to graph kernels in calculating similarity based on walks in the hypergraph. Experiments on challenging chemical datasets demonstrate that the kernel outperforms existing ILP methods, and is competitive with state-of-the-art graph kernels. The experiments also demonstrate that the encoding of graph data can affect performance dramatically, a fact that can be useful beyond kernel methods. 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. Kernel classifiers and regressors designed for structured data, such as sequences, trees and graphs, have significantly advanced a number of interdisciplinary areas such as computational biology and drug design. Typically, kernels are designed beforehand for a data type which either exploit statistics of the structures or make use of probabilistic generative models, and then a discriminative classifier is learned based on the kernels via convex optimization. However, such an elegant two-stage approach also limited kernel methods from scaling up to millions of data points, and exploiting discriminative information to learn feature representations. We propose, structure2vec, an effective and scalable approach for structured data representation based on the idea of embedding latent variable models into feature spaces, and learning such feature spaces using discriminative information. Interestingly, structure2vec extracts features by performing a sequence of function mappings in a way similar to graphical model inference procedures, such as mean field and belief propagation. In applications involving millions of data points, we showed that structure2vec runs 2 times faster, produces models which are @math times smaller, while at the same time achieving the state-of-the-art predictive performance. Learning from complex data is becoming increasingly important, and graph kernels have recently evolved into a rapidly developing branch of learning on structured data. However, previously proposed kernels rely on having discrete node label information. In this paper, we explore the power of continuous node-level features for propagation-based graph kernels. Specifically, propagation kernels exploit node label distributions from propagation schemes like label propagation, which naturally enables the construction of graph kernels for partially labeled graphs. In order to efficiently extract graph features from continuous node label distributions, and in general from continuous vector-valued node attributes, we utilize randomized techniques, which easily allow for deriving similarity measures based on propagated information. We show that propagation kernels utilizing locality-sensitive hashing reduce the runtime of existing graph kernels by several orders of magnitude. We evaluate the performance of various propagation kernels on real-world bioinformatics and image benchmark datasets. Data mining algorithms are facing the challenge to deal with an increasing number of complex objects. For graph data, a whole toolbox of data mining algorithms becomes available by defining a kernel function on instances of graphs. Graph kernels based on walks, subtrees and cycles in graphs have been proposed so far. As a general problem, these kernels are either computationally expensive or limited in their expressiveness. We try to overcome this problem by defining expressive graph kernels which are based on paths. As the computation of all paths and longest paths in a graph is NP-hard, we propose graph kernels based on shortest paths. These kernels are computable in polynomial time, retain expressivity and are still positive definite. In experiments on classification of graph models of proteins, our shortest-path kernels show significantly higher classification accuracy than walk-based kernels. In this paper, we present a hyper graph kernel computed using substructure isomorphism tests. Measuring the isomorphisms between hyper graphs straightforwardly tends to be elusive since a hyper graph may exhibit varying relational orders. We thus transform a hyper graph into a directed line graph. This not only accurately reflects the multiple relationships exhibited by the hyper graph but is also easier to manipulate isomorphism tests. To locate the isomorphisms between hyper graphs through their directed line graphs, we propose a new directed Weisfeiler-Lehman isomorphism test for directed graphs. The new isomorphism test precisely reflects the structure of the directed edges. By identifying the isomorphic substructures of directed graphs, the hyper graph kernel for a pair of hyper graphs is computed by counting the number of pair wise isomorphic substructures from their directed line graphs. We show that our kernel limits tottering that arises in the existing walk and sub tree based (hyper)graph kernels. Experiments on challenging (hyper)graph datasets demonstrate the effectiveness of our kernel. This paper presents a survey as well as an empirical comparison and evaluation of seven kernels on graphs and two related similarity matrices, that we globally refer to as ''kernels on graphs'' for simplicity. They are the exponential diffusion kernel, the Laplacian exponential diffusion kernel, the von Neumann diffusion kernel, the regularized Laplacian kernel, the commute-time (or resistance-distance) kernel, the random-walk-with-restart similarity matrix, and finally, a kernel first introduced in this paper (the regularized commute-time kernel) and two kernels defined in some of our previous work and further investigated in this paper (the Markov diffusion kernel and the relative-entropy diffusion matrix). The kernel-on-graphs approach is simple and intuitive. It is illustrated by applying the nine kernels to a collaborative-recommendation task, viewed as a link prediction problem, and to a semisupervised classification task, both on several databases. The methods compute proximity measures between nodes that help study the structure of the graph. Our comparisons suggest that the regularized commute-time and the Markov diffusion kernels perform best on the investigated tasks, closely followed by the regularized Laplacian kernel. In this article, we propose fast subtree kernels on graphs. On graphs with n nodes and m edges and maximum degree d, these kernels comparing subtrees of height h can be computed in O(mh), whereas the classic subtree kernel by Ramon & Gartner scales as O(n24dh). Key to this efficiency is the observation that the Weisfeiler-Lehman test of isomorphism from graph theory elegantly computes a subtree kernel as a byproduct. Our fast subtree kernels can deal with labeled graphs, scale up easily to large graphs and outperform state-of-the-art graph kernels on several classification benchmark datasets in terms of accuracy and runtime. The availability of large scale data sets of manually annotated predicate-argument structures has recently favored the use of machine learning approaches to the design of automated semantic role labeling (SRL) systems. The main research in this area relates to the design choices for feature representation and for effective decompositions of the task in different learning models. Regarding the former choice, structural properties of full syntactic parses are largely employed as they represent ways to encode different principles suggested by the linking theory between syntax and semantics. The latter choice relates to several learning schemes over global views of the parses. For example, re-ranking stages operating over alternative predicate-argument sequences of the same sentence have shown to be very effective. In this article, we propose several kernel functions to model parse tree properties in kernel-based machines, for example, perceptrons or support vector machines. In particular, we define different kinds of tree kernels as general approaches to feature engineering in SRL. Moreover, we extensively experiment with such kernels to investigate their contribution to individual stages of an SRL architecture both in isolation and in combination with other traditional manually coded features. The results for boundary recognition, classification, and re-ranking stages provide systematic evidence about the significant impact of tree kernels on the overall accuracy, especially when the amount of training data is small. As a conclusive result, tree kernels allow for a general and easily portable feature engineering method which is applicable to a large family of natural language processing tasks. While graphs with continuous node attributes arise in many applications, state-of-the-art graph kernels for comparing continuous-attributed graphs suffer from a high runtime complexity. For instance, the popular shortest path kernel scales as O(n4), where n is the number of nodes. In this paper, we present a class of graph kernels with computational complexity O(n2(m + log n + δ2 + d)), where δ is the graph diameter, m is the number of edges, and d is the dimension of the node attributes. Due to the sparsity and small diameter of real-world graphs, these kernels typically scale comfortably to large graphs. In our experiments, the presented kernels outperform state-of-the-art kernels in terms of speed and accuracy on classification benchmark datasets. Current graph kernels suffer from two limitations: graph kernels based on counting particular types of subgraphs ignore the relative position of these subgraphs to each other, while graph kernels based on algebraic methods are limited to graphs without node labels. In this paper we present the graphlet spectrum, a system of graph invariants derived by means of group representation theory that capture information about the number as well as the position of labeled subgraphs in a given graph. In our experimental evaluation the graphlet spectrum outperforms state-of-the-art graph kernels. Kernel methods provide a powerful and unified framework for pattern discovery, motivating algorithms that can act on general types of data (e.g. strings, vectors or text) and look for general types of relations (e.g. rankings, classifications, regressions, clusters). The application areas range from neural networks and pattern recognition to machine learning and data mining. This book, developed from lectures and tutorials, fulfils two major roles: firstly it provides practitioners with a large toolkit of algorithms, kernels and solutions ready to use for standard pattern discovery problems in fields such as bioinformatics, text analysis, image analysis. Secondly it provides an easy introduction for students and researchers to the growing field of kernel-based pattern analysis, demonstrating with examples how to handcraft an algorithm or a kernel for a new specific application, and covering all the necessary conceptual and mathematical tools to do so. In this paper we have designed and experimented novel convolution kernels for automatic classification of predicate arguments. Their main property is the ability to process structured representations. Support Vector Machines (SVMs), using a combination of such kernels and the flat feature kernel, classify Prop-Bank predicate arguments with accuracy higher than the current argument classification state-of-the-art.Additionally, experiments on FrameNet data have shown that SVMs are appealing for the classification of semantic roles even if the proposed kernels do not produce any improvement. Motivation: Kernel methods such as support vector machines require a kernel function between objects to be defined a priori. Several works have been done to derive kernels from probability distributions, e.g., the Fisher kernel. However, a general methodology to design a kernel is not fully developed. Results: We propose a reasonable way of designing a kernel when objects are generated from latent variable models (e.g., HMM). First of all, a joint kernel is designed for complete data which include both visible and hidden variables. Then a marginalized kernel for visible data is obtained by taking the expectation with respect to hidden variables. We will show that the Fisher kernel is a special case of marginalized kernels, which gives another viewpoint to the Fisher kernel theory. Although our approach can be applied to any object, we particularly derive several marginalized kernels useful for biological sequences (e.g., DNA and proteins). The effectiveness of marginalized kernels is illustrated in the task of classifying bacterial gyrase subunit B (gyrB) amino acid sequences.