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183
A Survey of Kernels for Structured Data
, 2003
"... Kernel methods in general and support vector machines in particular have been successful in various learning tasks on data represented in a single table. Much ‘realworld’ data, however, is structured – it has no natural representation in a single table. Usually, to apply kernel methods to ‘realwor ..."
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Cited by 146 (2 self)
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Kernel methods in general and support vector machines in particular have been successful in various learning tasks on data represented in a single table. Much ‘realworld’ data, however, is structured – it has no natural representation in a single table. Usually, to apply kernel methods to ‘realworld’ data, extensive preprocessing is performed to embed the data into a real vector space and thus in a single table. This survey describes several approaches of defining positive definite kernels on structured instances directly.
Graph Kernels
, 2007
"... We present a unified framework to study graph kernels, special cases of which include the random walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexit ..."
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Cited by 94 (9 self)
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We present a unified framework to study graph kernels, special cases of which include the random walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexity of kernel computation between unlabeled graphs with n vertices from O(n 6) to O(n 3). We find a spectral decomposition approach even more efficient when computing entire kernel matrices. For labeled graphs we develop conjugate gradient and fixedpoint methods that take O(dn 3) time per iteration, where d is the size of the label set. By extending the necessary linear algebra to Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for ddimensional edge kernels, and O(n 4) in the infinitedimensional case; on sparse graphs these algorithms only take O(n 2) time per iteration in all cases. Experiments on graphs from bioinformatics and other application domains show that these techniques can speed up computation of the kernel by an order of magnitude or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to Rconvolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment kernel of Fröhlich et al. (2006) yet provably positive semidefinite.
Kernels and Distances for Structured Data
 Machine Learning
, 2004
"... This paper brings together two strands of machine learning of increasing importance: kernel methods and highly structured data. We propose a general method for constructing a kernel following the syntactic structure of the data, as defined by its type signature in a higherorder logic. Our main theo ..."
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Cited by 65 (3 self)
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This paper brings together two strands of machine learning of increasing importance: kernel methods and highly structured data. We propose a general method for constructing a kernel following the syntactic structure of the data, as defined by its type signature in a higherorder logic. Our main theoretical result is the positive definiteness of any kernel thus defined. We report encouraging experimental results on a range of realworld datasets. By converting our kernel to a distance pseudometric for 1nearest neighbour, we were able to improve the best accuracy from the literature on the Diterpene dataset by more than 10%.
ShortestPath Kernels on Graphs
 In Proceedings of the 2005 International Conference on Data Mining
, 2005
"... 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 bee ..."
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Cited by 61 (5 self)
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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 NPhard, 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 shortestpath kernels show significantly higher classification accuracy than walkbased kernels. 1
Graph Kernels for Chemical Informatics
, 2005
"... Increased availability of large repositories of chemical compounds is creating new challenges and opportunities for the application of machine learning methods to problems in computational chemistry and chemical informatics. Because chemical compounds are often represented by the graph of their cova ..."
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Cited by 58 (7 self)
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Increased availability of large repositories of chemical compounds is creating new challenges and opportunities for the application of machine learning methods to problems in computational chemistry and chemical informatics. Because chemical compounds are often represented by the graph of their covalent bonds, machine learning methods in this domain must be capable of processing graphical structures with variable size. Here we first briefly review the literature on graph kernels and then introduce three new kernels (Tanimoto, MinMax, Hybrid) based on the idea of molecular fingerprints and counting labeled paths of depth up to d using depthfirst search from each possible vertex. The kernels are applied to three classification problems to predict mutagenicity, toxicity, and anticancer activity on three publicly available data sets. The kernels achieve performances at least comparable, and most often superior, to those previously reported in the literature reaching accuracies of 91.5 % on the Mutag dataset, 6567 % on the PTC (Predictive Toxicology Challenge) dataset, and 72 % on the NCI (National Cancer Institute) dataset. Properties and tradeoffs of these kernels, as well as other proposed kernels that leverage 1D or 3D representations of molecules, are briefly discussed.
Expressivity versus efficiency of graph kernels
 Proceedings of the First International Workshop on Mining Graphs, Trees and Sequences
, 2003
"... Abstract. Recently, kernel methods have become a popular tool for machine learning and data mining. As most ‘realworld ’ 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 dat ..."
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Cited by 54 (0 self)
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Abstract. Recently, kernel methods have become a popular tool for machine learning and data mining. As most ‘realworld ’ 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 tradeoff 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. 1
Characterizing structural relationships in scenes using graph kernels
 In ACM TOG
, 2011
"... Modeling virtual environments is a time consuming and expensive task that is becoming increasingly popular for both professional and casual artists. The model density and complexity of the scenes representing these virtual environments is rising rapidly. This trend suggests that datamining a 3D sc ..."
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Cited by 51 (5 self)
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Modeling virtual environments is a time consuming and expensive task that is becoming increasingly popular for both professional and casual artists. The model density and complexity of the scenes representing these virtual environments is rising rapidly. This trend suggests that datamining a 3D scene corpus to facilitate collaborative content creation could be a very powerful tool enabling more efficient scene design. In this paper, we show how to represent scenes as graphs that encode models and their semantic relationships. We then define a kernel between these relationship graphs that compares common virtual substructures in two graphs and captures the similarity between their corresponding scenes. We apply this framework to several scene modeling problems, such as finding similar scenes, relevance feedback, and contextbased model search. We show that incorporating structural relationships allows our method to provide a more relevant set of results when compared against previous approaches to model context search.
Efficient graphlet kernels for large graph comparison
, 2009
"... Stateoftheart graph kernels do not scale to large graphs with hundreds of nodes and thousands of edges. In this article we propose to compare graphs by counting graphlets, i.e., subgraphs with k nodes where k ∈ {3, 4, 5}. Exhaustive enumeration of all graphlets being prohibitively expensive, we i ..."
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Cited by 50 (10 self)
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Stateoftheart graph kernels do not scale to large graphs with hundreds of nodes and thousands of edges. In this article we propose to compare graphs by counting graphlets, i.e., subgraphs with k nodes where k ∈ {3, 4, 5}. Exhaustive enumeration of all graphlets being prohibitively expensive, we introduce two theoretically grounded speedup schemes, one based on sampling and the second one specifically designed for bounded degree graphs. In our experimental evaluation, our novel kernels allow us to efficiently compare large graphs that cannot be tackled by existing graph kernels.
Image classification with segmentation graph kernels
 In Proc. CVPR
, 2007
"... We propose a family of kernels between images, defined as kernels between their respective segmentation graphs. The kernels are based on soft matching of subtreepatterns of the respective graphs, leveraging the natural structure of images while remaining robust to the associated segmentation proces ..."
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Cited by 47 (12 self)
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We propose a family of kernels between images, defined as kernels between their respective segmentation graphs. The kernels are based on soft matching of subtreepatterns of the respective graphs, leveraging the natural structure of images while remaining robust to the associated segmentation process uncertainty. Indeed, output from morphological segmentation is often represented by a labelled graph, each vertex corresponding to a segmented region, with edges joining neighboring regions. However, such image representations have mostly remained underused for learning tasks, partly because of the observed instability of the segmentation process and the inherent hardness of inexact graph matching with uncertain graphs. Our kernels count common virtual substructures amongst images, which enables to perform efficient supervised classification of natural images with a support vector machine. Moreover, the kernel machinery allows us to take advantage of recent advances in kernelbased learning: i) semisupervised learning reduces the required number of labelled images, while ii) multiple kernel learning algorithms efficiently select the most relevant similarity measures between images within our family. 1.