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.

The advantages of discriminative learning algorithms and kernel machines are combined with generative modeling using a novel kernel between distributions. In the probability product kernel, data points in the input space are mapped to distributions over the sample space and a general inner product is then evaluated as the integral of the product of pairs of distributions. The kernel is straightforward to evaluate for all exponential family models such as multinomials and Gaussians and yields interesting nonlinear kernels. Furthermore, the kernel is computable in closed form for latent distributions such as mixture models, hidden Markov models and linear dynamical systems. For intractable models, such as switching linear dynamical systems, structured mean-field approximations can be brought to bear on the kernel evaluation. For general distributions, even if an analytic expression for the kernel is not feasible, we show a straightforward sampling method to evaluate it. Thus, the kernel permits discriminative learning methods, including support vector machines, to exploit the properties, metrics and invariances of the generative models we infer from each datum. Experiments are shown using multinomial models for text, hidden Markov models for biological data sets and linear dynamical systems for time series data.

Relational reinforcement learning is a Q-learning technique for relational state-action spaces. It aims to enable agents to learn how to act in an environment that has no natural representation as a tuple of constants. In this case, the learning algorithm used to approximate the mapping between state-action pairs and their so called Q(uality)-value has to be not only very reliable, but it also has to be able to handle the relational representation of state-action pairs. In this paper we investigate...

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.

Abstract. We derive a family of kernels on dynamical systems by applying the Binet-Cauchy theorem to trajectories of states. Our derivation provides a unifying framework for all kernels on dynamical systems currently used in machine learning, including kernels derived from the behavioral framework, diffusion processes, marginalized kernels, kernels on graphs, and the kernels on sets arising from the subspace angle approach. In the case of linear time-invariant systems, we derive explicit formulae for computing the proposed Binet-Cauchy kernels by solving Sylvester equations, and relate the proposed kernels to existing kernels based on cepstrum coefficients and subspace angles. Besides their theoretical appeal, these kernels can be used efficiently in the comparison of video sequences of dynamic scenes that can be modeled as the output of a linear time-invariant dynamical system. One advantage of our kernels is that they take the initial conditions of the dynamical systems into account. As a first example, we use our kernels to compare video sequences of dynamic textures. As a second example, we apply our kernels to the problem of clustering short clips of a movie. Experimental evidence shows superior performance of our kernels. Keywords: Binet-Cauchy theorem, ARMA models and dynamical systems, Sylvester

Reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high-dimensional data analysis. Fisher discriminant analysis (FDA) is a traditional technique for supervised dimensionality reduction, but it tends to give undesired results if samples in a class are multimodal. An unsupervised dimensionality reduction method called localitypreserving projection (LPP) can work well with multimodal data due to its locality preserving property. However, since LPP does not take the label information into account, it is not necessarily useful in supervised learning scenarios. In this paper, we propose a new linear supervised dimensionality reduction method called local Fisher discriminant analysis (LFDA), which effectively combines the ideas of FDA and LPP. LFDA has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem. We demonstrate the practical usefulness and high scalability of the LFDA method in data visualization and classification tasks through extensive simulation studies. We also show that LFDA can be extended to non-linear dimensionality reduction scenarios by applying the kernel trick.

We review recent methods for learning with positive definite kernels. All these methods formulate learning and estimation problems as linear tasks in a reproducing kernel Hilbert space (RKHS) associated with a kernel. We cover a wide range of methods, ranging from simple classifiers to sophisticated methods for estimation with structured data.

Abstract. In recent years the development of computational techniques that build models to correctly assign chemical compounds to various classes or to retrieve potential drug-like compounds has been an active area of research. Many of the bestperforming techniques for these tasks utilize a descriptor-based representation of the compound that captures various aspects of the underlying molecular graph’s topology. In this paper we compare five different set of descriptors that are currently used for chemical compound classification. We also introduce four different descriptors derived from all connected fragments present in the molecular graphs primarily for the purpose of comparing them to the currently used descriptor spaces and analyzing what properties of descriptor spaces are helpful in providing effective representation for molecular graphs. In addition, we introduce an extension to existing vector-based kernel functions to take into account the length of the fragments present in the descriptors. We experimentally evaluate the performance of the previously introduced and the new descriptors in the context of SVM-based classification and ranked-retrieval on 28 classification and retrieval problems derived from 18 datasets. Our experiments show that for both of these tasks, two of the four descriptors introduced in this paper along with the extended connectivity fingerprint based descriptors consistently and statistically outperform previously developed schemes based on the widely used fingerprint- and Maccs keys-based descriptors, as well as recently introduced descriptors obtained by mining and analyzing the structure of the molecular graphs.

Attributed graphs are increasingly more common in many application domains such as chemistry, biology and text processing. A central issue in graph mining is how to collect informative subgraph patterns for a given learning task. We propose an iterative mining method based on partial least squares regression (PLS). To apply PLS to graph data, a sparse version of PLS is developed first and then it is combined with a weighted pattern mining algorithm. The mining algorithm is iteratively called with different weight vectors, creating one latent component per one mining call. Our method, graph PLS, is efficient and easy to implement, because the weight vector is updated with elementary matrix calculations. In experiments, our graph PLS algorithm showed competitive prediction accuracies in many chemical datasets and its efficiency was significantly superior to graph boosting (gBoost) and the naive method based on frequent graph mining.

It has become a promising direction to measure similarity of Web search queries by mining the increasing amount of clickthrough data logged by Web search engines, which record the interactions between users and the search engines. Most existing approaches employ the click-through data for similarity measure of queries with little consideration of the temporal factor, while the click-through data is often dynamic and contains rich temporal information. In this paper we present a new framework of time-dependent query semantic similarity model on exploiting the temporal characteristics of historical click-through data. The intuition is that more accurate semantic similarity values between queries can be obtained by taking into account the timestamps of the log data. With a set of user-defined calendar schema and calendar patterns, our time-dependent query similarity model is constructed using the marginalized kernel technique, which can exploit both explicit similarity and implicit semantics from the click-through data effectively. Experimental results on a large set of click-through data acquired from a commercial search engine show that our time-dependent query similarity model is more accurate than the existing approaches. Moreover, we observe that our time-dependent query similarity model can, to some extent, reflect real-world semantics such as real-world events that are happening over time.