Kernel Methods for Tree Structured Data

Machine learning comprises a series of techniques for automatic extraction of meaningful information from large collections of noisy data. In many real world applications, data is naturally represented in structured form. Since traditional methods in machine learning deal with vectorial information, they require an a priori form of preprocessing. Among all the learning techniques for dealing with structured data, kernel methods are recognized to have a strong theoretical background and to be effective approaches. They do not require an explicit vectorial representation of the data in terms of features, but rely on a measure of similarity between any pair of objects of a domain, the kernel function. Designing fast and good kernel functions is a challenging problem. In the case of tree structured data two issues become relevant: kernel for trees should not be sparse and should be fast to compute. The sparsity problem arises when, given a dataset and a kernel function, most structures of the dataset are completely dissimilar to one another. In those cases the classifier has too few information for making correct predictions on unseen data. In fact, it tends to produce a discriminating function behaving as the nearest neighbour rule. Sparsity is likely to arise for some standard tree kernel functions, such as the subtree and subset tree kernel, when they are applied to datasets with node labels belonging to a large domain. A second drawback of using tree kernels is the time complexity required both in learning and classification phases. Such a complexity can sometimes prevents the kernel application in scenarios involving large amount of data. This thesis proposes three contributions for resolving the above issues of kernel for trees. A first contribution aims at creating kernel functions which adapt to the statistical properties of the dataset, thus reducing its sparsity with respect to traditional tree kernel functions. Specifically, we propose to encode the input trees by an algorithm able to project the data onto a lower dimensional space with the property that similar structures are mapped similarly. By building kernel functions on the lower dimensional representation, we are able to perform inexact matchings between different inputs in the original space. A second contribution is the proposal of a novel kernel function based on the convolution kernel framework. Convolution kernel measures the similarity of two objects in terms of the similarities of their subparts. Most convolution kernels are based on counting the number of shared substructures, partially discarding information about their position in the original structure. The kernel function we propose is, instead, especially focused on this aspect. A third contribution is devoted at reducing the computational burden related to the calculation of a kernel function between a tree and a forest of trees, which is a typical operation in the classification phase and, for some algorithms, also in the learning phase. We propose a general methodology applicable to convolution kernels. Moreover, we show an instantiation of our technique when kernels such as the subtree and subset tree kernels are employed. In those cases, Direct Acyclic Graphs can be used to compactly represent shared substructures in different trees, thus reducing the computational burden and storage requirements.

Learning the kernel matrix with semidefinite programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive semidefinite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space - classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -using the labeled part of the data one can learn an embedding also for the unlabeled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method for learning the 2-norm soft margin parameter in support vector machines, solving an important open problem.

Active Hidden Models for Tracking with Kernel Projections

We introduce Active Hidden Models (AHM) that utilize kernel methods traditionally associated with classification. We use AHMs to track deformable objects in video sequences by leveraging kernel projections. We introduce the "subset projection" method which improves the efficiency of our tracking approach by a factor of ten. We successfully tested our method on facial tracking with extreme head movements (including full 180-degree head rotation), facial expressions, and deformable objects. Given a kernel and a set of training observations, we derive unbiased estimates of the accuracy of the AHM tracker. Kernels are generally used in classification methods to make training data linearly separable. We prove that the optimal (minimum variance) tracking kernels are those that make the training observations linearly dependent.

A comparison of statistical methods for genomic selection in a mice population

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Transitive state alignment for the quantum jensen-shannon kernel

Kernel methods provide a convenient way to apply a wide range of learning techniques to complex and structured data by shifting the representational problem from one of finding an embedding of the data to that of defining a positive semidefinite kernel. One problem with the most widely used kernels is that they neglect the locational information within the structures, resulting in less discrimination. Correspondence-based kernels, on the other hand, are in general more discriminating, at the cost of sacrificing positive-definiteness due to their inability to guarantee transitivity of the correspondences between multiple graphs. In this paper we generalize a recent structural kernel based on the Jensen-Shannon divergence between quantum walks over the structures by introducing a novel alignment step which rather than permuting the nodes of the structures, aligns the quantum states of their walks. This results in a novel kernel that maintains localization within the structures, but still guarantees positive definiteness. Experimental evaluation validates the effectiveness of the kernel for several structural classification tasks. © 2014 Springer-Verlag Berlin Heidelberg.

A continuous-time quantum walk kernel for unattributed graphs

Kernel methods provide a way to apply a wide range of learning techniques to complex and structured data by shifting the representational problem from one of finding an embedding of the data to that of defining a positive semidefinite kernel. In this paper, we propose a novel kernel on unattributed graphs where the structure is characterized through the evolution of a continuous-time quantum walk. More precisely, given a pair of graphs, we create a derived structure whose degree of symmetry is maximum when the original graphs are isomorphic. With this new graph to hand, we compute the density operators of the quantum systems representing the evolutions of two suitably defined quantum walks. Finally, we define the kernel between the two original graphs as the quantum Jensen-Shannon divergence between these two density operators. The experimental evaluation shows the effectiveness of the proposed approach. © 2013 Springer-Verlag.

Multiview point cloud kernels for semisupervised learning [Lecture Notes]

In semisupervised learning (SSL), a predictive model is learn from a collection of labeled data and a typically much larger collection of unlabeled data. These paper presented a framework called multi-view point cloud regularization (MVPCR), which unifies and generalizes several semisupervised kernel methods that are based on data-dependent regularization in reproducing kernel Hilbert spaces (RKHSs). Special cases of MVPCR include coregularized least squares (CoRLS), manifold regularization (MR), and graph-based SSL. An accompanying theorem shows how to reduce any MVPCR problem to standard supervised learning with a new multi-view kernel.

Sparseness vs estimating conditional probabilities : some asymptotic results

One of the nice properties of kernel classifiers such as SVMs is that they often produce sparse solutions. However, the decision functions of these classifiers cannot always be used to estimate the conditional probability of the class label. We investigate the relationship between these two properties and show that these are intimately related: sparseness does not occur when the conditional probabilities can be unambiguously estimated. We consider a family of convex loss functions and derive sharp asymptotic results for the fraction of data that becomes support vectors. This enables us to characterize the exact trade-off between sparseness and the ability to estimate conditional probabilities for these loss functions.

Trading spaces : on the lore and limitations of latent semantic analysis

Two decades after its inception, Latent Semantic Analysis(LSA) has become part and parcel of every modern introduction to Information Retrieval. For any tool that matures so quickly, it is important to check its lore and limitations, or else stagnation will set in. We focus here on the three main aspects of LSA that are well accepted, and the gist of which can be summarized as follows: (1) that LSA recovers latent semantic factors underlying the document space, (2) that such can be accomplished through lossy compression of the document space by eliminating lexical noise, and (3) that the latter can best be achieved by Singular Value Decomposition. For each aspect we performed experiments analogous to those reported in the LSA literature and compared the evidence brought to bear in each case. On the negative side, we show that the above claims about LSA are much more limited than commonly believed. Even a simple example may show that LSA does not recover the optimal semantic factors as intended in the pedagogical example used in many LSA publications. Additionally, and remarkably deviating from LSA lore, LSA does not scale up well: the larger the document space, the more unlikely that LSA recovers an optimal set of semantic factors. On the positive side, we describe new algorithms to replace LSA (and more recent alternatives as pLSA, LDA, and kernel methods) by trading its l2 space for an l1 space, thereby guaranteeing an optimal set of semantic factors. These algorithms seem to salvage the spirit of LSA as we think it was initially conceived.

Prediction of transcriptional regulatory interactions in bacteria : a comparative genomics approach

Exponential growth of genomic data in the last two decades has made manual analyses impractical for all but trial studies. As genomic analyses have become more sophisticated, and move toward comparisons across large datasets, computational approaches have become essential. One of the most important biological questions is to understand the mechanisms underlying gene regulation. Genetic regulation is commonly investigated and modelled through the use of transcriptional regulatory network (TRN) structures. These model the regulatory interactions between two key components: transcription factors (TFs) and the target genes (TGs) they regulate. Transcriptional regulatory networks have proven to be invaluable scientific tools in Bioinformatics. When used in conjunction with comparative genomics, they have provided substantial insights into the evolution of regulatory interactions. Current approaches to regulatory network inference, however, omit two additional key entities: promoters and transcription factor binding sites (TFBSs). In this study, we attempted to explore the relationships among these regulatory components in bacteria. Our primary goal was to identify relationships that can assist in reducing the high false positive rates associated with transcription factor binding site predictions and thereupon enhance the reliability of the inferred transcription regulatory networks. In our preliminary exploration of relationships between the key regulatory components in Escherichia coli transcription, we discovered a number of potentially useful features. The combination of location score and sequence dissimilarity scores increased de novo binding site prediction accuracy by 13.6%. Another important observation made was with regards to the relationship between transcription factors grouped by their regulatory role and corresponding promoter strength. Our study of E.coli ��70 promoters, found support at the 0.1 significance level for our hypothesis | that weak promoters are preferentially associated with activator binding sites to enhance gene expression, whilst strong promoters have more repressor binding sites to repress or inhibit gene transcription. Although the observations were specific to �70, they nevertheless strongly encourage additional investigations when more experimentally confirmed data are available. In our preliminary exploration of relationships between the key regulatory components in E.coli transcription, we discovered a number of potentially useful features { some of which proved successful in reducing the number of false positives when applied to re-evaluate binding site predictions. Of chief interest was the relationship observed between promoter strength and TFs with respect to their regulatory role. Based on the common assumption, where promoter homology positively correlates with transcription rate, we hypothesised that weak promoters would have more transcription factors that enhance gene expression, whilst strong promoters would have more repressor binding sites. The t-tests assessed for E.coli �70 promoters returned a p-value of 0.072, which at 0.1 significance level suggested support for our (alternative) hypothesis; albeit this trend may only be present for promoters where corresponding TFBSs are either all repressors or all activators. Nevertheless, such suggestive results strongly encourage additional investigations when more experimentally confirmed data will become available. Much of the remainder of the thesis concerns a machine learning study of binding site prediction, using the SVM and kernel methods, principally the spectrum kernel. Spectrum kernels have been successfully applied in previous studies of protein classification [91, 92], as well as the related problem of promoter predictions [59], and we have here successfully applied the technique to refining TFBS predictions. The advantages provided by the SVM classifier were best seen in `moderately'-conserved transcription factor binding sites as represented by our E.coli CRP case study. Inclusion of additional position feature attributes further increased accuracy by 9.1% but more notable was the considerable decrease in false positive rate from 0.8 to 0.5 while retaining 0.9 sensitivity. Improved prediction of transcription factor binding sites is in turn extremely valuable in improving inference of regulatory relationships, a problem notoriously prone to false positive predictions. Here, the number of false regulatory interactions inferred using the conventional two-component model was substantially reduced when we integrated de novo transcription factor binding site predictions as an additional criterion for acceptance in a case study of inference in the Fur regulon. This initial work was extended to a comparative study of the iron regulatory system across 20 Yersinia strains. This work revealed interesting, strain-specific difierences, especially between pathogenic and non-pathogenic strains. Such difierences were made clear through interactive visualisations using the TRNDifi software developed as part of this work, and would have remained undetected using conventional methods. This approach led to the nomination of the Yfe iron-uptake system as a candidate for further wet-lab experimentation due to its potential active functionality in non-pathogens and its known participation in full virulence of the bubonic plague strain. Building on this work, we introduced novel structures we have labelled as `regulatory trees', inspired by the phylogenetic tree concept. Instead of using gene or protein sequence similarity, the regulatory trees were constructed based on the number of similar regulatory interactions. While the common phylogentic trees convey information regarding changes in gene repertoire, which we might regard being analogous to `hardware', the regulatory tree informs us of the changes in regulatory circuitry, in some respects analogous to `software'. In this context, we explored the `pan-regulatory network' for the Fur system, the entire set of regulatory interactions found for the Fur transcription factor across a group of genomes. In the pan-regulatory network, emphasis is placed on how the regulatory network for each target genome is inferred from multiple sources instead of a single source, as is the common approach. The benefit of using multiple reference networks, is a more comprehensive survey of the relationships, and increased confidence in the regulatory interactions predicted. In the present study, we distinguish between relationships found across the full set of genomes as the `core-regulatory-set', and interactions found only in a subset of genomes explored as the `sub-regulatory-set'. We found nine Fur target gene clusters present across the four genomes studied, this core set potentially identifying basic regulatory processes essential for survival. Species level difierences are seen at the sub-regulatory-set level; for example the known virulence factors, YbtA and PchR were found in Y.pestis and P.aerguinosa respectively, but were not present in both E.coli and B.subtilis. Such factors and the iron-uptake systems they regulate, are ideal candidates for wet-lab investigation to determine whether or not they are pathogenic specific. In this study, we employed a broad range of approaches to address our goals and assessed these methods using the Fur regulon as our initial case study. We identified a set of promising feature attributes; demonstrated their success in increasing transcription factor binding site prediction specificity while retaining sensitivity, and showed the importance of binding site predictions in enhancing the reliability of regulatory interaction inferences. Most importantly, these outcomes led to the introduction of a range of visualisations and techniques, which are applicable across the entire bacterial spectrum and can be utilised in studies beyond the understanding of transcriptional regulatory networks.