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.

Automotive diesel engine transient operation: modeling, optimization and control

Traditionally, the study of internal combustion engines operation has focused on the steady-state performance. However, the daily driving schedule of automotive engines is inherently related to unsteady conditions. There are various operating conditions experienced by (diesel) engines that can be classified as transient. Besides the variation of the engine operating point, in terms of engine speed and torque, also the warm up phase can be considered as a transient condition. Chapter 2 has to do with this thermal transient condition; more precisely the main issue is the performance of a Selective Catalytic Reduction (SCR) system during cold start and warm up phases of the engine. The proposal of the underlying work is to investigate and identify optimal exhaust line heating strategies, to provide a fast activation of the catalytic reactions on SCR. Chapters 3 and 4 focus the attention on the dynamic behavior of the engine, when considering typical driving conditions. The common approach to dynamic optimization involves the solution of a single optimal-control problem. However, this approach requires the availability of models that are valid throughout the whole engine operating range and actuator ranges. In addition, the result of the optimization is meaningful only if the model is very accurate. Chapter 3 proposes a methodology to circumvent those demanding requirements: an iteration between transient measurements to refine a purpose-built model and a dynamic optimization which is constrained to the model validity region. Moreover all numerical methods required to implement this procedure are presented. Chapter 4 proposes an approach to derive a transient feedforward control system in an automated way. It relies on optimal control theory to solve a dynamic optimization problem for fast transients. From the optimal solutions, the relevant information is extracted and stored in maps spanned by the engine speed and the torque gradient.