Exploiting the structure of feature spaces in kernel learning

The problem of learning the optimal representation for a specific task recently became an important and not trivial topic in the machine learning community. In this field, deep architectures are the current gold standard among the machine learning algorithms by generating models with several levels of abstraction discovering very complicated structures in large datasets. Kernels and Deep Neural Networks (DNNs) are the principal methods to handle the representation problem in a deep manner. A DNN uses the famous back-propagation algorithm improving the state-of-the-art performance in several different real world applications, e.g. speech recognition, object detection and signal processing. Nevertheless, DNN algorithms have some drawbacks, inherited from standard neural networks, since they are theoretically not well understood. The main problems are: the complex structure of the solution, the unclear decoupling between the representation learning phase and the model generation, long training time, and the convergence to a sub-optimal solution (because of local minima and vanishing gradient). For these reasons, in this thesis, we propose new ideas to obtain an optimal representation by exploiting the kernels theory. Kernel methods have an elegant framework that decouples learning algorithms from data representations. On the other hand, kernels also have some weaknesses, for example they do not scale and they generally bring a shallow representation. In this thesis, we propose new theory and algorithms to fill this gap and make kernel learning able to generate deeper representation and to be more scalable. Considering this scenario we propose a different point of view regarding the Multiple Kernel Learning (MKL) framework, starting from the idea of a deeper kernel. An algorithm able to combine thousands of weak kernels with low computational and memory complexities is proposed. This procedure, called EasyMKL, outperforms the state-of-the-art methods combining the fragmented information in order to create an optimal kernel for the given task. Pursuing the idea to create an optimal family of weak kernels, we create a new measure for the evaluation of the kernel expressiveness, called spectral complexity. Exploiting this measure we are able to generate families of kernels with a hierarchical structure of the features by defining a new property concerning the monotonicity of the spectral complexity. We prove the quality of these weak families of kernels developing a new methodology for the Multiple Kernel Learning (MKL). Firstly we are able to create an optimal family of weak kernels by using the monotonically spectral-complex property; then we combine the optimal family of kernels by exploiting EasyMKL, obtaining a new kernel that is specific for the task; finally, we are able to generate the model by using a kernel machine. Moreover, we highlight the connection among distance metric learning, feature learning and kernel learning by proposing a method to learn the optimal family of weak kernels for a MKL algorithm in the different context in which the combination rule is the product element-wise of kernel matrices. This algorithm is able to generate the best parameters for an anisotropic RBF kernel and, therefore, a connection naturally appears among feature weighting, combinations of kernels and metric learning. Finally, the importance of the representation is also taken into account in three tasks from real world problems where we tackle different issues such as noise data, real-time application and big data