Identification of Data Structure with Machine Learning: From Fisher to Bayesian networks

This thesis proposes a theoretical framework to thoroughly analyse the structure of a dataset in terms of a) metric, b) density and c) feature associations. To look into the first aspect, Fisher's metric learning algorithms are the foundations of a novel manifold based on the information and complexity of a classification model. When looking at the density aspect, the Probabilistic Quantum clustering, a Bayesian version of the original Quantum Clustering is proposed. The clustering results will depend on local density variations, which is a desired feature when dealing with heteroscedastic data. To address the third aspect, the constraint-based PC-algorithm is the starting point of many structure learning algorithms, it is focused on finding feature associations by means of conditional independent tests. This is then used to select Bayesian networks, based on a regularized likelihood score. These three topics of data structure analysis were fully tested with synthetic data examples and real cases, which allowed us to unravel and discuss the advantages and limitations of these algorithms. One of the biggest challenges encountered was related to the application of these methods to a Big Data dataset that was analysed within the framework of a collaboration with a large UK retailer, where the interest was in the identification of the data structure underlying customer shopping baskets

Music genre profiling based on Fisher manifolds and Probabilistic Quantum Clustering

Probabilistic classifiers induce a similarity metric at each location in the space of the data. This is measured by the Fisher Information Matrix. Pairwise distances in this Riemannian space, calculated along geodesic paths, can be used to generate a similarity map of the data. The novelty in the paper is twofold; to improve the methodology for visualisation of data structures in low-dimensional manifolds, and to illustrate the value of inferring the structure from a probabilistic classifier by metric learning, through application to music data. This leads to the discovery of new structures and song similarities beyond the original genre classification labels. These similarities are not directly observable by measuring Euclidean distances between features of the original space, but require the correct metric to reflect similarity based on genre. The results quantify the extent to which music from bands typically associated with one particular genre can, in fact, crossover strongly to another genre

The Role of Riemannian Manifolds in Computer Vision: From Coding to Deep Metric Learning

A diverse number of tasks in computer vision and machine learning enjoy from representations of data that are compact yet discriminative, informative and robust to critical measurements. Two notable representations are offered by Region Covariance Descriptors (RCovD) and linear subspaces which are naturally analyzed through the manifold of Symmetric Positive Definite (SPD) matrices and the Grassmann manifold, respectively, two widely used types of Riemannian manifolds in computer vision. As our first objective, we examine image and video-based recognition applications where the local descriptors have the aforementioned Riemannian structures, namely the SPD or linear subspace structure. Initially, we provide a solution to compute Riemannian version of the conventional Vector of Locally aggregated Descriptors (VLAD), using geodesic distance of the underlying manifold as the nearness measure. Next, by having a closer look at the resulting codes, we formulate a new concept which we name Local Difference Vectors (LDV). LDVs enable us to elegantly expand our Riemannian coding techniques to any arbitrary metric as well as provide intrinsic solutions to Riemannian sparse coding and its variants when local structured descriptors are considered. We then turn our attention to two special types of covariance descriptors namely infinite-dimensional RCovDs and rank-deficient covariance matrices for which the underlying Riemannian structure, i.e. the manifold of SPD matrices is out of reach to great extent. %Generally speaking, infinite-dimensional RCovDs offer better discriminatory power over their low-dimensional counterparts. To overcome this difficulty, we propose to approximate the infinite-dimensional RCovDs by making use of two feature mappings, namely random Fourier features and the Nystrom method. As for the rank-deficient covariance matrices, unlike most existing approaches that employ inference tools by predefined regularizers, we derive positive definite kernels that can be decomposed into the kernels on the cone of SPD matrices and kernels on the Grassmann manifolds and show their effectiveness for image set classification task. Furthermore, inspired by attractive properties of Riemannian optimization techniques, we extend the recently introduced Keep It Simple and Straightforward MEtric learning (KISSME) method to the scenarios where input data is non-linearly distributed. To this end, we make use of the infinite dimensional covariance matrices and propose techniques towards projecting on the positive cone in a Reproducing Kernel Hilbert Space (RKHS). We also address the sensitivity issue of the KISSME to the input dimensionality. The KISSME algorithm is greatly dependent on Principal Component Analysis (PCA) as a preprocessing step which can lead to difficulties, especially when the dimensionality is not meticulously set. To address this issue, based on the KISSME algorithm, we develop a Riemannian framework to jointly learn a mapping performing dimensionality reduction and a metric in the induced space. Lastly, in line with the recent trend in metric learning, we devise end-to-end learning of a generic deep network for metric learning using our derivation