Technical Report PHD-2014-04

Title: Learning Methods for Modeling High-Dimensional Distributions
Authors: Assaf Glazer
Supervisors: Shaul Markovitch, Michael Lindenbaum
Abstract: A reliable density estimation is hard to obtain in problems of high-dimensional data, especially when the sample used for estimation is small. As a result, various studies have tried to find approximate solutions to this problem by reducing it to a less general, and hopefully solvable, form. One prominent approach in this direction is estimating the \emph{minimum-volume set (MV-set)} of a distribution at level $\alpha$ instead of its density function. (An MV-set at level $\alpha$ is a subset of the input space with probability mass of at least $\alpha$ that has the smallest volume.) However, even a perfectly estimated MV-set reveals only partial information about the distribution. Can we define a problem whose solution is more informative than MV-set estimation, yet is easier to solve than density estimation?

In this dissertation we introduce novel methods that do just that. Our methods, which can also be regarded as generalized quantile functions, are based on the idea of estimating (or approximating) hierarchical MV-sets for distribution representation in high-dimensional data. In most of our proposed methods, we use the \emph{one-class SVM (OCSVM)} algorithm to estimate the hierarchical MV-sets. Note that a straightforward approach of training a set of \emph{OCSVMs}, one for each MV-set, would not necessarily satisfy the hierarchy requirement. We thus introduce novel variants of the \emph{OCSVM} algorithm that find all estimated MV-sets such that the hierarchy constraint is fulfilled.

We provide theoretical and empirical justifications for our methods in the general context of estimating hierarchical MV-sets. In addition, we apply our methods and show their superiority over competitors in various domains including concept drift detection, topic change detection in document streams, background subtraction in image sequences, and hierarchical clustering.

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