TR#:  PHD201404 
Class:  PHD 
Title:  Learning Methods for Modeling HighDimensional Distributions 
Authors:  Assaf Glazer 
Supervisors:  Shaul Markovitch, Michael Lindenbaum 
PHD201404.pdf  
Abstract:  A reliable density estimation is hard to obtain in problems of highdimensional 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{minimumvolume set (MVset)} of a distribution at level $\alpha$ instead of its density function. (An MVset 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 MVset reveals only partial information about the distribution. Can we define a problem whose solution is more informative than MVset 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 MVsets for distribution representation in highdimensional data. In most of our proposed methods, we use the \emph{oneclass SVM (OCSVM)} algorithm to estimate the hierarchical MVsets. Note that a straightforward approach of training a set of \emph{OCSVMs}, one for each MVset, would not necessarily satisfy the hierarchy requirement. We thus introduce novel variants of the \emph{OCSVM} algorithm that find all estimated MVsets such that the hierarchy constraint is fulfilled. We provide theoretical and empirical justifications for our methods in the general context of estimating hierarchical MVsets. 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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