Robustness in Dimensionality Reduction
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Date
2016-04-14
Authors
Liang, Jiaxi
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
Dimensionality reduction is widely used in many statistical applications, such as image analysis, microarray analysis, or text mining. This thesis focuses on three problems that relate to the robustness in dimension reduction.
The first topic is the performance analysis in dimension reduction, that is, quantitatively assessing the performance of a algorithm on a given dataset. A criterion for success is established from the geometric point of view to address this issues. A family of goodness measures, called \textsl{local rank correlation}, is developed to assess the performance of dimensionality reduction methods. The potential application of the local rank correlation in selecting tuning parameters of dimension reduction algorithms is also explored. The second topic is the sensitivity analysis in dimension reduction. Two types of influence functions are developed as measures of robustness, based on which we develop graphical display strategies for visualizing the robustness of a dimension reduction method, and flagging potential outliers. In the third part of the thesis, a novel robust PCA framework, called \textsl{Performance-Weighted Bagging PCA}, is proposed from the perspective of model averaging. It obtains a robust linear subspace by weighted averaging a collection of subspaces produced by subsamples. The robustness against outliers is achieved by a proper weighting scheme, and possible choices of weighting scheme are investigated.
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Keywords
Robustness, Dimensionality reduction, Influence function, Performance measure, PCA