Supervised linear dimension reduction

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Supervised linear dimension reduction (SLDR) is one of the most effective methods for complexity reduction, which has been widely applied in pattern recognition, computer vision, information retrieval, and multimedia data processing. This thesis explores SLDR by enriching the theory of existing methods and by proposing new methods. In the first part of this thesis, we present theoretical analysis of Fisher’s linear discriminant analysis (LDA), one of the most representative methods for SLDR. 1) Classical asymptotic analysis of LDA is based on a fixed dimensionality, and thus does not apply in the case where the dimensionality and the training sample number are proportionally large. Besides, the classical result does not provide quantitative information on the performance of LDA. To address these limitations, we present an asymptotic generalization analysis of LDA, allowing both the dimensionality and the training sample number to be proportionally large, from which we principally obtain an asymptotic generalization bound that quantitatively describes the performance of LDA in terms of the dimensionality and the training sample number. 2) We study a new regularization method for LDA, termed the block-diagonal regularization. By partitioning variables into small groups and treating them independently, block-diagonal regularization effectively reduces the dimensionality to training sample number ratio and thus improves the generalization ability of LDA. We present a theoretical justification of the block-diagonally regularized LDA by investigating its approximation and sample errors. We show that the block-diagonally regularized LDA performs competitively compared to other types of regularized LDA, e.g., with the Tikhonov regularization and the banded regularization. In the second part of this thesis, we propose two new methods for SLDR. 1) The first method is for parametric SLDR, termed max-min distance analysis (MMDA). MMDA optimizes the projection matrix by maximizing the minimum pairwise distance of all class pairs in the dimension reduced space. Thus, it duly considers the separation of all classes and overcomes the “class separation” problem of existing parametric SLDR methods that close class pairs tend to merge in the dimension reduced space. 2) The second method is for nonparametric SLDR, which uses minimizing the asymptotic nearest neighbor classification error (MNNE) as the criterion for optimizing the projection matrix. Theoretically, we compare MNNE with other criteria, e.g., maximizing mutual information (MMI) and minimizing Bhattacharyya bound. We show that MMNE is superior to these two criteria in terms of the closeness to the Bayes optimal criterion. Empirical studies show that the proposed methods, MMDA and MNNE, achieve state-of-the-art performance for parametric and nonparametric SLDR, respectively.
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