Date of Degree
affine feature extraction, Exponential Family, Hybrid learning, maximization-minimization, QDA, trust region
In this thesis, we revisit quadratic discriminant analysis (QDA), a standard classification method. Specifically, we investigate the parameter estimation and dimension reduction problems for QDA.
Traditionally, the parameters of QDA are estimated generatively; that is the parameters are estimated by maximizing the joint likelihood of observations and their labels. In practice, classical QDA, though computationally efficient, often underperforms discriminative classifiers, such as SVM, Boosting methods, and logistic regression. Motivated by recent research on hybrid generative/discriminative learning, we propose to estimate the parameters of QDA by minimizing a convex combination of negative joint log-likelihood and negative conditional log-likelihood of observations and their labels. For this purpose, we propose an iterative majorize-minimize (MM) algorithm for classifiers of which conditional distributions are from the exponential family; in each iteration of the MM algorithm, a convex optimization problem needs to be solved. To solve the convex problem specially derived for QDA, we propose a block-coordinate descent algorithm that sequentially updates the parameters of QDA; in each update, we present a trust region method for solving optimal estimations, of which we have closed form solutions in each iteration. Numerical experiments show: 1) the hybrid approach to QDA is competitive with, and in some cases significant better than other approaches to QDA, SVM with polynomial kernel ($d=2$) and logistic regression with linear and quadratic features; 2) in many cases, our optimization method converges faster to equal or better optimums than the conjugate gradient method used in the literature.
Dimension reduction methods are commonly used to extract more compact features in the hope to build more efficient and possibly more robust classifiers. It is well known that Fisher's discriminant analysis generates optimal lower dimensional features for linear discriminant analysis. However, "...for QDA, where so far there has been no universally accepted dimension-reduction technique in the literature'', though considerable efforts have been made. To construct a dimension reduction method for QDA, we generalize the Fukunaga-Koontz transformation, and propose novel affine feature extraction (AFE) methods for binary QDA. The proposed AFE methods have closed-form solutions and thus can be solved efficiently. We show that 1) the AFE methods have desired geometrical, statistical and information-theoretical properties; and 2) the AFE methods generalize dimension reduction methods for LDA and QDA with equal means. Numerical experiments show that the new proposed AFE method is competitive with, and in some cases significantly better than some commonly used linear dimension reduction techniques for QDA in the literature.
Cao, Wenbo, "Quadratic Discriminant Analysis Revisited" (2015). CUNY Academic Works.