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We present a novel method of nonlinear discriminant analysis involving a set of locally linear transformations called "Locally Linear Discriminant Analysis (LLDA).” The underlying idea is that global nonlinear data structures are locally linear and local structures can be linearly aligned. Input vectors are projected into each local feature space by linear transformations found to yield locally linearly transformed classes that maximize the between-class covariance while minimizing the within-class covariance. In face recognition, linear discriminant analysis (LDA) has been widely adopted owing to its efficiency, but it does not capture nonlinear manifolds of faces which exhibit pose variations. Conventional nonlinear classification methods based on kernels such as generalized discriminant analysis (GDA) and support vector machine (SVM) have been developed to overcome the shortcomings of the linear method, but they have the drawback of high computational cost of classification and overfitting. Our method is for multiclass nonlinear discrimination and it is computationally highly efficient as compared to GDA. The method does not suffer from overfitting by virtue of the linear base structure of the solution. A novel gradient-based learning algorithm is proposed for finding the optimal set of local linear bases. The optimization does not exhibit a local-maxima problem. The transformation functions facilitate robust face recognition in a low-dimensional subspace, under pose variations, using a single model image. The classification results are given for both synthetic and real face data.
Linear discriminant analysis, generalized discriminant analysis, support vector machine, dimensionality reduction, face recognition, feature extraction, pose invariance, subspace representation.

T. Kim and J. Kittler, "Locally Linear Discriminant Analysis for Multimodally Distributed Classes for Face Recognition with a Single Model Image," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 27, no. , pp. 318-327, 2005.
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