Summary: Dimension reduction has been applied in various areas of pattern recognition and data mining. While a traditional dimension reduction method, Principal Component Analysis (PCA) finds projective directions to maximize the global scatter in data, Locality Preserving Projection (LPP) pursues linear dimension reduction to minimize the local scatter. However, the discriminative power by either global or local scatter optimization is not guaranteed to be effective for classification. A recently proposed method, Unsupervised Discriminant Projection (UDP) aims to minimize the local scatter among near points and maximize the global scatter of distant points at the same time. Although its performance has been proven to be comparable to other dimension reduction methods, PCA preprocessing step due to the singularity of global and local scatter matrices may degrade the performance of UDP. In this paper, we propose several algorithms to improve the performances of UDP greatly. An improved algorithm for UDP is presented which applies the Generalized Singular Value Decomposition (GSVD) to overcome singularities of scatter matrices in UDP. Two-dimensional UDP and nonlinear extension of UDP are also proposed. Extensive experimental results demonstrate superiority of the proposed algorithms.