Abstract: For high-dimensional data, extraction of effective features is important for pattern recognition. Unsupervised discriminant projection shows desirable performance by maximizing the ratio of non-local scatter to the local scatter, but it is an unsupervised method and suffers from the singularity problem, which is also called the small sample size (SSS) problem. To solve these problems, supervised Laplacian discriminant analysis (SLDA) is proposed, which takes the class information into account while calculating the local and non-local scatter matrix. The null space of total Laplacian scatter matrix is discarded firstly, then the intra-class Laplacian scatter matrix is projected onto the range space of the total Laplacian scatter matrix, and the solution is reduced to the eigen-problem in that space, thus the SSS is artfully avoided. Theoretical analysis shows that no discriminative information is lost and it is also computational efficient. Experiments on face recognition confirm the correctness and effectiveness of the proposed algorithm.