Discriminant feature extraction plays a fundamental role in pattern recognition. In this paper, we propose the Linear Laplacian Discrimination (LLD) algorithm for discriminant feature extraction. LLD is an extension of Linear Discriminant Analysis (LDA). Our motivation is to address the issue that LDA cannot work well in cases where sample spaces are non-Euclidean. Specifically, we define the within-class scatter and the between-class scatter using similarities which are based on pairwise distances in sample spaces. Thus the structural information of classes is contained in the within-class and the between-class Laplacian matrices which are free from metrics of sample spaces. The optimal discriminant subspace can be derived by controlling the structural evolution of Laplacian matrices. Experiments are performed on the facial database for FRGC version 2. Experimental results show that LLD is effective in extracting discriminant features.