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In this paper, we propose a new local discriminative feature learning method for finger vein recognition. Unlike most previous finger vein recognition methods, which use hand-crafted descriptors such as local binary pattern, local line binary pattern, and Gabor features, this paper aims to learn a feature mapping to enhance the discriminative ability of local features. To achieve this goal, we first extract multi-directional pixel difference vectors (MDPDVs) for each pixel in a training finger vein image by computing the difference between each pixel and its straight-line neighboring pixels. Second, we learn a feature mapping to project these MDPDVs into low-dimensional binary codes in a supervised manner, where: 1) the loss between the original real-valued codes and learned binary vectors is minimized; 2) the between-class variation of the local binary features is maximized; and 3) the within-class variation of the local binary features is minimized. Last, we represent each finger vein image as a histogram feature by clustering and pooling these binary codes. Experiments on SDUMLA-FV and PolyU databases verify the superior performance of the proposed method over other existing finger vein recognition methods.