Semantic gap is a common problem for most distance metric learning (DML) algorithms. Because of this problem, the semantic information may be inconsistent with the image features, which negatively affects the image classification accuracy. To solve the problem, this paper puts forward a new supervised DML method called semantic discriminative metric learning (SDML). The SDML maximizes the geometric mean of the normalized dispersion, making dispersions between different classes as identical as possible. Moreover, the Log function was combined with the geometric mean to further balance the dispersion between classes, and the maximum-margin criterion (MMC) was introduced to further enhance the discrimination ability of the distance metric. Finally, two constraints were applied to optimize the distance metric matrix. The effectiveness of the SDML algorithm was fully proved through experiments on actual datasets. The experimental results show that our algorithm outperformed many other typical DML methods in classification accuracy. This research provides an effectively way to measure the similarity between image samples and classify high-dimensional images.