In this paper, we propose a novel supervised learning method, Fast Flux Discriminant (FFD), for large-scale nonlinear classification. Compared with other existing methods, FFD has unmatched advantages, as it attains the efficiency and interpretability of linear models as well as the accuracy of nonlinear models. It is also sparse and naturally handles mixed data types. It works by decomposing the kernel density estimation in the entire feature space into selected low-dimensional subspaces. Since there are many possible subspaces, we propose a submodular optimization framework for subspace selection. The selected subspace predictions are then transformed to new features on which a linear model can be learned. Besides, since the transformed features naturally expect non-negative weights, we only require smooth optimization even with the L1 regularization. Unlike other nonlinear models such as kernel methods, the FFD model is interpretable as it gives importance weights on the original features. Its training and testing are also much faster than traditional kernel models. We carry out extensive empirical studies on real-world datasets and show that the proposed model achieves state-of-the-art classification results with sparsity, interpretability, and exceptional scalability. Our model can be learned in minutes on datasets with millions of samples, for which most existing nonlinear methods will be prohibitively expensive in space and time.