RBF-Softmax: Learning Deep Representative Prototypes with Radial Basis Function Softmax

Deep neural networks have achieved remarkable successes in learning feature representations for visual classification. However, deep features learned by the softmax cross-entropy loss generally show excessive intra-class variations. We argue that, because the traditional softmax losses aim to optimize only the relative differences between intra-class and inter-class distances (logits), it cannot obtain representative class prototypes (class weights/centers) to regularize intra-class distances, even when the training is converged. Previous efforts mitigate this problem by introducing auxiliary regularization losses. But these modified losses mainly focus on optimizing intra-class compactness, while ignoring keeping reasonable relations between different class prototypes. These lead to weak models and eventually limit their performance. To address this problem, this paper introduces a novel Radial Basis Function (RBF) distances to replace the commonly used inner products in the softmax loss function, such that it can adaptively assign losses to regularize the intra-class and inter-class distances by reshaping the relative differences, and thus creating more representative prototypes of classes to improve optimization. The proposed RBF-Softmax loss function not only effectively reduces intra-class distances, stabilizes the training behavior, and reserves ideal relations between prototypes, but also significantly improves the testing performance. Experiments on visual recognition benchmarks including MNIST, CIFAR-10/100, and ImageNet demonstrate that the proposed RBF-Softmax achieves better results than cross-entropy and other state-of-the-art classification losses. The code is at https://github.com/2han9x1a0release/RBF-Softmax .