Deep Hierarchical-Hyperspherical Learning (DH^2L)

Regularization is known to be an inexpensive and reasonable solution to alleviate over-fitting problems of inference models, including deep neural networks. In this paper, we propose a hierarchical regularization which preserves the semantic structure of a sample distribution. At the same time, this regularization promotes diversity by imposing distance between parameter vectors enlarged within semantic structures. To generate evenly distributed parameters, we constrain them to lie on \emph{hierarchical hyperspheres}. Evenly distributed parameters are considered to be less redundant. To define hierarchical parameter space, we propose to reformulate the topology space with multiple hypersphere space. On each hypersphere space, the projection parameter is defined by two individual parameters. Since maximizing groupwise pairwise distance between points on hypersphere is nontrivial (generalized Thomson problem), we propose a new discrete metric integrated with continuous angle metric. Extensive experiments on publicly available datasets (CIFAR-10, CIFAR-100, CUB200-2011, and Stanford Cars), our proposed method shows improved generalization performance, especially when the number of super-classes is larger.