Kernel methods in hyperbolic spaces

Embedding data in hyperbolic spaces has proven beneficial for many advanced machine learning applications such as image classification and word embeddings. However, working in hyperbolic spaces is not without difficulties as a result of its curved geometry (e.g., computing the Frechet mean of a set of points requires an iterative algorithm). Furthermore, in Euclidean spaces, one can resort to kernel machines that not only enjoy rich theoretical properties but that can also lead to superior representational power (e.g., infinite-width neural networks). In this paper, we introduce positive definite kernel functions for hyperbolic spaces. This brings in two major advantages, 1. kernelization will pave the way to seamlessly benefit from kernel machines in conjunction with hyperbolic embeddings, and 2. the rich structure of the Hilbert spaces associated with kernel machines enables us to simplify various operations involving hyperbolic data. That said, identifying valid kernel functions on curved spaces is not straightforward and is indeed considered an open problem in the learning community. Our work addresses this gap and develops several valid positive definite kernels in hyperbolic spaces, including the universal ones (e.g., RBF). We comprehensively study the proposed kernels on a variety of challenging tasks including few-shot learning, zero-shot learning, person re-identification and knowledge distillation, showing the superiority of the kernelization for hyperbolic representations.