Poincaré Kernels for Hyperbolic Representations.

Embedding data in hyperbolic spaces has proven beneficial for many advanced machine learning applications. However, working in hyperbolic spaces is not without difficulties as a result of its curved geometry (e.g., computing the Fréchet mean of a set of points requires an iterative algorithm). 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 valid kernel functions for hyperbolic representations. This brings in two major advantages, 1. kernelization will pave the way to seamlessly benefit the representational power 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 positive definite kernels in hyperbolic spaces (modeled by a Poincaré ball), the proposed kernels include the rich universal ones (e.g., Poincaré RBF kernel), or realize the multiple kernel learning scheme (e.g., Poincaré radial kernel). We comprehensively study the proposed kernels on a variety of challenging tasks including few-shot learning, zero-shot learning, person re-identification, deep metric learning, knowledge distillation and self-supervised learning. The consistent performance gain over different tasks shows the benefits of the kernelization for hyperbolic representations. [ABSTRACT FROM AUTHOR]