Hyperbolic Positive Definite Kernels Based on Möbius Gyrovector Space
YANG Meimei1,2, FANG Pengfei1,2, ZHU Shipeng1,2, XUE Hui1,2
1. School of Computer Science and Engineering, Southeast University, Nanjing 211189; 2. Key Laboratory of New Generation Artificial Intelligence Technology and Its Interdisciplinary Applications of Ministry of Education, Southeast University, Nanjing 211189
Abstract：Hierarchical data is widely present in various machine learning scenarios and the data can be encoded in hyperbolic spaces with very low distortion. Kernel methods are introduced to further enhance the representation capability of hyperbolic space. However, the existing hyperbolic kernels still have the drawbacks of low adaptive capacity or data distortion. To address these issues, hyperbolic positive definite kernels based on Möbius gyrovector space is proposed in this paper. By leveraging the relationship between the Möbius gyrovector space and the Poincaré model, a class of hyperbolic kernel functions, the Möbius radial basis kernels, are constructed. Specifically, the Möbius gyrodistance is employed in place of the Euclidean distance to construct the Möbius Gaussian kernel and the Möbius Laplacian kernel, with the positive definiteness of the kernel functions further demonstrated. Moreover, kernel functions are transformed from complex space to real space, and thus they are more suitable for most machine learning tasks. Experiments on several real-world social network datasets validate the effectiveness of the proposed method.
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2023, Vol. 36 
