Abstract—Kernels as similarity measures are key components of machine learning algorithms such as Support Vector Machine and Gaussian Process. Invariant kernels are an effective way to incorporate prior knowledge in applications with the invariance property. In this paper, we characterize all the rotation invariant kernels on spheres. We show that such a kernel is a function of the dot product of the input vectors alone. This function can be expanded as a series of Chebyshev polynomials with non-negative coefficients. In a 2-D space this condition is also sufficient. On a 3-D sphere the function can be expanded as a series of Legendre polynomials with non-negative coefficients. In general, a necessary and sufficient condition for the rotation invariant kernel is that the function on dot products can be expanded as a series of Gegenbauer polynomials with non-negative coefficients.

Index Terms—Bochner’s theorem, Fourier transform, Gegenbauer polynomial, invariant kernel.

Hongjun Su is with Department of Electrical and Computer Engineering, Georgia Southern University, Savannah, GA, USA. Hong Zhang is with Department of Computer Science, Georgia Southern University, Savannah, GA, USA.

Cite:Hongjun Su, Hong Zhang, "Rotation Invariant Kernels on Spheres," International Journal of Applied Physics and Mathematics vol. 11, no. 3, pp. 43-49, 2021.

Copyright © 2021 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).