We prove that kernel density estimation on symmetric spaces of non-compact type, whose L2-risk was bounded above in previous work (Asta,2021), in fact achieves a minimax rate of convergence. With this result, the story for kernel density estimation on all symmetric spaces is completed. The idea in adapting the proof for Euclidean space is to suitably abstract vector space operations on Euclidean space to both actions of symmetric groups and reparametrizations of Helgason-Fourier transforms and to use the fact that the exponential map for symmetric spaces of non-compact type defines a diffeomorphism.

翻译：我们证明了非紧型对称空间上的核密度估计——其L2风险的上界已在先前工作（Asta,2021）中得到控制——实际上达到了极小化最优收敛速率。这一结果完善了所有对称空间上核密度估计的理论体系。在将欧氏空间中的证明方法进行适应性推广时，核心思路是：将欧氏空间中的向量空间运算适当地抽象为对称群的群作用以及Helgason-Fourier变换的重新参数化，并利用非紧型对称空间的指数映射构成微分同胚这一事实。