Lower Bounds for Kernel Density Estimation on Symmetric Spaces

Dena Marie Asta

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arXiv · 2403.10480

Lower Bounds for Kernel Density Estimation on Symmetric Spaces

Dena Marie Asta

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Abstract

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.

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Dena Marie Asta. 2024-03-15. Lower Bounds for Kernel Density Estimation on Symmetric Spaces. https://arxiv.org/abs/2403.10480

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