Kernel Density Estimation on Symmetric Spaces of Non-Compact Type

TL;DR

This paper develops a kernel density estimator for symmetric spaces of non-compact type, providing convergence rates and simplified formulas for special cases like the space of normal distributions.

Contribution

It introduces a kernel density estimation method tailored for non-Euclidean symmetric spaces and derives convergence bounds similar to Euclidean cases.

Findings

Established an upper bound for the estimator's convergence rate.

Derived a simplified formula for the case of the space of normal distributions.

Extended kernel density estimation techniques to non-compact symmetric spaces.

Abstract

We construct a kernel density estimator on symmetric spaces of non-compact type and establish an upper bound for its convergence rate, analogous to the minimax rate for classical kernel density estimators on Euclidean space. Symmetric spaces of non-compact type include hyperboloids of constant negative curvature and spaces of symmetric positive definite matrices. This paper obtains a simplified formula in the special case when the symmetric space is the space of normal distributions, a 2-dimensional hyperboloid.