# Estimating densities with nonlinear support using Fisher-Gaussian kernels

## TL;DR

This paper introduces Fisher-Gaussian kernels, a new class of density estimators that better capture data supported on nonlinear manifolds by incorporating curvature, improving performance over traditional Gaussian-based methods.

## Contribution

It proposes a novel Fisher-Gaussian kernel class with an added curvature parameter, enabling more accurate density estimation on nonlinear supports.

## Key findings

- Fisher-Gaussian kernels effectively model data on curved manifolds.
- The method shows improved density estimation in simulations and real data.
- Analytic form facilitates Bayesian implementation with MCMC.

## Abstract

Current tools for multivariate density estimation struggle when the density is concentrated near a nonlinear subspace or manifold. Most approaches require choice of a kernel, with the multivariate Gaussian by far the most commonly used. Although heavy-tailed and skewed extensions have been proposed, such kernels cannot capture curvature in the support of the data. This leads to poor performance unless the sample size is very large relative to the dimension of the data. This article proposes a novel generalization of the Gaussian distribution, which includes an additional curvature parameter. We refer to the proposed class as Fisher-Gaussian (FG) kernels, since they arise by sampling from a von Mises-Fisher density on the sphere and adding Gaussian noise. The FG density has an analytic form, and is amenable to straightforward implementation within Bayesian mixture models using Markov chain Monte Carlo. We provide theory on large support, and illustrate gains relative to competitors in simulated and real data applications.