# On non-parametric density estimation on linear and non-linear manifolds using generalized Radon transforms

## TL;DR

This paper introduces a novel non-parametric density estimation method based on Radon transforms, providing stable, invertible, and computationally efficient algorithms applicable to data on manifolds, with competitive results on synthetic datasets.

## Contribution

The paper develops new algorithms for density estimation on manifolds using Radon transforms, extending classical inverse problem techniques to manifold learning.

## Key findings

- Algorithms perform well on synthetic 2D density problems
- Method offers stable invertibility and fast computation
- Competitive with state-of-the-art density estimation methods

## Abstract

Here we present a new non-parametric approach to density estimation and classification derived from theory in Radon transforms and image reconstruction. We start by constructing a "forward problem" in which the unknown density is mapped to a set of one dimensional empirical distribution functions computed from the raw input data. Interpreting this mapping in terms of Radon-type projections provides an analytical connection between the data and the density with many very useful properties including stable invertibility, fast computation, and significant theoretical grounding. Using results from the literature in geometric inverse problems we give uniqueness results and stability estimates for our methods. We subsequently extend the ideas to address problems in manifold learning and density estimation on manifolds. We introduce two new algorithms which can be readily applied to implement density estimation using Radon transforms in low dimensions or on low dimensional manifolds embedded in $\mathbb{R}^d$. We test our algorithms performance on a range of synthetic 2-D density estimation problems, designed with a mixture of sharp edges and smooth features. We show that our algorithm can offer a consistently competitive performance when compared to the state-of-the-art density estimation methods from the literature.