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 ${\mathbb{R}}^d.$ The code for our algorithms can be found here https://github.com/jameswebber1/On-nonparametric-density-estimation-on-linear-and-nonlinear-manifolds. 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.

摘要

在这里，我们提出了一种新的非参数方法来进行密度估计和分类，该方法源自 Radon 变换和图像重建中的理论。我们首先构建一个“前向问题”，其中未知密度映射到一组从原始输入数据计算的一维经验分布函数。用氡型投影解释这种映射提供了数据和密度之间的分析联系，具有许多非常有用的特性，包括稳定的可逆性、快速计算和重要的理论基础。使用几何逆问题中的文献结果，我们给出了我们方法的唯一性结果和稳定性估计。我们随后扩展了这些想法以解决流形学习和流形上的密度估计中的问题。${\mathbb{R}}^d.$我们算法的代码可以在这里找到 https://github.com/jameswebber1/On-nonparametric-density-estimation-on-linear-and-nonlinear-manifolds。我们在一系列合成二维密度估计问题上测试我们的算法性能，这些问题设计有锐利的边缘和平滑的特征。我们表明，与文献中最先进的密度估计方法相比，我们的算法可以提供始终如一的竞争性能。