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

TL;DR

This work introduces a non-parametric density-estimation framework based on generalized Radon transforms, mapping a density to one-dimensional empirical projections via spherical and half-space Radon operators. By discretizing the forward models on grids and solving TV-regularized inverse problems, the authors derive convergence guarantees and demonstrate robust, edge-preserving density reconstructions in low dimensions, outperforming some KDE-based and prior Radon-based methods. They extend the approach to densities on low-dimensional manifolds by locally embedding data into tangent spaces and applying the same reconstruction strategy, supported by theoretical bounds on local tangent-space approximations. The methods offer a scalable, non-parametric alternative for density estimation with clear pathways for further development in high dimensions and more sophisticated projection densities.

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.

Figures (17)

• Figure 1: Point cloud of IID samples from an unknown density $f$ (left). Half space projection of $f$ along the $x$ axis (right). Here $\theta=(1,0)$ is in the direction of the $x$ axis. The values in the right hand plot are the density count in the right half space to the line shown in the left hand plot as it is translated in the direction $\theta$.
• Figure 1: A hyperplane in two dimensions parameterized by the perpendicular distance $s$ from the origin and a direction $\theta$ (right). A circle parameterized by a center point $x$ and radius $s$ (left).
• Figure 1: Comparisons of errors $\epsilon$ in density reconstructions 1--4, using the methods Sph, Hs, Os, Ker and Koen.
• Figure 1: Comparisons of errors $\epsilon$ in density reconstructions 1--4, using the methods ROF, Poi, Sph and Moh.
• Figure 1: Ground truth density functions. A Gaussian mixture with 100 Gaussians (density 1, top left), 5 overlapping uniform densities (density 2, top right), a Gaussian--uniform mixture density (density 3, bottom left) and a general mixture density (a sum of exponential, Gaussian, gamma and uniform density functions (density 4, bottom right)).

Theorems & Definitions (26)

• Definition 5.1: natterer
• Definition 5.2
• Definition 5.3
• Theorem 6.1
• Proof 1
• Corollary 6.2
• Proof 2
• Theorem 6.3
• Proof 3
• Theorem 7.1