Robust boundary detection and density estimation using doubly stochastic scaling of the Gaussian kernel

TL;DR

This paper introduces a novel method for boundary detection and density estimation on manifolds with boundary, leveraging doubly stochastic Gaussian kernel scaling and local PCA, especially effective in noisy data.

Contribution

It extends doubly stochastic Gaussian kernel scaling to manifolds with boundary and develops boundary-aware estimators outperforming standard methods.

Findings

Boundary direction estimator effectively identifies boundary points.

Boundary-corrected density estimator improves accuracy under noise.

Simulations show superior performance over standard Gaussian kernel methods.

Abstract

This paper addresses the problem of detecting boundary points and estimating the sampling density of a dataset derived from a compact manifold with boundary, potentially in the presence of noise. We extend recent advances in doubly stochastic scaling of the Gaussian heat kernel via Sinkhorn iterations to this setting. Our main contributions are: (a) deriving a characterization of the scaling factors for manifolds with boundary, (b) developing a boundary direction estimator aimed at identifying boundary points followed by a boundary-corrected kernel density estimator based on doubly stochastic kernel and local principal component analysis, and (c) demonstrating through simulations that the resulting estimates of the boundary points and the sampling density outperform the standard Gaussian kernel-based approach, particularly under noisy conditions.