On boundary detection

TL;DR

This paper introduces a new statistical test to determine if a manifold's boundary is empty, based on sample data, without needing prior support estimation, and demonstrates its effectiveness through theoretical guarantees and simulations.

Contribution

It proposes a boundary detection test that does not require support estimation and provides theoretical properties and heuristics for practical implementation.

Findings

The test reliably detects boundary presence with high probability for large samples.

The level of the test can be accurately estimated.

Simulation results validate the test's effectiveness.

Abstract

Given a sample of a random variable supported by a smooth compact manifold $M\subset \mathbb{R}^d$, we propose a test to decide whether the boundary of $M$ is empty or not with no preliminary support estimation. The test statistic is based on the maximal distance between a sample point and the average of its $k_n$-nearest neighbors. We prove that the level of the test can be estimated, that, with probability one, its power is one for $n$ large enough, and that there exists a consistent decision rule. Heuristics for choosing a convenient value for the $k_n$ parameter and identifying observations close to the boundary are also given. We provide a simulation study of the test.