Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction

TL;DR

This paper demonstrates that Tangential Delaunay Complexes can be used to reconstruct manifolds from noisy data with optimal convergence rates, even without prior tangent space information, and proves their stability under perturbations.

Contribution

It establishes the minimax optimality and stability of Tangential Delaunay Complexes for manifold reconstruction from noisy samples, extending existing algorithms with theoretical guarantees.

Findings

Estimator is asymptotically minimax optimal under noise.

Reconstruction achieves Hausdorff distance bounds.

Stability of Tangential Delaunay Complexes under perturbations.

Abstract

We consider the problem of optimality in manifold reconstruction. A random sample $\mathbb{X}_n = \left\{X_1,\ldots,X_n\right\}\subset \mathbb{R}^D$ composed of points close to a $d$-dimensional submanifold $M$, with or without outliers drawn in the ambient space, is observed. Based on the Tangential Delaunay Complex, we construct an estimator $\hat{M}$ that is ambient isotopic and Hausdorff-close to $M$ with high probability. The estimator $\hat{M}$ is built from existing algorithms. In a model with additive noise of small amplitude, we show that this estimator is asymptotically minimax optimal for the Hausdorff distance over a class of submanifolds satisfying a reach constraint. Therefore, even with no a priori information on the tangent spaces of $M$, our estimator based on Tangential Delaunay Complexes is optimal. This shows that the optimal rate of convergence can be achieved through existing algorithms. A similar result is also derived in a model with outliers. A geometric interpolation result is derived, showing that the Tangential Delaunay Complex is stable with respect to noise and perturbations of the tangent spaces. In the process, a decluttering procedure and a tangent space estimator both based on local principal component analysis (PCA) are studied.