Estimating the Reach of a Manifold via its Convexity Defect Function

Area of Science:

• Computational geometry
• Manifold learning
• Geometric inference

Background:

• The reach of a submanifold is a critical parameter for manifold learning and geometric inference from point clouds.
• Understanding and estimating the reach is essential for accurately reconstructing shapes from data.

Purpose of the Study:

• To develop a novel estimator for the reach of a submanifold.
• To analyze the theoretical performance of this estimator using convexity defect functions.
• To provide bounds on the estimation error and compare it with minimax rates.

Main Methods:

• Relating the submanifold reach to its convexity defect function.
• Utilizing stability properties of convexity defect functions and new theoretical bounds.
• Employing the recent submanifold estimator by Aamari and Levrard.
• Deriving uniform expected loss bounds and minimax lower bounds for reach estimation.

Main Results:

• An effective estimator for the reach of a submanifold is proposed.
• A uniform expected loss bound is established for a specific model.
• Minimax lower bounds for reach estimation are provided.
• The proposed estimator demonstrates near-optimal performance in certain cases, with a logarithmic gap.

Conclusions:

• The developed estimator for the submanifold reach shows strong theoretical guarantees.
• This work advances the field of manifold learning and geometric inference by providing better tools for shape analysis.
• The findings offer insights into the fundamental limits of estimating geometric properties from point cloud data.