# Estimating the reach of a manifold via its convexity defect function

arXiv: https://arxiv.org/abs/2001.08006 · HTML: https://mlsanity.com/paper/2001.08006/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jan 2020.

## Built on by (most cited first, 6 of 6)

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- [The Data Manifold under the Microscope](https://mlsanity.com/paper/2606.15760/) (2026)
- [MGPO: Manifold-Guided Diffusion Alignment for Task-Aware Dataset Distillation](https://mlsanity.com/paper/2610.05252/) (2026)

## Abstract (arXiv)

The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard [Ann. Statist. 47 177-204 (2019)], an estimator for the reach is given. A uniform expected loss bound over a C^k model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the C^3 and C^4 cases, with a gap given by a logarithmic factor.