# Efficient Weingarten Map and Curvature Estimation on Manifolds

## TL;DR

This paper introduces an efficient method for estimating the Weingarten map from point cloud data on manifolds, providing theoretical analysis and practical applications in curvature estimation and data simplification.

## Contribution

It presents a novel estimator for the Weingarten map with proven convergence properties and demonstrates its effectiveness on real and simulated data.

## Key findings

- Convergence rate of the estimator established theoretically.
- Successful application to curvature estimation.
- Effective point cloud simplification achieved.

## Abstract

In this paper, we propose an efficient method to estimate the Weingarten map for point cloud data sampled from manifold embedded in Euclidean space. A statistical model is established to analyze the asymptotic property of the estimator. In particular, we show the convergence rate as the sample size tends to infinity. We verify the convergence rate through simulated data and apply the estimated Weingarten map to curvature estimation and point cloud simplification to multiple real data sets.