# Data driven estimation of Laplace-Beltrami operator

arXiv: https://arxiv.org/abs/1612.09434 · HTML: https://mlsanity.com/paper/1612.09434/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Dec 2016.

## Built on by (most cited first, 4 of 4)

- [Neural Laplacian Operator for 3D Point Clouds](https://mlsanity.com/paper/2409.06506/) (2024)
- [Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression](https://mlsanity.com/paper/2311.00140/) (2023)
- [Non-Parametric Manifold Learning](https://mlsanity.com/paper/2107.08089/) (2021)
- [Learning Eigenstructures of Unstructured Data Manifolds](https://mlsanity.com/paper/2512.01103/) (2025)

## Abstract (arXiv)

Approximations of Laplace-Beltrami operators on manifolds through graph Lapla-cians have become popular tools in data analysis and machine learning. These discretized operators usually depend on bandwidth parameters whose tuning remains a theoretical and practical problem. In this paper, we address this problem for the unnormalized graph Laplacian by establishing an oracle inequality that opens the door to a well-founded data-driven procedure for the bandwidth selection. Our approach relies on recent results by Lacour and Massart [LM15] on the so-called Lepski's method.