# Non-Parametric Manifold Learning

arXiv: https://arxiv.org/abs/2107.08089 · HTML: https://mlsanity.com/paper/2107.08089/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 45 references that are in the map.
- First posted on arXiv 16 Jul 2021.

## Builds on (most cited first, 7 of 7)

- [A survey of dimensionality reduction techniques](https://mlsanity.com/paper/1403.2877/) (2014)
- [Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery](https://mlsanity.com/paper/1305.7255/) (2013)
- [Ollivier-Ricci curvature convergence in random geometric graphs](https://mlsanity.com/paper/2008.01209/) (2020)
- [Spectral Convergence Rate of Graph Laplacian](https://mlsanity.com/paper/1510.08110/) (2015)
- [Data driven estimation of Laplace-Beltrami operator](https://mlsanity.com/paper/1612.09434/) (2016)
- [Unconstrained and Curvature-Constrained Shortest-Path Distances and their Approximation](https://mlsanity.com/paper/1706.09441/) (2017)
- [Minimax Estimation of Distances on a Surface and Minimax Manifold Learning in the Isometric-to-Convex Setting](https://mlsanity.com/paper/2011.12478/) (2020)

## Abstract (arXiv)

We introduce an estimator for distances in a compact Riemannian manifold based on graph Laplacian estimates of the Laplace-Beltrami operator. We upper bound the error in the estimate of manifold distances, or more precisely an estimate of a spectrally truncated variant of manifold distance of interest in non-commutative geometry (cf. [Connes and Suijelekom, 2020]), in terms of spectral errors in the graph Laplacian estimates and, implicitly, several geometric properties of the manifold. A consequence is a proof of consistency for (untruncated) manifold distances. The estimator resembles, and in fact its convergence properties are derived from, a special case of the Kontorovic dual reformulation of Wasserstein distance known as Connes' Distance Formula.