# Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery

arXiv: https://arxiv.org/abs/1305.7255 · HTML: https://mlsanity.com/paper/1305.7255/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 May 2013.

## Built on by (most cited first, 14 of 14)

- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Effects of Data Geometry in Early Deep Learning](https://mlsanity.com/paper/2301.00008/) (2022)
- [A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations](https://mlsanity.com/paper/2201.09656/) (2021)
- [Nonlinear Isometric Manifold Learning for Injective Normalizing Flows](https://mlsanity.com/paper/2203.03934/) (2022)
- [On Manifold Learning in Plato's Cave: Remarks on Manifold Learning and Physical Phenomena](https://mlsanity.com/paper/2304.14248/) (2023)
- [Cryo-em images are intrinsically low dimensional](https://mlsanity.com/paper/2504.11249/) (2025)
- [Staying the course: Locating equilibria of dynamical systems on Riemannian manifolds defined by point-clouds](https://mlsanity.com/paper/2204.10413/) (2022)
- [Principal subbundles for dimension reduction](https://mlsanity.com/paper/2307.03128/) (2023)
- [Learning collective variables that preserve transition rates](https://mlsanity.com/paper/2506.01222/) (2025)
- [Reconstruction of Manifold Distances from Noisy Observations](https://mlsanity.com/paper/2511.13025/) (2025)
- [FloDR: An invertible dimensionality reduction method based on a normalising flow](https://mlsanity.com/paper/2607.26278/) (2026)
- [Learning Low-dimensional Manifolds for Scoring of Tissue Microarray Images](https://mlsanity.com/paper/2102.11396/) (2021)
- [Non-Parametric Manifold Learning](https://mlsanity.com/paper/2107.08089/) (2021)
- [Understanding Differentiable Embeddings Through Differential and Integral Geometry](https://mlsanity.com/paper/2608.06809/) (2026)

## Abstract (arXiv)

In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man ifold geometry using either local or global features of the data. Building on the Laplacian Eigenmap and Diffusionmaps framework, we propose a new paradigm that offers a guarantee, under reasonable assumptions, that any manifo ld learning algorithm will preserve the geometry of a data set. Our approach is based on augmenting the output of embedding algorithms with geometric informatio n embodied in the Riemannian metric of the manifold. We provide an algorithm for estimating the Riemannian metric from data and demonstrate possible application s of our approach in a variety of examples.