# Asymptotic theory for density ridges

arXiv: https://arxiv.org/abs/1406.5663 · HTML: https://mlsanity.com/paper/1406.5663/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2014.

## Built on by (most cited first, 9 of 9)

- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Quadratic Matrix Factorization with Applications to Manifold Learning](https://mlsanity.com/paper/2301.12965/) (2023)
- [Algorithms for ridge estimation with convergence guarantees](https://mlsanity.com/paper/2104.12314/) (2021)
- [Mode and Ridge Estimation in Euclidean and Directional Product Spaces: A Mean Shift Approach](https://mlsanity.com/paper/2110.08505/) (2021)
- [Latent Manifold Reconstruction and Representation with Topological and Geometrical Regularization](https://mlsanity.com/paper/2505.04412/) (2025)
- [Density Ridge Selective Prediction for LLM and VLM Hallucination Detection under Calibration Label Scarcity](https://mlsanity.com/paper/2606.10198/) (2026)
- [Statistical embedding: Beyond principal components](https://mlsanity.com/paper/2106.01858/) (2021)
- [Bump hunting through density curvature features](https://mlsanity.com/paper/2208.00174/) (2022)
- [Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift](https://mlsanity.com/paper/2608.05112/) (2026)

## Abstract (arXiv)

The large sample theory of estimators for density modes is well understood. In this paper we consider density ridges, which are a higher-dimensional extension of modes. Modes correspond to zero-dimensional, local high-density regions in point clouds. Density ridges correspond to s-dimensional, local high-density regions in point clouds. We establish three main results. First we show that under appropriate regularity conditions, the local variation of the estimated ridge can be approximated by an empirical process. Second, we show that the distribution of the estimated ridge converges to a Gaussian process. Third, we establish that the bootstrap leads to valid confidence sets for density ridges.