# Manifold Dimension Estimation via Local Graph Structure

arXiv: https://arxiv.org/abs/2510.15141 · HTML: https://mlsanity.com/paper/2510.15141/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 46 references that are in the map.
- First posted on arXiv 16 Oct 2025.

## Builds on (most cited first, 12 of 12)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Intrinsic Dimensionality Estimation within Tight Localities: A Theoretical and Experimental Analysis](https://mlsanity.com/paper/2209.14475/) (2022)
- [Intrinsic Dimension Estimation Using Wasserstein Distances](https://mlsanity.com/paper/2106.04018/) (2021)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [Non-Parametric Estimation of Manifolds from Noisy Data](https://mlsanity.com/paper/2105.04754/) (2021)
- [Tangent Space and Dimension Estimation with the Wasserstein Distance](https://mlsanity.com/paper/2110.06357/) (2021)
- [Hy-Facial: Hybrid Feature Extraction by Dimensionality Reduction Methods for Enhanced Facial Expression Classification](https://mlsanity.com/paper/2509.26614/) (2025)
- [CA-PCA: Manifold Dimension Estimation, Adapted for Curvature](https://mlsanity.com/paper/2309.13478/) (2023)

## Abstract (arXiv)

Most existing manifold dimension estimators rely on the assumption that the underlying manifold is locally flat within the neighborhoods under consideration. More recently, curvature-adjusted principal component analysis (CA-PCA) has emerged as a powerful alternative by explicitly accounting for the manifold's curvature. Motivated by these ideas, we propose a manifold dimension estimation framework that captures the local graph structure of the manifold through regression on local PCA coordinates. Within this framework, we introduce two representative estimators: quadratic embedding (QE) and total least squares (TLS). Experiments on both synthetic and real-world datasets demonstrate that these methods perform competitively with, and often outperform, state-of-the-art approaches.