Manifold Dimension Estimation via Local Graph Structure

arXiv:2510.15141 · stat.ML, cs.LG, stat.AP · Submitted 2025-10-16 · Read on arXiv

stat.ML, cs.LG, stat.AP

Submitted: 2025-10-16

Updated: 2026-09-14

Comments: multiple appendix sections added, main contents updated to incorporate new theoretical results

Code: https://github.com/loong-bi/manifold-dimension-estimation

License: http://creativecommons.org/licenses/by/4.0/

The gist: Most existing manifold dimension estimators rely on the assumption that the underlying manifold is locally flat within the neighborhoods under consideration.

Terminology

Abstract

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.

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