Beyond PCA: Manifold Dimension Estimation via Local Graph Structure

TL;DR

This work tackles intrinsic dimension estimation under the manifold hypothesis by reframing the problem as regression on PCA-derived coordinates to recover the local graph structure of the manifold. It introduces a general framework with curvature-aware local models and instantiates two estimators, Quadratic Embedding (QE) and Total Least Squares (TLS), both using quadratic representations to capture curvature. Through extensive experiments on synthetic and real datasets, QE and TLS show competitive or superior performance to leading methods, especially with small sample sizes and nonlinear embeddings, while robustly handling noise. The approach offers a flexible, regression-based path to improve intrinsic dimension estimation and can incorporate prior geometric knowledge via the local graph model class $G_j$.

Abstract

Local principal component analysis (Local PCA) has proven to be an effective tool for estimating the intrinsic dimension of a manifold. More recently, curvature-adjusted PCA (CA-PCA) has improved upon this approach by explicitly accounting for the curvature of the underlying manifold, rather than assuming local flatness. Building on these insights, we propose a general framework for manifold dimension estimation that captures the manifold's local graph structure by integrating PCA with regression-based techniques. 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 alternatives.

Figures (6)

• Figure 1: Local coordinate and tangent space.
• Figure 2: Local Graph Representation.
• Figure 3: Practical and theoretical local approximations.
• Figure 4: Distribution of $F$-statistics for different values of $j$.
• Figure 5: Total TLS error and its relative drops as a function of model dimension.