CA-PCA: Manifold Dimension Estimation, Adapted for Curvature

TL;DR

This paper introduces CA-PCA, a local PCA method that accounts for manifold curvature to improve dimension estimation in high-dimensional data analysis.

Contribution

It presents a novel curvature-aware calibration for local PCA, enhancing manifold dimension estimation accuracy over existing flat models.

Findings

CA-PCA outperforms traditional methods in various experiments.

Accounting for curvature improves dimension estimation accuracy.

The method is effective across diverse data settings.

Abstract

The success of algorithms in the analysis of high-dimensional data is often attributed to the manifold hypothesis, which supposes that this data lie on or near a manifold of much lower dimension. It is often useful to determine or estimate the dimension of this manifold before performing dimension reduction, for instance. Existing methods for dimension estimation are calibrated using a flat unit ball. In this paper, we develop CA-PCA, a version of local PCA based instead on a calibration of a quadratic embedding, acknowledging the curvature of the underlying manifold. Numerous careful experiments show that this adaptation improves the estimator in a wide range of settings.