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CA-PCA: Manifold Dimension Estimation, Adapted for Curvature

Anna C. Gilbert, Kevin O'Neill

TL;DR

CA-PCA is developed, a version of local PCA based instead on a calibration of a quadratic embedding, acknowledging the curvature of the underlying manifold, and shows that this adaptation improves the estimator in a wide range of 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.

CA-PCA: Manifold Dimension Estimation, Adapted for Curvature

TL;DR

CA-PCA is developed, a version of local PCA based instead on a calibration of a quadratic embedding, acknowledging the curvature of the underlying manifold, and shows that this adaptation improves the estimator in a wide range of 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.
Paper Structure (23 sections, 5 theorems, 88 equations, 9 figures, 1 algorithm)

This paper contains 23 sections, 5 theorems, 88 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Problem Setup
  3. Main Results and Proposed Method: CA-PCA
  4. First-Order Behavior of Eigenvalues
  5. Proposed Method: CA-PCA
  6. Proof of Proposition \ref{['prop:main prop']}
  7. Experiments
  8. Outline
  9. Some Synthetic Data
  10. Some Simulated Data
  11. Analysis of Non-Manifolds
  12. Robustness to Error
  13. Varying Curvature and Sample Size
  14. Conclusion
  15. Integral Computations
  16. ...and 8 more sections

Key Result

Lemma 2.1

\newlabellemma:distribution for ball0 Let $W$ be a $d$-dimensional subspace of $\mathbb{R}^D$ ($D\ge d$) and let $\nu$ denote the $d$-dimensional Lebesgue measure on $W$ intersected with the unit ball of $\mathbb{R}^D$. Then

Figures (9)

  • Figure 1: Results for Synthetic Data
  • Figure 2: Synthetic Manifolds in Higher Dimensions
  • Figure 3: Standard Deviation of Estimates for Synthetic Manifolds in Higher Dimensions
  • Figure 4: Standard Deviation of Estimates for Synthetic Manifolds in Higher Dimensions
  • Figure 5: Results for Simulated Data
  • ...and 4 more figures

Theorems & Definitions (7)

  • Lemma 2.1: Lemma 6.1 in lim2021tangent, Lemma 13 in arias2017spectral
  • Proposition 3.1
  • Lemma 3.2
  • Proof 1
  • Theorem A.1
  • Lemma A.2
  • Proof 2