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Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions

Zhengang Zhong, Yury Korolev, Matthew Thorpe

Abstract

Laplace learning is a semi-supervised method, a solution for finding missing labels from a partially labeled dataset utilizing the geometry given by the unlabeled data points. The method minimizes a Dirichlet energy defined on a (discrete) graph constructed from the full dataset. In finite dimensions the asymptotics in the large (unlabeled) data limit are well understood with convergence from the graph setting to a continuum Sobolev semi-norm weighted by the Lebesgue density of the data-generating measure. The lack of the Lebesgue measure on infinite-dimensional spaces requires rethinking the analysis if the data aren't finite-dimensional. In this paper we make a first step in this direction by analyzing the setting when the data are generated by a Gaussian measure on a Hilbert space and proving pointwise convergence of the graph Dirichlet energy.

Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions

Abstract

Laplace learning is a semi-supervised method, a solution for finding missing labels from a partially labeled dataset utilizing the geometry given by the unlabeled data points. The method minimizes a Dirichlet energy defined on a (discrete) graph constructed from the full dataset. In finite dimensions the asymptotics in the large (unlabeled) data limit are well understood with convergence from the graph setting to a continuum Sobolev semi-norm weighted by the Lebesgue density of the data-generating measure. The lack of the Lebesgue measure on infinite-dimensional spaces requires rethinking the analysis if the data aren't finite-dimensional. In this paper we make a first step in this direction by analyzing the setting when the data are generated by a Gaussian measure on a Hilbert space and proving pointwise convergence of the graph Dirichlet energy.
Paper Structure (19 sections, 11 theorems, 107 equations, 12 figures)

This paper contains 19 sections, 11 theorems, 107 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Setting and Main Results
  3. Background Material
  4. Gaussian Distributions
  5. Technical Preliminary Results
  6. Asymptotics of Discrete and Nonlocal Functionals
  7. Large Data Consistency Results
  8. Continuum Localization Limits
  9. Numerical Experiments
  10. Methods
  11. Laplace Learning in Finite Dimensions: the Unnormalized Laplacian
  12. Laplace Learning in Infinite Dimensions: the Normalized Laplacian
  13. Shifted White Noise
  14. Features and Labels
  15. Results
  16. ...and 4 more sections

Key Result

Theorem 2.2

Suppose Assumptions ass:Setting:X - ass:Setting:eta are satisfied. Let $\alpha\in (-1,+\infty)$ and $\beta\in\mathbb{R}$. Assume $u\in\mathrm{C}^{2}(X^\beta)$, $\sum_{i=1}^\infty \lambda_i^{\beta+\frac{1}{2}-\frac{\alpha}{2}}<+\infty$, $\sum_{i=1}^\infty\lambda_i^{\alpha+1}<+\infty$, $\sum_{i=1}^\in almost surely.

Figures (12)

  • Figure 1: Results for shifted white noise data for different resolutions (row labels) and different methods (column labels). See Section \ref{['sec:white_noise']} for details. Whilst at low resolutions both methods perform similarly, the labeling accuracy of the unnormalized Laplacian with the Euclidean geometry drops at higher resolutions while the normalized Laplacian with the $\mathrm{H}^{-s}$ geometry ($s=1.01$) performs consistently well at all resolutions.
  • Figure 2: Labeling error for shifted white noise data at different resolutions. Solid line: mean error; dashed lines: maximum and minimum errors. See Section \ref{['sec:white_noise']} for details. The accuracy of the normalized Laplacian with $\mathrm{H}^{-s}$ geometry ($s=1.01$) remains good across all resolutions whilst that of the unnormalised Laplacian with the Euclidean geometry and the normalized Laplacian with $\mathrm{L}^{2}$ geometry deteriorates at high resolutions.
  • Figure : Avg value labeling: True labels
  • Figure : Avg value labeling: True labels
  • Figure : Avg value labeling: Predicted labels
  • ...and 7 more figures

Theorems & Definitions (29)

  • Remark 2.1
  • Theorem 2.2: Pointwise Consistency
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1: Gaussian measures on separable Hilbert space
  • Definition 3.2
  • Definition 3.3: Cameron-Martin space
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.6
  • ...and 19 more