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Hodge Laplacians and Hodge Diffusion Maps

Alvaro Almeida Gomez, Jorge Duque Franco

TL;DR

The paper presents Hodge Diffusion Maps, a higher-order manifold learning framework that approximates the exterior derivative on $k$-forms to estimate the Hodge Laplacian $\Delta^k$ from sample data. Building a matrix-based pipeline with Local PCA tangent estimation and differential-array representations, it derives a practical algorithm to compute $\mathbf{d}_k$ via an operator $\mathbf{P}_t$ and a sparse block structure $\mathbf{ED}_k$, yielding the Hodge-Laplacian matrix $\mathbf{H}_{k,t}$. The authors prove that, as $t\to 0$, the projected operator recovers $\hat{\mathbf{d}}_k(W)$, and they provide normalization and truncation error controls for the resulting diffusion embedding. Numerical experiments on a torus and a sphere show that Hodge Diffusion Maps uncover topological partitions (vertical sections) and outperform several baseline methods in revealing higher-order structure, with potential applications in pre-processing for Cryo-EM and regularization in learning systems.

Abstract

We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms, thereby providing an approximation of the Hodge Laplacian operator. Hodge Diffusion Maps extend existing non-linear dimensionality reduction techniques, including vector diffusion maps, as well as the theories behind diffusion maps and Laplacian Eigenmaps. Our approach captures higher-order topological features of the data-set by projecting it into lower-dimensional Euclidean spaces using the Hodge Laplacian. We develop a theoretical framework to estimate the approximation error of the exterior derivative, based on sample points distributed over a real manifold. Numerical experiments support and validate the proposed methodology.

Hodge Laplacians and Hodge Diffusion Maps

TL;DR

The paper presents Hodge Diffusion Maps, a higher-order manifold learning framework that approximates the exterior derivative on -forms to estimate the Hodge Laplacian from sample data. Building a matrix-based pipeline with Local PCA tangent estimation and differential-array representations, it derives a practical algorithm to compute via an operator and a sparse block structure , yielding the Hodge-Laplacian matrix . The authors prove that, as , the projected operator recovers , and they provide normalization and truncation error controls for the resulting diffusion embedding. Numerical experiments on a torus and a sphere show that Hodge Diffusion Maps uncover topological partitions (vertical sections) and outperform several baseline methods in revealing higher-order structure, with potential applications in pre-processing for Cryo-EM and regularization in learning systems.

Abstract

We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms, thereby providing an approximation of the Hodge Laplacian operator. Hodge Diffusion Maps extend existing non-linear dimensionality reduction techniques, including vector diffusion maps, as well as the theories behind diffusion maps and Laplacian Eigenmaps. Our approach captures higher-order topological features of the data-set by projecting it into lower-dimensional Euclidean spaces using the Hodge Laplacian. We develop a theoretical framework to estimate the approximation error of the exterior derivative, based on sample points distributed over a real manifold. Numerical experiments support and validate the proposed methodology.

Paper Structure

This paper contains 21 sections, 8 theorems, 183 equations, 13 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diffusion Maps
  4. Hodge Laplacians
  5. Exterior derivative approximation
  6. Differential forms and differential arrays
  7. Approximation of the exterior derivative using sample points
  8. Matrix-based computations
  9. Matrix representation of differential arrays
  10. Matrix-based computation of the exterior derivative
  11. Implementation of the Algorithm
  12. Hodge Diffusion-Maps
  13. Hodge Laplacians approximation
  14. Hodge Diffusion-Maps and Hodge Diffusion-Distance
  15. Numerical Experiments
  16. ...and 6 more sections

Key Result

Theorem 3.1

Let $W \in \Theta^k(\mathcal{M})$ be a $k$-differential array, and let $x \in \mathcal{M}.$ For any $\delta$ satisfying where $d$ is the dimension of $\mathcal{M},$ the following estimate holds: where the exponent $f$ is given by In particular, taking the limit as $t\to 0^+,$ we obtain

Figures (13)

  • Figure 1: Top: The first and second coordinates of the parametrization system for the torus. Bottom: The dataset $X$ plotted on the torus $T^2$, with the colorbar indicating the order of the sample points.
  • Figure 2: Top: The first and second coordinates of the parametrization system for the sphere. Bottom: The dataset $X$ plotted on the sphere $S^2$, with the colorbar indicating the order of the sample points.
  • Figure 3: Plot of the $(c_1, c_2)$ components, where $1 \leq c_1 \leq c_2 \leq 3$, of the first order normalized Hodge Diffusion Maps $\eta^{\textbf{1}}_{1,3}$ as given in Eq. \ref{['embnormalizada']}.
  • Figure 4: Plot of the $(c_1, c_2)$ components, where $1 \leq c_1 \leq c_2 \leq 3$, of the second order normalized Hodge Diffusion Maps $\eta^{\textbf{1}}_{2,3}$ as given in Eq. \ref{['embnormalizada']}
  • Figure 5: Plot of the $(c_1, c_2)$ components, where $1 \leq c_1 \leq c_2 \leq 3$, of vector diffusion maps.
  • ...and 8 more figures

Theorems & Definitions (16)

  • Remark 3.1
  • Theorem 3.1
  • Remark 4.1
  • Remark 5.1
  • Definition 1
  • Definition 2
  • Proposition A.1
  • Example 1
  • Remark A.1
  • Remark A.2
  • ...and 6 more