[Paper Review] The Diffusion Geometry of Fibre Bundles: Horizontal Diffusion Maps

Tingran Gao|arXiv (Cornell University)|Feb 7, 2016

Topological and Geometric Data Analysis70 references3 citations

TL;DR

This paper introduces Horizontal Diffusion Maps (HDM), a nonparametric framework that extends diffusion maps to data with structural correspondences by modeling datasets as fibre bundles with connections. By incorporating horizontal diffusion processes on the unit tangent bundle, HDM captures sub-Riemannian geometry and asymptotically converges to a horizontal Laplacian, enabling improved dimensionality reduction for complex, high-dimensional data with functional pairwise relations.

ABSTRACT

Kernel-based non-linear dimensionality reduction methods, such as Local Linear Embedding (LLE) and Laplacian Eigenmaps, rely heavily upon pairwise distances or similarity scores, with which one can construct and study a weighted graph associated with the dataset. When each individual data object carries additional structural details, however, the correspondence relations between these structures provide extra information that can be leveraged for studying the dataset using the graph. Based on this observation, we generalize Diffusion Maps (DM) in manifold learning and introduce the framework of Horizontal Diffusion Maps (HDM). We model a dataset with pairwise structural correspondences as a fibre bundle equipped with a connection. We demonstrate the advantage of incorporating such additional information and study the asymptotic behavior of HDM on general fibre bundles. In a broader context, HDM reveals the sub-Riemannian structure of high-dimensional datasets, and provides a nonparametric learning framework for datasets with structural correspondences.

Motivation & Objective

• To address the limitation of standard diffusion maps in capturing complex, non-scalar pairwise relations in high-dimensional data with internal structure.
• To model data objects as points on a base manifold and their internal structures (e.g., shape landmarks, image pixels) as fibres in a fibre bundle with a connection.
• To develop a graph-based method that leverages correspondence relations between structural elements—not just scalar distances—for improved geometric learning.
• To establish the asymptotic convergence of HDM to the horizontal Laplacian on the unit tangent bundle, ensuring theoretical consistency.
• To provide a framework for analyzing datasets with multiplex, heterogeneous, or time-varying relations beyond simple graph abstractions.

Proposed method

• Model the dataset as a fibre bundle where data objects are base manifold points and internal data points (e.g., landmarks) are fibres, with a connection encoding structural correspondences.
• Define horizontal random walks on the unit tangent bundle (UTM) of the base manifold, using parallel transport to propagate information across fibres.
• Construct graph horizontal Laplacians using a kernel function that depends on both base manifold distance and the distance between transported fibre points.
• Use a double-scale kernel $ K_{ ho, ho}( au_i, au_j) $ combining base distance $ \|\xi_i - \xi_j\|^2 / \epsilon $ and horizontal transport error $ \|P_{\xi_j,\xi_i} \bar{\tau}_{i,r} - \bar{\tau}_{j,s}\|^2 / \delta $.
• Derive infinitesimal generators for horizontal and vertical diffusion operators on UTM, showing convergence to the horizontal Laplacian $ \Delta_{UTM}^H $ under asymptotic conditions.
• Establish convergence rates via law of large numbers and error bounds involving $ \epsilon_{\text{PCA}}^{1/2} + \epsilon^{3/2} $, with correction terms scaling as $ \delta^{-1}(\epsilon_{\text{PCA}}^{1/2} + \epsilon^{3/2}) $.

Experimental results

Research questions

• RQ1 How can diffusion maps be generalized to incorporate non-scalar, functional pairwise relations between data structures?
• RQ2 What is the asymptotic behavior of the proposed Horizontal Diffusion Map (HDM) on general fibre bundles with connections?
• RQ3 How does the inclusion of structural correspondences improve the geometric representation of high-dimensional data compared to standard diffusion maps?
• RQ4 What is the convergence rate of the empirical HDM operator to its population counterpart under finite sampling?
• RQ5 Can HDM recover the underlying sub-Riemannian geometry of data with complex internal structures?

Key findings

• HDM asymptotically converges to the horizontal Laplacian $ \Delta_{UTM}^H $ on the unit tangent bundle, with correction terms scaling as $ \epsilon \cdot \frac{m_{21}}{2m_0} \Delta_{UTM}^H $ and $ \delta \cdot \frac{m_{22}}{2m_0} \Delta_{UTM}^V $.
• The empirical HDM operator $ \hat{H}_{\epsilon,\delta}^{(\alpha)} $ converges to the true operator $ \tilde{H}_{\epsilon,\delta}^{(\alpha)} $ with error $ O(\delta^{-1}(\epsilon_{\text{PCA}}^{1/2} + \epsilon^{3/2})) $, which vanishes as $ \epsilon \to 0 $ under the condition $ \delta^{-1}(\epsilon_{\text{PCA}}^{1/2} + \epsilon^{3/2}) \to 0 $.
• The framework successfully captures sub-Riemannian structure in high-dimensional data by leveraging horizontal diffusion on the unit tangent bundle.
• Numerical experiments confirm the theoretical convergence rates and demonstrate improved performance on geometric morphometrics data with complex shape correspondences.
• The method outperforms standard diffusion maps by preserving richer geometric and structural information encoded in optimal correspondences between data objects.
• Theoretical analysis shows that the kernel-based HDM operator on UTM asymptotically approximates the horizontal diffusion process, with error bounded by PCA noise and bandwidth terms.