Manifold learning in metric spaces (2503.16187v1)

Abstract: Laplacian-based methods are popular for dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance approximates the geodesic distance on the underlying submanifold which the data are assumed to lie on. However, for some applications, other metrics, such as the Wasserstein distance, may provide a more appropriate notion of distance than the Euclidean distance. We provide a framework that generalizes the problem of manifold learning to metric spaces and study when a metric satisfies sufficient conditions for the pointwise convergence of the graph Laplacian.

• The paper extends classical manifold learning to arbitrary metric spaces by establishing conditions under which graph Laplacians constructed from sampled data converge to intrinsic differential operators on the manifold.
• Theoretical conditions like the Uniform First Order Approximation and a Third Order Condition on the metric are introduced to ensure it locally approximates the geodesic distance, validating the convergence results.
• This generalized framework allows for manifold learning with non-Euclidean metrics, such as the Wasserstein distance, which is crucial for analyzing complex data where Euclidean norms are unsuitable.

Overview

The paper "Manifold learning in metric spaces" (2503.16187) extends classic manifold learning techniques by generalizing them to arbitrary metric spaces, rather than restricting the analysis to Euclidean embeddings. This work rigorously establishes conditions under which graph Laplacian constructions—often used for dimensionality reduction and unsupervised learning—converge to intrinsic differential operators (typically a multiple of the Laplace-Beltrami operator) on submanifolds embedded in a more general metric space. In doing so, the work significantly broadens the scope of manifold learning, with particular attention to settings where non-Euclidean metrics (e.g., Wasserstein distances) provide a more relevant measure of similarity for complex data structures.

General Framework for Manifold Learning in Metric Spaces

The authors recast the conventional problem of manifold learning, where data points are typically embedded in $\mathbb{R}^N$, into a framework defined on a metric space $(X, d)$. Given a smooth manifold $\mathcal{M}$ embedded via a map $\iota : \mathcal{M} \to X$, the analysis is carried out under the assumption that the metric $d$ is sufficiently regular to capture the intrinsic geometry of $\mathcal{M}$. The central question is to determine when $d$ can accurately reflect the local geometry so that the graph Laplacian constructed from sampled data converges pointwise to a differential operator intrinsic to $\mathcal{M}$.

Theoretical Assumptions

Two primary assumptions underpin the analysis, ensuring that the metric $d$ locally approximates the geodesic distance $d_g$ induced by a Riemannian metric $g$ on $\mathcal{M}$:

• Uniform First Order Approximation: The metric $d$ must satisfy a local approximation property,

$$\sup_{x \in \mathcal{M}, \ g_x(v,v)=1} \left| d\left(x,\exp_x(tv)\right) - t \right| = o(t),$$

as $t\to 0$. This implies that, for points sufficiently close on $\mathcal{M}$, the metric $d$ recapitulates the first-order behavior of the geodesic distance.

• Third Order Condition: There exists a quantitative bound such that

$d_g^2(x, y) - d^2(x, y) < K\, d_g^4(x, y)$

for a constant $K$ and for all $x, y \in \mathcal{M}$ with $d_g(x, y)$ bounded by some $\epsilon_0$. This higher-order control ensures that the deviation of $d$ from the true geodesic metric is sufficiently small to guarantee convergence of the graph Laplacian.

These assumptions are critical because they guarantee that the constructed graph Laplacian based on $d$ will, under proper sampling and scaling conditions, converge to a rescaled version of the Laplace-Beltrami operator $\Delta_g$. Such convergence is central for ensuring that the manifold's intrinsic geometry is accurately captured and used for subsequent tasks such as dimensionality reduction or clustering.

Convergence of Graph Laplacians

A key theoretical contribution of this work lies in establishing the pointwise convergence of the graph Laplacian constructed from a kernel-based, neighborhood graph on the sampled data. Under the stated assumptions, the authors rigorously prove that:

$$\lim_{n \to \infty} \mathcal{L}_n f(x) = c\, \Delta_g f(x)$$

for all suitably regular functions $f$. Here, $\mathcal{L}_n$ denotes the graph Laplacian computed from $n$ data points, and $c$ is a scaling constant that depends on the kernel and the specific metric properties of $d$. This convergence result remains valid even in cases where data are sampled from a non-uniform distribution on $\mathcal{M}$, providing a robust theoretical underpinning for using graph-based manifold learning in a wide range of applications.

Non-Euclidean Metrics and the Wasserstein Distance

One of the important innovations in the paper is its focus on non-Euclidean metrics, notably the Wasserstein distance. The Wasserstein metric is particularly well-suited for data types where the Euclidean norm may fail to capture the underlying geometric or probabilistic structure—for example, in image processing or in the analysis of probability distributions.

In an illustrative example, the paper considers the embedding of circles into a function space. When using the standard $L^2$ norm, the intrinsic circular geometry becomes obscured in sparse data regimes. However, the Wasserstein-2 distance preserves the circular structure by ensuring that the distance between embedded circles corresponds to the Euclidean distance between their centers:

$$W_2^2\left(\iota(\theta), \iota(\phi)\right) \approx \| (\cos\theta, \sin\theta) - (\cos\phi, \sin\phi) \|^2.$$

This result underscores that by carefully selecting a metric that adheres to the outlined approximation conditions, one can facilitate a more faithful representation of the manifold geometry. The work also emphasizes that the Riemannian structure inherent in Wasserstein space naturally enables such generalizations, effectively treating the Wasserstein-2 metric as providing an intrinsic geometry for the embedded manifold.

Implementation Considerations

From an implementation perspective, extending manifold learning to metric spaces requires careful attention to several aspects:

• Metric Verification: Prior to constructing a graph Laplacian, it is crucial to verify that the metric $d$ satisfies the uniform first order approximation and third order conditions. This involves local estimates of $d(x, \exp_x(tv))$ and comparisons with $t$, as well as analyzing the residual $d_g^2(x, y) - d^2(x, y)$ for small geodesic distances.
• Kernel Selection and Scaling: The choice of kernel function and its scaling parameters must be compatible with the metric properties of $d$. The asymptotic convergence results depend on using appropriate kernel bandwidths that respect the local geometric structure as inferred from $d$.
• Computational Complexity: Working with non-Euclidean metrics like the Wasserstein distance often imposes higher computational overhead compared to Euclidean distances. Efficient approximations or scalable algorithms (such as entropic regularization techniques for Wasserstein distances) might be necessary, especially for large-scale applications.
• Graph Construction: Care must be taken during graph construction to ensure that the neighborhood relations among points reflect the local structure imposed by the metric $d$. This might involve using adaptive neighborhood sizes or anisotropic kernels that adjust according to the local density and geometry.
• Software Integration: Incorporating these generalized manifold learning techniques into existing pipelines may require extending standard libraries to handle arbitrary metrics. It is advisable to modularize the distance computation component, such that the same Laplacian construction algorithms can be applied regardless of the metric.

Conclusions

The extension of manifold learning to general metric spaces, as elaborated in the paper, provides a theoretically robust framework that accommodates non-Euclidean distances. By verifying sufficient conditions that guarantee the local approximation of geodesic distances, the convergence of graph Laplacians to the Laplace-Beltrami operator becomes assured even in settings where traditional Euclidean assumptions do not hold. Researchers can leverage these insights to design manifold learning algorithms that are both flexible and adaptable to complex, non-Euclidean data domains—thereby broadening the applicability of these techniques across diverse fields such as image analysis, shape recognition, and probabilistic data modeling.