MANIFOLD LEARNING IN METRIC SPACES

Area of Science:

• Machine Learning
• Data Science
• Topology

Background:

• Laplacian-based methods are widely used for dimensionality reduction of data in Euclidean space (ℝN).
• Theoretical guarantees for these methods often rely on the Euclidean distance approximating geodesic distances on data submanifolds.
• Alternative distance metrics, such as the Wasserstein distance, may be more suitable for certain datasets than the Euclidean distance.

Purpose of the Study:

• To generalize manifold learning to arbitrary metric spaces.
• To establish theoretical conditions for the convergence of graph Laplacians when using non-Euclidean metrics.
• To explore the applicability of metrics beyond Euclidean distance in dimensionality reduction.

Main Methods:

• Development of a generalized theoretical framework for manifold learning in metric spaces.
• Analysis of the pointwise convergence of the graph Laplacian operator.
• Investigation of metric properties required for convergence guarantees.

Main Results:

• A framework is presented that extends manifold learning to general metric spaces.
• Sufficient conditions are identified for the pointwise convergence of the graph Laplacian in these generalized settings.
• The study demonstrates the theoretical possibility of using metrics like Wasserstein distance for dimensionality reduction.

Conclusions:

• The proposed framework broadens the applicability of Laplacian-based dimensionality reduction techniques.
• The findings provide theoretical underpinnings for using diverse distance metrics in manifold learning.
• This research opens avenues for applying advanced dimensionality reduction to complex data structures where Euclidean distance is suboptimal.