Theoretical Foundations of Principal Manifold Estimation with Non-Euclidean Templates

TL;DR

This paper establishes a rigorous theoretical framework for estimating principal manifolds with non-Euclidean topology from high-dimensional data, including convergence, consistency, and complexity selection methods.

Contribution

It introduces a novel theoretical foundation for principal manifold estimation accommodating complex topologies using Sobolev spaces on Riemannian manifolds.

Findings

Proves convergence of the iterative principal manifold estimation algorithm.

Shows consistency of the finite-sample estimator.

Provides a new method for selecting the complexity level of fitted manifolds.

Abstract

We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with general, potentially non-Euclidean topology, which can be inferred using tools from topological data analysis. Using the theory of Sobolev spaces on Riemannian manifolds, we establish that the proposed principal manifolds are well defined, prove convergence of the iterative algorithm used to compute them, and show consistency of the finite-sample estimator. Furthermore, we introduce a novel method for selecting the complexity level of a fitted manifold, which addresses the shortcomings of the classical fitting-error criterion. We also provide a detailed geometric interpretation of the penalty term in our framework. In addition to the theoretical developments, we present extensive numerical experiments supporting our results. This article provides theoretical foundations for approaches that have been used in applications such as robotics. More importantly, it extends these approaches to general topological settings with potential applications across a broad range of disciplines, including neuroimaging and shape data analysis.