Manifold unwrapping using density ridges

TL;DR

This paper introduces two novel algorithms for unwrapping manifolds by leveraging density ridge estimation and principal curves, enabling effective unfolding of manifolds through gradient flow and parallel transport, with promising results on real and synthetic data.

Contribution

The paper presents new algorithms for manifold unwrapping based on density ridge estimation, addressing a gap in existing density ridge manifold learning methods.

Findings

Algorithms successfully unwrap manifolds in experiments

Results are comparable to state-of-the-art manifold learning methods

Effective unwrapping demonstrated on both real and synthetic data

Abstract

Research on manifold learning within a density ridge estimation framework has shown great potential in recent work for both estimation and de-noising of manifolds, building on the intuitive and well-defined notion of principal curves and surfaces. However, the problem of unwrapping or unfolding manifolds has received relatively little attention within the density ridge approach, despite being an integral part of manifold learning in general. This paper proposes two novel algorithms for unwrapping manifolds based on estimated principal curves and surfaces for one- and multi-dimensional manifolds respectively. The methods of unwrapping are founded in the realization that both principal curves and principal surfaces will have inherent local maxima of the probability density function. Following this observation, coordinate systems that follow the shape of the manifold can be computed by following the integral curves of the gradient flow of a kernel density estimate on the manifold. Furthermore, since integral curves of the gradient flow of a kernel density estimate is inherently local, we propose to stitch together local coordinate systems using parallel transport along the manifold. We provide numerical experiments on both real and synthetic data that illustrates clear and intuitive unwrapping results comparable to state-of-the-art manifold learning algorithms.