A Tangent Distance Preserving Dimensionality Reduction Algorithm

TL;DR

This paper introduces TDPM, a nonlinear dimensionality reduction method that preserves the manifold's folded structure by using tangent distances and MDS, offering an alternative to unfolding-based methods.

Contribution

The paper proposes a novel algorithm, TDPM, which preserves the nonlinear structure of manifolds using tangent distances instead of geodesic distances.

Findings

TDPM effectively preserves manifold structure in low-dimensional embeddings.

It outperforms traditional unfolding methods in certain nonlinear scenarios.

The method provides insights into the folded structure of high-dimensional data.

Abstract

This paper considers the problem of nonlinear dimensionality reduction. Unlike existing methods, such as LLE, ISOMAP, which attempt to unfold the true manifold in the low dimensional space, our algorithm tries to preserve the nonlinear structure of the manifold, and shows how the manifold is folded in the high dimensional space. We call this method Tangent Distance Preserving Mapping (TDPM). TDPM uses tangent distance instead of geodesic distance, and then applies MDS to the tangent distance matrix to map the manifold into a low dimensional space in which we can get its nonlinear structure.