Tangent Bundle Manifold Learning via Grassmann&Stiefel Eigenmaps

TL;DR

This paper introduces Tangent Bundle Manifold Learning, which improves manifold reconstruction by aligning tangent spaces, providing a new algorithm that enhances traditional methods.

Contribution

It proposes a novel Tangent Bundle ML approach that incorporates tangent space proximity, along with a new algorithm for improved manifold learning.

Findings

Derived a local lower bound for reconstruction error based on tangent space distances

Proposed an amplification of ML that aligns tangent spaces for better manifold estimation

Presented a new algorithm implementing Tangent Bundle ML

Abstract

One of the ultimate goals of Manifold Learning (ML) is to reconstruct an unknown nonlinear low-dimensional manifold embedded in a high-dimensional observation space by a given set of data points from the manifold. We derive a local lower bound for the maximum reconstruction error in a small neighborhood of an arbitrary point. The lower bound is defined in terms of the distance between tangent spaces to the original manifold and the estimated manifold at the considered point and reconstructed point, respectively. We propose an amplification of the ML, called Tangent Bundle ML, in which the proximity not only between the original manifold and its estimator but also between their tangent spaces is required. We present a new algorithm that solves this problem and gives a new solution for the ML also.