Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment

TLDR

The authors introduce a novel algorithm for manifold learning and nonlinear dimensionality reduction, and outline theoretical and algorithmic directions for future work. The method learns local geometry from noisy samples by estimating tangent spaces at each point and aligning them to recover global coordinates, demonstrated on curves and surfaces in 2D/3D and higher-dimensional Euclidean spaces. Error analysis indicates that reconstruction errors can be very small in certain cases.

Abstract

We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in two-dimensional/three-dimensional (2D/3D) Euclidean spaces and in higher-dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.