2004/01/01 by Zhenyue Zhang, Hongyuan Zha · 5 citations
Computer Science · Engineering · Mathematics · #Advanced Vision and Imaging #Advanced Numerical Analysis Techniques #Human Pose and Action Recognition #Tangent space #Mathematics #Nonlinear dimensionality reduction #Manifold (fluid mechanics) #Dimensionality reduction #Diffusion map #Euclidean space #Manifold alignment #Parameterized complexity #Tangent cone #Tangent #Tangent vector #Curse of dimensionality #Algorithm #Isomap #Pseudo-Riemannian manifold #Nonlinear system #Mathematical analysis #Geometry #Computer science #Curvature #Artificial intelligence
paper · doi:10.1137/s1064827502419154
openalex publication_date 2004/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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.