2019/09/30 by Zhigang Yao, Yao, Zhigang, Bingjie Li +3
Computer Science · Mathematics · #Advanced Vision and Imaging #FOS: Computer and information sciences #Human Pose and Action Recognition #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Video Surveillance and Tracking Methods #cs.LG #stat.ML
paper · pdf · doi:10.48550/arxiv.1909.13492
34 pages
openalex publication_date 2019/09/30 · arxiv created 2022/02/04 · arxiv updated 2022/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure, namely a manifold. Manifold learning is an emerging field for learning the underlying structure. The study of manifold learning can be split into two main branches: dimension reduction and manifold fitting. With the aim of combining statistics and geometry, we address the problem of manifold fitting in the ambient space. Inspired by the relation between the eigenvalues of the Laplace-Beltrami operator and the geometry of a manifold, we aim to find a small set of points that preserve the geometry of the underlying manifold. From this relationship, we extend the idea of subsampling to sample points in high-dimensional space and employ the Moving Least Squares (MLS) approach to approximate the underlying manifold. We analyze the two core steps in our proposed method theoretically and also provide the bounds for the MLS approach. Our simulation results and theoretical analysis demonstrate the superiority of our method in estimating the underlying manifold.