2008/07/29 by Nikolay Balov, Nikolay H. Balov, Balov, Nikolay H.
Mathematics · Physics and Astronomy · #Computation (stat.CO) #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Morphological variations and asymmetry #Numerical methods in inverse problems #Statistical Mechanics and Entropy #Statistics Theory (math.ST) #math.DG #math.ST #stat.CO #stat.TH
paper · pdf · doi:10.48550/arxiv.0807.4690
28 pages, core thesis paper
openalex publication_date 2008/07/29 · arxiv created 2009/01/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and study covariance fields of distributions on a Riemannian manifold. At each point on the manifold, covariance is defined to be a symmetric and positive definite (2,0)-tensor. Its product with the metric tensor specifies a linear operator on the respected tangent space. Collectively, these operators form a covariance operator field. We show that, in most circumstances, covariance fields are continuous. We also solve the inverse problem: recovering distribution from a covariance field. Surprisingly, this is not possible on Euclidean spaces. On non-Euclidean manifolds however, covariance fields are true distribution representations.