1. Hodge Theory on Metric Spaces
- Author
-
Bartholdi, Laurent, Schick, Thomas, Smale, Nat, Smale, Steve, and Baker, Anthony W.
- Subjects
Computational Geometry (cs.CG) ,FOS: Computer and information sciences ,Mathematics(all) ,Machine Learning (stat.ML) ,Homology (mathematics) ,Theoretical Computer Science ,Separable space ,Mathematics - Geometric Topology ,Statistics - Machine Learning ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Mathematics ,Probability measure ,Applied Mathematics ,Hodge theory ,K-Theory and Homology (math.KT) ,Geometric Topology (math.GT) ,Numerical Analysis (math.NA) ,Algebra ,Computational Mathematics ,Metric space ,Compact space ,Computational Theory and Mathematics ,Computer Science::Computer Vision and Pattern Recognition ,Mathematics - K-Theory and Homology ,Computer Science - Computational Geometry ,Analysis ,58A14, 54E05, 55P55, 57M50 - Abstract
Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional \alpha-scale homology., Comment: appendix by Anthony W. Baker, 48 pages, AMS-LaTeX. v2: final version, to appear in Foundations of Computational Mathematics. Minor changes and additions
- Published
- 2011