Geometric and topological inference

Description

Geometric and topological inference deals with the retrieval of information about a geometric object using only a finite set of possibly noisy sample points. It has connections to manifold learning and provides the mathematical and algorithmic foundations of the rapidly evolving field of topological data analysis. Building on a rigorous treatment of simplicial complexes and distance functions, this self-contained book covers key aspects of the field, from data representation and combinatorial questions to manifold reconstruction and persistent homology. It can serve as a textbook for graduate students or researchers in mathematics, computer science and engineering interested in a geometric approach to data science.  Read more... Abstract: This book offers a rigorous introduction to geometric and topological inference, a rapidly evolving field that intersects computational geometry, applied topology, and data analysis. It can serve as a textbook for graduate students or researchers in mathematics, computer science and engineering interested in a geometric approach to data science.  Read more... Content: Part I. Topological Preliminaries : 1. Topological Spaces -- 2. Simplicial Complexes -- Part II. Display Complexes : 3. Convex Polytopes -- 4. Delaunay Complexes -- 5. Good Triangulations -- 6. Delaunay Filtrations -- Part III. Reconstruction of Smooth Submanifolds : 7. Triangulation of Submanifolds -- 8. Reconstruction of Submanifolds -- Part IV. Distance-Based Inference : 9. Stability of Distance Functions -- 19. Distance to Probability Measures -- 11. Homology Inference.