Geometric and Topological Inference

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.

Part I. Topological Preliminaries: 1. Topological spaces; 2. Simplicial complexes; Part II. Delaunay 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; 10. Distance to probability measures; 11. Homology inference.

Jean-Daniel Boissonnat is a Research Director at the Institut national de recherche en informatique et en automatique, France. His research interests are in computational geometry and topology. He has published several books and more than 180 research papers, and is on the editorial board of the Journal of the ACM and of Discrete and Computational Geometry. He received the IBM award in Computer Science in 1987, the EADS award in Information Sciences in 2006 and was awarded an advanced grant from the European Research Council in 2014. He has taught at several universities in Paris and at the Collège de France.
Frédéric Chazal is a Research Director at the Institut national de recherche en informatique et en automatique, France, where he is heading the DataShape team, a pioneering and world leading group in computational geometry and topological data analysis. His current primary research is on topological data analysis and its connections with statistics and machine learning, and he has authored several reference papers in this domain. He is an associate editor of four international journals and he teaches topological data analysis in various universities and engineering schools in the Paris area.
Mariette Yvinec was a Researcher at the Institut national de recherche en informatique et en automatique, France. She is a specialist in the field of shape reconstruction and meshing, and taught master's courses on the subject in various universities in Paris. She co-authored a reference book on computational geometry with Jean-Daniel Boissonnat, and played an active role in the design and development of the software library CGAL.