Geometric and Topological Inference

A rigorous introduction to geometric and topological inference, for anyone interested in a geometric approach to data science.

Jean-Daniel Boissonnat (Author), Frédéric Chazal (Author), Mariette Yvinec (Author)

9781108410892, Cambridge University Press

Paperback / softback, published 27 September 2018

246 pages
22.9 x 15.3 x 1.4 cm, 0.35 kg

'… it is clear that this book addresses issues that are likely to be of some interest to budding researchers. I suspect that it provides them with as accessible an introduction to this material as is currently available.' Mark Hunacek, The Mathematical Gazette

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.

Subject Areas: Mathematical theory of computation [UYA], Topology [PBP], Geometry [PBM], Data analysis: general [GPH]