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.
Author(s): Boissonnat, Jean-Daniel; Chazal, Frédéric; Yvinec, Mariette
Publisher: Cambridge University Press
Year: 2018
Language: English
Pages: 247
Tags: Geometry; Topology
01.0_pp_i_iv_frontmatter......Page 2
02.0_pp_v_vii_Contents......Page 6
03.0_pp_viii_xii_Introduction......Page 9
04.0_pp_1_2_Topological_Preliminaries......Page 14
04.1_pp_3_9_Topological_Spaces......Page 16
04.2_pp_10_20_Simplicial_Complexes......Page 23
05.0_pp_21_22_Delaunay_Complexes......Page 34
05.1_pp_23_43_Convex_Polytopes......Page 36
05.2_pp_44_68_Delaunay_Complexes......Page 57
05.3_pp_69_97_Good_Triangulations......Page 82
05.4_pp_98_112_Delaunay_Filtrations......Page 111
06.0_pp_113_114_Reconstruction_of_Smooth_Submanifolds......Page 126
06.1_pp_115_136_Triangulation_of_Submanifolds......Page 128
06.2_pp_137_160_Reconstruction_of_Submanifolds......Page 150
07.0_pp_161_162_Distance-Based_Inference......Page 174
07.1_pp_163_179_Stability_of_Distance_Functions......Page 176
07.2_pp_180_196_Distance_to_Probability_Measures......Page 193
07.3_pp_197_223_Homology_Inference......Page 210
08.0_pp_224_230_Bibliography......Page 237
09.0_pp_231_234_Index......Page 244
