Model Selection for Simplicial Approximation

• Published in: Foundations of computational mathematics Vol. 11; no. 6; pp. 707 - 731
• Main Authors: Caillerie, Claire, Michel, Bertrand
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.12.2011; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Approximation; Calibration; Complex systems; Computational mathematics; Computer Science; Computer simulation; Criteria; Economics; Foundations; Geometry; Heuristic; Least squares method; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; 55U10; Computational geometry; Penalization; Model selection; 62H30; Spectral clustering; Simplicial complexes; Slope heuristics; 05E45; 62H12; Geometric inference
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-011-9103-7

In the computational geometry field, simplicial complexes have been used to describe an underlying geometric shape knowing a point cloud sampled on it. In this article, an adequate statistical framework is first proposed for the choice of a simplicial complex among a parametrized family. A least-squares penalized criterion is introduced to choose a complex, and a model selection theorem states how to select the “best” model, from a statistical point of view. This result gives the shape of the penalty, and then the “slope heuristics method” is used to calibrate the penalty from the data. Some experimental studies on simulated and real datasets illustrate the method for the selection of graphs and simplicial complexes of dimension two.