Topological Complexity of Smooth Random Functions

Ecole d'Ete de Probabilites de Saint-Flour XXXIX-2009

By: Jonathan E. Taylor, Robert Adler

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Paperback | 18 May 2011 | Edition Number 2011

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These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors' 2007 Springer monograph "Random Fields and Geometry." While not as exhaustive as the full monograph, they are also less exhausting, while still covering the basic material, typically at a more intuitive and less technical level. They also cover some more recent material relating to random algebraic topology and statistical applications. The notes include an introduction to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness. This is followed by a quick review of geometry, both integral and Riemannian, with an emphasis on tube formulae, to provide the reader with the material needed to understand and use the Gaussian kinematic formula, the main result of the notes. This is followed by chapters on topological inference and random algebraic topology, both of which provide applications of the main results.

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"These little lecture notes are a rare delight. The authors succeed in an impressive manner to combine a writing style that focuses on the main ideas and intuitions while still stating all the results in full mathematical rigor. They take the reader on an exciting journey through the theories of Gaussian processes and differential topology and geometry and then show how fascinating mathematics arises when combining these fields not to speak about the wide range of applications." (H. M. Mai, Zentralblatt MATH, Vol. 1230, 2012)

"This concise book is written for graduate students as well as researchers who want to learn the state of the art of geometry of smooth Gaussian (and Gaussian-related) random fields and their significant applications. ... The authors have done an excellent job in showing not only the mathematical beauty and the essence of the 'Gaussian Kinematic Formulae', but also their powerful applicability. The book is very interesting to read." (Yimin Xiao, Mathematical Reviews, Issue 2012 h)

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Introduction	p. 1
Random Fields on Stratified Manifolds	p. 1
opological Complexity	p. 3
Random Fields and Complexity	p. 7
Statistical Implications	p. 10
Connections with Sample Path Behaviour	p. 10
Geometry	p. 11
Gaussian Processes	p. 13
The Cosine Process	p. 14
The Cosine Field	p. 16
Constructing Gaussian Processes	p. 18
The Canonical Process on S(R^l)	p. 20
The Canonical Processes and Exceedence Probabilities	p. 22
The Canonical Process and Geometry	p. 24
The Basic Theory of Gaussian Fields	p. 25
Regularity for Gaussian Process	p. 26
Gaussian Fields on R^N	p. 28
Differentiability	p. 29
Stationarity, Isotropy, and Constant Variance	p. 31
Spectral Moments and Derivatives of Random Fields	p. 31
Local Isotropy and the Induced Metric	p. 33
Three Gaussian Facts	p. 34
Some Geometry and Some Topology	p. 37
Some Notation for Riemannian Manifolds	p. 37
Coarea Formula	p. 39
Stratified Manifolds	p. 40
Tube Formulae and Lipschitz-Killing Curvatures	p. 42
Describing Tubes	p. 43
Computing Volumes	p. 46
Intrinsic Volumes	p. 50
Probabilities of Tubes: Gaussian Minkowski Functionals	p. 51
Kinematic Formulae	p. 54
Crofton's Formula	p. 55
Morse's Theorem	p. 56
The Gaussian Kinematic Formula	p. 59
The Kac-Rice Metatheorem	p. 60
Real Isotropic Fields on Rectangles: Euler Characteristic	p. 65
Real Isotropic Fields: Lipschitz-Killing Curvatures	p. 70
Real Stationary Fields on Rectangles: Euler Characteristic	p. 71
The Induced Metric and the Need for Riemannian Geometry	p. 73
The Canonical Isotropic Process on the Sphere	p. 77
A Model Process	p. 77
The GKF for the Model Process	p. 79
Back to the Canonical Process	p. 81
Fields with Finite Expansions	p. 82
The GKF in the General Case	p. 83
Not Just Excursion Sets	p. 84
Infinite Dimensions	p. 84
On Applications: Topological Inference	p. 87
Local Structure of Extrema and the Euler Characteristic Heuristic	p. 88
Gaussian Related Random Fields	p. 91
² Fields	p. 93
Brain Imaging	p. 94
Estimating Lipschitz-Killing Curvatures	p. 98
Cosmology	p. 102
Algebraic Topology of Excursion Sets: A New Challenge	p. 107
Persistent Homology and Barcodes	p. 108
Barcodes of Excursion Sets	p. 108
Barcode Distributions	p. 110
The Mean Euler Characteristic of the Barcodes of Gaussian Excursion Sets	p. 112
References	p. 115
Notation Index	p. 118
Subject Index	p. 121
Table of Contents provided by Ingram. All Rights Reserved.

Topological Complexity of Smooth Random Functions

Ecole d'Ete de Probabilites de Saint-Flour XXXIX-2009

At a Glance

Paperback

Industry Reviews

Shipping

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