Euler Integration of Gaussian Random Fields and Persistent Homology

Abstract

In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.

21 pages, 1 figure

References in corpus (3)

• Validity of the expected Euler characteristic heuristic
• A Gaussian kinematic formula
• Euler integration over definable functions

Cited by in corpus (13)

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• Is the Observable Universe Consistent with the Cosmological Principle?
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• Topology and Geometry of Gaussian random fields I: on Betti Numbers, Euler characteristic and Minkowski functionals
• A fractal dimension for measures via persistent homology
• DTM-based Filtrations
• Persistent Magnitude
• Probing multipartite entanglement through persistent homology
• Hybrid transforms of constructible functions
• Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds
• The self-similar evolution of stationary point processes via persistent homology
• Expected Lipschitz-Killing curvatures for spin random fields and other non-isotropic fields
• Hadwiger's Theorem for Definable Functions