Euler Integration of Gaussian Random Fields and Persistent Homology

Abstract

<jats:p>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.</jats:p>

Persistent homology and applied homotopy theory

“…The paper [9] proves a result concerning the persistent homology of the sublevel sets of functions sampled from Gaussian random fields. We consider the real where it is understood that for any i such that x ≤ a i , the interval [a i , b i ] is simply deleted.…”

“…In [9], the following result is proved concerning the distribution of χ pers (M, f, x) for Gaussian random fields on Riemannian manifolds which produce Morse functions with probability one.…”

“…Then for any x ∈ R, we have where Remark 10.1 The point of this result is that it gives a theoretical estimate for the persistent Euler characteristic of sublevel sets in terms of classical invariants of the manifols. Also, the result in [9] is actually proved in a much more general context, that of regular stratified spaces and stratified Morse theory, which in particular permits the study of manifolds with boundary. It also includes the study of random fields that are of the form G • (F 1 , .…”

A central limit theorem for the Euler integral of a Gaussian random field

“…However, it is somewhat more delicate, since although (1.1) extends to a finite inclusion-exclusion form, it does not typically extend to the countably infinite case needed for a standard measure based theory of integration. The definition that we have chosen above avoids these issues, and in taking it we follow the lead of Bobrowski and Borman (2012) who, by taking (1.2) as a definition rather than a property, save often irritating but unimportant (for our needs) technicalities.…”

“…For example, we need not assume that they are all convex or even have the same number of connected components. While everything in (1.3) is deterministic, Bobrowski and Borman (2012) raises the question as to what happens when the deterministic 'signal' x = M h⌈dχ⌉, is observed via a noisy measurement Y = M (h + X)⌈dχ⌉, where X is a smooth random process on M . They show that, although Euler integrals are not always additive, in this case it is true that…”