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Presented by Roei Zohar

EULER INTEGRATION OF GAUSSIAN RANDOM FIELDS AND PERSISTENT HOMOLOGY Omer Bobrowski & Matthew Strom Borman. Presented by Roei Zohar. The euler Integral- reasoning. As we know, the Euler characteristic is an additive operator on compact sets: which reminds us of a measure

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Presented by Roei Zohar

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  1. EULER INTEGRATION OF GAUSSIAN RANDOM FIELDSAND PERSISTENT HOMOLOGYOmer Bobrowski & Matthew Strom Borman Presented by Roei Zohar

  2. The euler Integral- reasoning • As we know, the Euler characteristic is an additive operator oncompact sets: which reminds us of a measure • That is why it seems reasonable to define an integral with respect to this “measure”.

  3. The euler Integral- drawback • The main problem with this kind of integration is that the Euler characteristic is only finitely additive. • This is why under some conditions it can be defined for “constructible functions” as • But we can’t go on from here approximating other functions using CF functions

  4. Euler integral - Extensions • We shall define another form of integration that will be more useful for calculations: • For a tame function the limits are well defined, but generally not equal. • This definition enable us to use the following useful proposition

  5. Euler integral - Extensions • The proof appears in [3] • We will only use the upper Euler integral

  6. Euler integral - Extensions • We continue to derive the following Morse like expression for the integral:

  7. Further helpful development

  8. Gaussian Random Fields and the Gaussian Kinematic Formula • Now we turn to show the GK formula which is an explicit expression for the mean value of all Lipschitz-Killing curvatures of excursion sets for zero mean, constant value variance, c², Gaussian random fields. We shall not go into details, you can take a look in [2].

  9. The metric here, under certain conditions is • Under this metric the L-K curvatures are computed in the GKF, and in it the manifold M is bounded

  10. When taking we receive and: • Now we are interested in computing the Euler integral of a Gaussian random field:Let M be a stratified space and let be a Gaussian or Gaussian related random field.We are interested in computing the expected value of the Euler integral of the field g over M.

  11. Theorem: Let M be a compact d-dimensional stratified space, and let be a k-dimensional Gaussian random field satisfying the GKF conditions. For piecewise c² functionlet setting , we have

  12. The proof:

  13. The difficulty in evaluating the expression above lies in computing the Minkowskifunctionals • In the article few cases where they have been computed are presented, which allows us to simplify , I will mention one of them Real Valued Fields:

  14. And if G is strictly monotone:

  15. Weighted Sum of Critical Values • Taking in Theorem and using Proposition yields the following compact formula

  16. Weighted Sum of Critical Values • The thing to note about the last result, is that the expected value of a weighted sum of the critical values scales like , a 1-dimensional measure of M and not the volume , as one might have expected. • Remark: if we scale the metric by , then scales by

  17. We’ll need this one up ahead The proof relies on

  18. ∂ ∂ ∂ ∂ ∂ ∂ Ck(X) Ck-1(X) C1(X) C0(X) 0 INTRODUCTION : Persistent Homology The Usual Homology • Have a single topological space, X • Get a chain complex • For k=0, 1, 2, … compute Hk(X) • Hk=Zk/Bk

  19. INTRODUCTION : Persistent Homology • Persistent homology is a way of tracking how the homology of a sequence of spaces changes • Given a filtration of spaces such that if s < t, the persistent homology of , ,consists of families of homology classes that ‘persist’ through time.

  20. INTRODUCTION : Persistent Homology • Explicitly an element of is a family of homology classes • for • The map induced by the inclusion , maps Given a tame function

  21. Barcode demonstration (1)

  22. Barcode demonstration (2)

  23. t=0 t=1 t=2 t=3 t=4 t=5 a a a a a a b b b b b b d c d c d c d c d c a, b c, d, ab,bc cd, ad ac abc acd INTRODUCTION : Persistent Homology A filtration of spaces (Simplicial complexes example): X1  X2  X3  …Xn

  24. Getting to the point… 1. 2. I’ll leave the proof of the second claim to you

  25. The Expected Euler Characteristic of the Persistent Homology of aGaussian Random Field • In light of the connection between the Euler integral of a function and the Euler characteristic of the function’s persistent homology in place, we will now reinterpret our computations about the expected Euler integral of a Gaussian random field

  26. 1. 2. The proof makes use of the following two expressions we saw before:

  27. Computing is not usually possible, but in the case of a real random Gaussian field we can get around it and it comes out that:

  28. An application – target enumeration

  29. The end

  30. Definitions • A “tame” function :

  31. References • Paper presented here, by Omer Bobrowski & Matthew Strom Borman • R.J. Adler and J.E. Taylor. Random Fields and Geometry. Springer, 2007. • Y . Baryshikov and R. Ghrist. Euler integration over definable functions. In print, 2009.

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