Topological Persistence

E0 244 : Lecture 15March 3, 2020

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Intuition in Curveland

f

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Intuition in Curveland

f

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Intuition in Curveland

Persistent components

f

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Intuition in Curveland

Upward sweep• empty

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Intuition in Curveland

Upward sweep• empty• birth

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Intuition in Curveland

Upward sweep• empty• birth• birth

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Intuition in Curveland

Upward sweep• empty• birth• birth• birth

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Intuition in Curveland

Upward sweep• empty• birth• birth• birth• death

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Intuition in Curveland

Upward sweep• empty• birth• birth• birth• death• death

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Intuition in Curveland

Upward sweep• empty• birth• birth• birth• death• death

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Next step : Flatland

€

s2€

M€

s1

€

[m2, s1]

€

[s2,M]Pairs:

m1m2

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Topological Simplification

Order the pairs of critical points based on their persistence

CANCELLING HANDLES THEOREM. A smoother Morsefunction g can be obtained from f bycanceling two critical points that differ inindex by 1

Pairing Critical Points

» Cycle creators and destroyers» Creator-destroyer pair» Persistence (ci)» Topological Simplification

§ Repeated removal of critical point pairs§ Ordered by persistence

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Persistence (Filtration)

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Filtration

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∅ = K0 ⊂ K1 ⊂…⊂ Kn = K

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Ki = Ki−1∪σ i

Positive simplices create cycles

Negative simplices kill cycles

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Betti Number Computation

» Set bj = 0 for j = 0,1,2» for i = 1 to n

§ k = dim (si)§ if si lies in a k-cycle then bk ++§ else bk-1 --§ endif

» endfor

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Topological Persistence

» Lifetime analogy» Label cycle after its creator» Cycle persists till its destroyer is

included» Age of a cycle is its persistence» Pair creator and destroyer

Cycle Search Algorithm

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Data Structure» Linear array T[0..n-1] » T behaves like a hash table» Initially, T is empty» After processing sj

§ T[i] contains j if si is positive§ si and sj are paired§ T[i].bdylist contains list of positive simplices§ T[i].bdylist is cycle destroyed by sj

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Cycle Search» Process negative simplex sj in filtration

§ Let B0 be ordered list of positive simplices in boundaryof sj

§ Let sy be youngest simplex in B0§ y is largest label in B0§ if T[y] is empty then set T[y] = j, T[y].bdylist = B0§ if T[y] is occupied

» we have a collision» merge B0 and T[y].bdylist into an ordered list B1» find youngest simplex in B1» continue till an empty slot i is found after m steps» set T[i] = j and T[i].bdylist = Bm

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Analysis» Let pi = j - i be the persistence of si» si is found in O(pi) steps» Nesting property

§ if we have a collision at sg then i < g < h < j,where sg is paired with sh

§ Size of T[i].bdylist is at most O(pi)§ Each merge takes O(pi) time

» si is found in O(n2) time» Total time is O(n3)

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Pairing Critical Points

» Filtration using lower stars» Simplices in lower star of regular point paired with locals» One simplex in lower star of critical point paired with a

foreigner» Dimension of above simplex equal to Morse index of critical

point» Run cycle search to pair simplices» If two paired simplices do not lie in lower star of a common

vertex then pair corresponding critical points