Stability Theorem Xinyi Kong, 1281909 Eindhoven, university of - PowerPoint PPT Presentation

Stability Theorem Xinyi Kong, 1281909 Eindhoven, university of Technology 31 May 2018

Overview ◮ Motivation ◮ Persistence Diagram ◮ Main Theory ◮ Hausdorff Stability ◮ Bottleneck Stability

Motivation

Motivation

Motivation Stability means that if the input changes a tiny bit, the output should not change by much either.

Recap Let f : R → R be a smooth function. We call that x is a critical ′ ( x ) = 0. A critical point and f ( x ) id a critical value of f if f ′′ ( x ) � = 0. point x is non-degenerate if f Example

Homological critical value Definition: Let X be a topological space and f : X → R be a function. A homological critical value of f is a real number b for which there exists an integer k such that for all sufficiently small ε > 0 the map H k ( f − 1 ( −∞ , b − ε ]) → H k ( f − 1 ( −∞ , b + ε ]) is not an isomorphism.

A tame function Definition: A function f : X → R is tame if it has a finite number of homological critical values and the homology groups H k ( f − 1 ( −∞ , a ]) are finite-dimensional for all k ∈ Z and a ∈ R . Figure 1: tame Figure 2: non-tame

Persistence Diagram How to draw a persistence diagram of f ? ◮ Pair the critical points of f . ◮ Map each pair to the corresponding point in persistence diagram.

Persistence Diagram Pair the critical points of f : When we pass a local maximum and merge two components, we pair the maximum with the higher (younger) of the two local minima that represent the two components. The other minimum is now the representative of the component resulting from the merger. Example

Persistence Diagram of a tame function Let f : X → R be a tame function, ( a i ) i =1 ... n its homological critical values, and ( b i ) i =0 ... n an interleaved sequence, namely b i − 1 < a i < b i for all i . We set b − 1 = a 0 = −∞ and b n +1 = a n = + ∞ . We consider the corresponding sequence of homology groups, 0 = H k ( X b − 1 ) → H k ( X b 0 ) → ... → H k ( X b n +1 ) = H k ( X ) and the maps between them. We define the multiplicity of the pair ( a i , a j ) by i = β b j b i − 1 − β b j b i + β b j − 1 − β b j − 1 µ j b i b i − 1

Persistence Diagram of a tame function R 2 of f is the set of Definition: The persistence diagram D( f ) ⊂ ¯ points ( a i , a j ), counted with multiplicity µ j i for 0 ≤ i < j ≤ n + 1, union all points on the diagonal, counted with infinite multiplicity. Example

♯ ( A ) The total multiplicity of a multiset A is written as ♯ ( A ) example The total multiplicity of the persistence diagram without the diagonal is written as follow: i < j µ j ♯ (D( f ) − ∆) = � i

L ∞ − norm For points p = ( p 1 , p 2 , ..., p n ) and q = ( q 1 , q 2 , ..., q n ) in ¯ R n , � p − q � ∞ := max( | p 1 − q 1 | , | p 2 − q 2 | , ..., | p n − q n | ). For function f and g , � f − g � ∞ = sup x | f ( x ) − g ( x ) | . Figure 3: � p − q � ∞ Figure 4: � f − g � ∞

Hausdorff distance and bottleneck distance Definition: Let X and Y be multisets of points. The Hausdorff distance and bottleneck distance between X and Y are � � d H ( X , Y ) = max sup x inf y � x − y � ∞ , sup y inf x � y − x � ∞ d B ( X , Y ) = inf γ sup x � x − γ ( x ) � ∞ where x ∈ X and y ∈ Y range over all points and γ ranges over all bijections from X to Y .

Hausdorff distance and bottleneck distance � � d H ( X , Y ) = max sup x inf y � x − y � ∞ , sup y inf x � y − x � ∞ Figure 6: sup y inf x � y − x � ∞ Figure 5: sup x inf y � x − y � ∞ It is the greatest of all the distances from a point in one set to the closest point in the other set.

Hausdorff distance and bottleneck distance d B ( X , Y ) = inf γ sup x � x − γ ( x ) � ∞ Figure 7: sup x � x − γ 1 ( x ) � ∞ Figure 8: sup x � x − γ 2 ( x ) � ∞

Hausdorff distance and bottleneck distance d H ( X , Y ) ≤ d B ( X , Y ) Bottleneck distance has one more constrain which makes it cannot map all the points in one set to the closest point in the other set. Figure 10: sup y inf x � y − x � ∞ Figure 9: sup x inf y � x − y � ∞ Figure 11: sup x � x − γ 1 ( x ) � ∞ Figure 12: sup x � x − γ 2 ( x ) � ∞

Main Theory Main Theory: Let X be a triangulable space with continuous tame functions f , g : X → R . Then the persistence diagrams satisfy d B (D( f ) , D( g )) ≤ � f − g � ∞ .

To proof the main theorem: I will first show what we can get from d H ( D ( f ) , D ( g )) ≤ � f − g � ∞ Then strengthen the result to proof d B ( D ( f ) , D ( g )) ≤ � f − g � ∞

Hausdorff Stability If the inequality d H ( D ( f ) , D ( g )) ≤ � f − g � ∞ = ε is true, we can find a point p ( x , y ) ∈ D ( f ), then there must be a point of D ( g ) at the distance less than or equal to ε from p ( x , y ). That means there must be a point q of D ( g ) inside the square [ x − ε, x + ε ] × [ y − ε, y + ε ].

Box lemma R 2 and let For a < b < c < d , let R = [ a , b ] × [ c , d ] be a box in ¯ R ε = [ a + ε, b − ε ] × [ c + ε, d − ε ] be the box obtained by shrinking R at all four sides. Box Lemma: ♯ ( D ( f ) ∩ R ε ) ≤ ♯ ( D ( g ) ∩ R )

Recall Main Theory: Let X be a triangulable space with continuous tame functions f , g : X → R . Then the persistence diagrams satisfy d B (D( f ) , D( g )) ≤ � f − g � ∞ .

Bottleneck stability Let’s start with a special case. Given a tame function f : X → R , we consider the minimum distance between off-diagonal points or between ans off-diagonal point and the diagonal: δ f = min {� p − q � ∞ | ( D ( f ) − ∆) ∋ p � = q ∈ D ( f ) }

Bottleneck Stability δ f = min {� p − q � ∞ | ( D ( f ) − ∆) ∋ p � = q ∈ D ( f ) } We get Figure 1 by drawing squares of radius r = δ f / 2 around the points of D ( f ). Figure 13: D ( f )

Bottleneck Stability δ f = min {� p − q � ∞ | ( D ( f ) − ∆) ∋ p � = q ∈ D ( f ) } Then we add another tame function g : X → R which is very close to f . That means f and g satisfy � f − g � ∞ ≤ δ f / 2 Figure 14: D ( f ) Figure 15: D ( f ) and D ( g )

Bottleneck Stability δ f = min {� p − q � ∞ | ( D ( f ) − ∆) ∋ p � = q ∈ D ( f ) } f and g satisfy � f − g � ∞ ≤ δ f / 2 Writing µ for the multiplicity of the point p ∈ ( D ( f ) − ∆) and � ε for the square with center p and radius ε = � f − g � ∞

Bottleneck Stability Recall the box lemma R 2 and let For a < b < c < d , let R = [ a , b ] × [ c , d ] be a box in ¯ R ε = [ a + ε, b − ε ] × [ c + ε, d − ε ] be the box obtained by shrinking R at all four sides. Box Lemma: ♯ ( D ( f ) ∩ R ε ) ≤ ♯ ( D ( g ) ∩ R ) From the box lemma we get: µ ≤ ♯ ( D ( g ) ∩ � ε ) ≤ ♯ ( D ( f ) ∩ � 2 ε )

Bottleneck Stability From the box lemma we get: µ ≤ ♯ ( D ( g ) ∩ � ε ) ≤ ♯ ( D ( f ) ∩ � 2 ε ) Since 2 ε ≤ δ f , p is the only point of D ( f ) in � 2 ε , which implies µ = ♯ ( D ( g ) ∩ � ε ).

Bottleneck Stability Since 2 ε ≤ δ f , p is the only point of D ( f ) in � 2 ε , which implies µ = ♯ ( D ( g ) ∩ � ε ). Therefore, we can map all points of D ( g ) in � ε to p . And the rest of point will be mapped to the nearest point on the diagonal, because d H ( D ( f ) , D ( g )) ≤ ε

Bottleneck Stability Easy Bijection lemma: Let f , g : X → R be tame functions and g very close to f . then the persistence diagrams satisfy d B ( D ( f ) , D ( g )) ≤ � f − g � ∞

Summary ◮ Persistence diagram of tame functions ◮ Hausdorff distance and bottleneck distance of two persistence diagrams ◮ Use the box lemma to proof the bottleneck distance ◮ If a function change a little bit, its persistence diagram is stable.

Source ◮ D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of Persistence Diagrams, Disc. Comp. Geom. 37: 103–120, 2007. ◮ H. Edelsbrunner, J. L. Harer, Persistent homology – a Survey, Surveys on discrete and computational geometry, 257–282, Contemp. Math., 453, Amer. Math. Soc., Providence, RI, 2008.