28 February 2017 Hierarchies and Ranks for Persistence Pairs Bastian Rieck 1 Heike Leitte 1 Filip Sadlo 2 1 TU Kaiserslautern, Germany 2 Heidelberg University, Germany
Motivation Different functions may have identical persistence diagrams 1 / 23 4 4 3 3 2 2 1 1 0 0 0 1 2 3 4 0 1 2 3 4
Motivation Different functions may have identical persistence diagrams 1 / 23 4 4 3 3 2 2 1 1 0 0 0 1 2 3 4 0 1 2 3 4
Motivation Different functions may have identical persistence diagrams 1 / 23 4 4 3 3 2 2 1 1 0 0 0 1 2 3 4 0 1 2 3 4
Motivation, continued Identical persistence diagrams 2 / 23 • Generic issue: occurs both in sublevel set and superlevel set calculations • Solution: add additional (geometrical) information, e.g. merge trees
Assumptions one with the smaller index with respect to the fjltration 3 / 23 • Pairing of connected components (zero-dimensional persistent homology) • Pairing uses “elder rule”: The “older” connected component persists, i.e. the • In the example below, component a persists, but component b is destroyed by the merge at c c b b a a a
Regular persistence hierarchy else end for end if component 4 / 23 hierarchy uses directed edges. Require: A domain D Require: A function f : D → R U ← ∅ Sort the function values of f in ascending order c for function value y of f do b if y is a local minimum then a Create a new connected component in U else if y is a local maximum or a saddle then Use U to merge the two connected components Let y ′ refer to the creator of the older component Create the edge ( y ′ , y ) in the hierarchy Add b → a to the hierarchy. Notice that the Use U to add y to the current connected
Regular persistence hierarchy, continued 5 / 23 • Introduced by Bauer, 2011, “Persistence in discrete Morse theory” • By defjnition, the hierarchy forms a directed acyclic graph • Original motivation: determining cancellation sequences of Morse functions
Problem Lack of expressiveness 6 / 23 e e d d c c b b a a ( a , ∞ ) ( a , ∞ ) ( b , c ) ( d , e ) ( b , c ) ( d , e )
Problem Lack of expressiveness Key observation 6 / 23 e e d d c c b b a a • Not all merges in the sublevel sets are equal! • Take connectivity with respect to other critical points into account.
Sublevel set connectivity Use interlevel set 7 / 23 e e 4 4 d d 3 3 c c 2 2 y y b b 1 1 a a 0 0 x x
Sublevel set connectivity Use interlevel set 7 / 23 e e 4 4 d d 3 3 c c 2 2 y y b b 1 1 a a 0 0 x x
Sublevel set connectivity Use interlevel set 7 / 23 e e 4 4 d d 3 3 c c 2 2 y y b b 1 1 a a 0 0 x x L l , u ( f ) : = L − u ( f ) \ L − l ( f ) = { x ∈ D | l ≤ f ( x ) ≤ u }
Sublevel set connectivity Use interlevel set function for the function, but insert pair on lower level function the function, but only one for 7 / 23 e e 4 4 d d 3 3 c c 2 2 y y b b 1 1 a a 0 0 x x L l , u ( f ) : = L − u ( f ) \ L − l ( f ) = { x ∈ D | l ≤ f ( x ) ≤ u } • L b , e ( f ) has two connected components for the • Hence: use the same level for the
Sublevel set connectivity Use interlevel set 7 / 23 e e 4 4 d d 3 3 c c 2 2 y y b b 1 1 a a 0 0 x x ( a , ∞ ) ( a , ∞ ) ( b , c ) ( d , e ) ( b , c ) ( d , e )
Algorithm else 15: end for end if 14: end if 13: end if 12: 11: 10: 9: Excerpt; shortened notation 8: 7: 5: 2: 3: 4: 8 / 23 6: if both components have a trivial critical value then 1: for function value y of f do if y is a local maximum then Use U to merge the two connected components Let C 1 and C 2 be the two components at y (w.l.o.g. let C 1 be the older one) Create the edge ( C 1 , C 2 ) in the hierarchy Let c 1 , c 2 be the critical values of C 1 , C 2 Create the interlevel set L : = L c 2 , y ( f ) if shortest path between c 1 , c 2 in L contains no other critical points then Create edge ( c 1 , y ) in the hierarchy
Necessity of the connectivity check In one dimension (segments), a simple connectivity check is suffjcient. In two dimensions (isolines), both interlevel sets are connected, though! 9 / 23 y y x x
Necessity of the connectivity check In one dimension (segments), a simple connectivity check is suffjcient. In two dimensions (isolines), both interlevel sets are connected, though! 9 / 23 y y x x
Implications Open questions 10 / 23 • Extended persistence hierarchy usually has more levels than the regular one • The calculation incorporates a modicum of geometrical information • Is this connectivity check suffjciently distinctive? • What is the relation to “basins of attraction” in discrete Morse theory?
Comparison with other tree-based concepts In the paper coincide for different functions 11 / 23 • Regular persistence hierarchy can be obtained via branch decomposition • Merge trees are discriminative, but their branch decomposition may still • Hence, extended persistence hierarchy cannot be derived that way
Robustness Merge tree vs. extended persistence hierarchy, colored by persistence Merge tree Extended persistence hierarchy 12 / 23
Application Ranks 13 / 23 How many nodes can be reached from a given node u in the (extended) persistence hierarchy H ? rank ( u ) : = card { v ∈ H | u ∼ v } 1 1 0 0 − 1 − 1 − 1 0 − 1 0 1 1
Application Ranks 13 / 23 How many nodes can be reached from a given node u in the (extended) persistence hierarchy H ? rank ( u ) : = card { v ∈ H | u ∼ v } 1 1 0 0 − 1 − 1 − 1 0 − 1 0 1 1
Application Stability measure Overarching question How stable is the location of a critical point? Persistence pairs are a continuous function of the input data, but their location is not. Previous work Bendich & Bubenik, 2015, “Stabilizing the output of persistent homology computations”. 14 / 23
Stability measure Critical points: Critical points: Example (superlevel sets) 15 / 23 f f e e y y d d c c b b a a a a x x • ( f , − ∞ ) • ( f , − ∞ ) • ( e , c ) • ( e , c ) • ( d , b ) • ( d , b )
Stability measure Critical points: Critical points: Example, perturbed 16 / 23 f f d b d e e y y b c c a a a a x x • ( f , − ∞ ) • ( f , − ∞ ) • ( e , c ) • ( d , c ) • ( d , b ) • ( e , b )
Stability measure Formal defjnition worst-case assumption! Using the minimum of all stability values is an extremely conservative for the second hierarchy. e e (2) (1) 17 / 23 For an edge e : = { ( σ , τ ) , ( σ ′ , τ ′ ) } in the hierarchy H : � | f ( σ ) − f ( σ ′ ) | , | f ( τ ) − f ( τ ′ ) | � stab ( e ) : = max For a vertex v : stab ( v ) : = min � min e =( v , w ) ∈H stab ( e ) , pers ( v ) � � ≪ pers � � � Here: stab
Implications Another criterion for distinguishing between functions with equal persistence diagrams, based on worst-case location stability of creators of critical pairs. Open questions 18 / 23 • How useful is this assumption? • Does it characterize all perturbations of critical points?
Application Dissimilarity measure Use existing tree edit distance algorithms. Cost function for relabeling a node: (3) Cost function for deleting or inserting a node: (4) bottleneck distance calculations, for example. 19 / 23 cost 1 = max � | c 1 − c 2 | , | d 1 − d 2 | � cost 2 = pers ( c , d ) = | d − c | , The choice of these costs is somewhat “natural” as the L ∞ -distance is used for
Advantages of this dissimilarity measure hierarchy. persistence diagrams”) (Kerber et al., “Geometry helps to compare (naïve) 20 / 23 n 2 m log m � � • Complexity of O , where n is number of nodes in smaller • Bottleneck distance � ( n + m ) 3 � • O � ( n + m ) 1.5 log ( n + m ) � • O
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Time-varying scalar fjeld (climate model simulation) 21 / 23
Results Dissimilarity measure in comparison with second Wasserstein distance Second Wasserstein distance Dissimilarity Tree edit distance 22 / 23 340 12 320 10 300 0 12 24 36 48 60 72 0 12 24 36 48 60 72 t t
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