Probabilistic Fr´ echet Means on Persistence Diagrams Paul Bendich Duke University :: Dept of Mathematics July 15, 2013 Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 1 / 33
Collaborators This is joint work with: ◮ Liz Munch (Duke) ◮ Kate Turner (Chicago) ◮ John Harer (Duke) ◮ Sayan Mukherjee (Duke) ◮ Jonathan Mattingly (Duke) Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 2 / 33
Main Idea and Results New definition of mean for a set X of diagrams in ( D p , W p ) Mileyko et. al.: ◮ µ X is itself a (set of) diagram(s) in D p . ◮ Problem: non-uniqueness leads to discontinuity issues. Our approach: ◮ Definition: µ X ∈ P ( D p ): (atomic) prob. dist. on diagrams. ◮ Theorem: X → µ X is H¨ older continuous (with exponent 1 2 ) x 1 b 1 b u v a a 2 2 y Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 3 / 33
Persistence Review 1 Why Means? 2 Frechet Means of Diagrams 3 Probabilistic Frechet Means 4 Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 4 / 33
Persistence Review 1 Why Means? 2 Frechet Means of Diagrams 3 Probabilistic Frechet Means 4 Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 4 / 33
Persistence modules A persistence module F is: ◮ family of vector spaces { F α } , α ∈ R , over a fixed field ◮ family of linear transformations f β α : F α → F β , for all α ≤ β , s.t α ≤ γ ≤ β implies f β α = f β γ ◦ f γ α . The number α is a regular value of the module if: ◮ There exists δ > 0 such that f α + ǫ α − ǫ is iso. for all ǫ < δ . If α is not a r.v., then it is a critical value of the module. Module is tame if only finitely many c.v’s, and each v.s is of finite rank. Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 5 / 33
Persistence Modules Given finitely many c.v’s c 1 < c 2 < . . . < c n . Interleave r.v’s a 0 < c 1 < a 1 < . . . < c n < a n . Set F i = F a i : F 0 → F 1 → F 2 . . . → F n − 1 → F n Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 6 / 33
Birth and Death A vector v ∈ F i is born at c i if v �∈ im f i i − 1 Such a v dies at c j if: ◮ f j i ( v ) ∈ im f j i − 1 ◮ f j − 1 ( v ) �∈ im f j − 1 i − 1 . i The persistence of v is c j − c i . F i F j − 1 F j F i − 1 v f j f i f j − 1 im f i im f j − 1 i − 1 j − 1 i − 1 i i − 1 Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 7 / 33
Persistence Diagrams Let P i , j be v.s of classes born at c i and dead at c j , and β i , j its rank. Plot a dot of multiplicity β i , j at ( c i , c j ) in plane. Plot a dot of infinite multiplicity at all y = x diagonal points. Result is Dgm ( F ). death death 0 0 birth birth Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 8 / 33
Example: persistent homology Let Y ⊆ R D be compact space. For α ≥ 0, define Y α = d − 1 Y [0 , α ] For each k , get module { H k ( Y α ) } , with maps induced by inclusion. death death 0 0 birth birth Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 9 / 33
Persistence Review 1 Why Means? 2 Frechet Means of Diagrams 3 Probabilistic Frechet Means 4 Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 10 / 33
Relate Multiple Samples
Relate Multiple Samples
Relate Multiple Samples How do we give a summary of the data? Will it play nicely with time varying persistence diagrams? Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 11 / 33
Significance Testing Suppose we obtain N points X in unit d -ball. We compute the diagram and are impressed with a feature. Should we be impressed? Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 12 / 33
Towards Topological Null Hypothesis Experiment: draw N points uniformly from d -ball and compute diagram. Question: what is expected diagram? Hope: repeat experiment many times, take mean diagram as answer. Mean of 500 1−D PDs generated from a sample of 50 points. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 13 / 33
Towards Topological Null Hypothesis Experiment: draw N points uniformly from d -cube and compute diagram. Question: what is expected diagram? Hope: repeat experiment many times, take mean diagram as answer. Mean of 500 1−D PDs generated from a sample of 510 points. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 14 / 33
Persistence Review 1 Why Means? 2 Frechet Means of Diagrams 3 Probabilistic Frechet Means 4 Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 15 / 33
Diagrams in the Abstract Abstract Persistence Diagram An abstract persistence diagram is a countable multiset of points along with the diagonal, ∆ = { ( x , x ) ∈ R 2 | x ∈ R } , with points in ∆ having infinite multiplicity. Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 16 / 33
Wasserstein Distance on D p c y d z b x a p -Wasserstein distance for diagrams Given diagrams X and Y , the distance between them is �� � 1 / p ( � x − ϕ ( x ) � q ) p W p [ L q ]( X , Y ) = inf . ϕ : X → Y x ∈ X Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 17 / 33
Discrete vs continuous Wasserstein Discrete Given diagrams X and Y , the distance between them is �� � 1 / p ( � x − ϕ ( x ) � q ) p W p [ L q ]( X , Y ) = inf . ϕ : X → Y x ∈ X Continuous Given probability distributions, ν and η , on metric space ( X , d X ) is � � 1 / p � d X ( x , y ) p d γ ( x , y ) W p [ d X ]( ν, η ) = inf γ ∈ Γ( ν,η ) X × X where Γ( ν, η ) is the space of distributions on X × X with marginals ν and η respectively. Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 18 / 33
The metric space ( D p , W p ) The space of persistence diagrams is D p = { X | W p [ L 2 ]( X , d ∅ ) < ∞} along with the p -Wasserstein metric, W p [ L 2 ]. Theorem (Mileyko et. al.): ( D p , W p ) is complete and separable. Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 19 / 33
Fr´ echet means Let ν be a measure on a metric space ( Y , d ). The Fr´ echet variance of ν is: � � � d ( x , y ) 2 d ν ( y ) < ∞ Var ν = inf F ν ( x ) = x ∈ Y Y The set at which the value is obtained E ( ν ) = { x | F ν ( X ) = Var ν } is the Fr´ echet expectation of ν , also called Fr´ echet mean. Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 20 / 33
Fr´ echet means in D p : Existence Theorem (Mileyko et. al.): Let ν be a probability measure on ( D p , B ( D p )) with a finite second moment. If ν has compact support, then E ( ν ) � = ∅ . In particular, Fr´ echet means of finite sets of diagrams exist. f c b z x g y h a Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 21 / 33
Algorithm for Computation - Selections and Matchings f c b z x g y h a Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 22 / 33
Algorithm for Computation - Selections and Matchings f c b z x Definition Given a set of diagrams g X 1 , · · · , X N , a selection is a y choice of one point from each diagram, where that point could h be ∆. a Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 22 / 33
Algorithm for Computation - Selections and Matchings f c b z x Definition The trivial selection for a g particular off-diagonal point y x ∈ X i is the selection s x which chooses x for X i and ∆ for every h other diagram. a Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 22 / 33
Algorithm for Computation - Selections and Matchings d ⋆ d � d • f c b 1 b x f z 2 ∆ ∆ a x 3 ∆ y g 4 ∆ ∆ z 5 ∆ ∆ h g y 6 ∆ ∆ c Definition h A matching is a set of selections so that every off-diagonal point a of every diagram is part of exactly one selection. Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 22 / 33
Algorithm for Computation - Selections and Matchings Definition The mean of a selection is the point which minimizes the sum of the square distances to the elements of the selection. Paul Bendich (Duke) Probabilistic Fr´ echet Means on Persistence Diagrams July 15, 2013 22 / 33
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