Confidence Sets for Persistent Diagrams Aleksandr Popov Eindhoven University of Technology 12th June 2018 1 / 31
What to expect 1. Introduction and motivation 2. Formal definition 3. Computation methods 2 / 31
Introduction and motivation 3 / 31
Back to the origins Main idea of TDA: determine topology of underlying space, based on a point cloud. Point cloud can be viewed as a sample of the true space. Random sample of 10 points. 4 / 31
Back to the origins Main idea of TDA: determine topology of underlying space, based on a point cloud. Point cloud can be viewed as a sample of the true space. Random sample of 10 points. 4 / 31
Čech complex Čech complex on the 10 points. 5 / 31
Čech complex Čech complex on the 10 points. 5 / 31
Čech complex Čech complex on the 10 points. 5 / 31
Čech complex Čech complex on the 10 points. 5 / 31
Čech complex Čech complex on the 10 points. 5 / 31
Homology Betui numbers: These will change if we take difgerent radius of balls! 6 / 31 β 0 = β 1 =
Homology Betui numbers: These will change if we take difgerent radius of balls! 6 / 31 β 0 = 1 β 1 = 1
Homology Betui numbers: These will change if we take difgerent radius of balls! 6 / 31 β 0 = 1 β 1 = 1
A difgerent Čech complex 7 / 31 A difgerent Čech complex on the 10 points: β 0 = 6 , β 1 = 0 .
A difgerent Čech complex 7 / 31 A difgerent Čech complex on the 10 points: β 0 = 6 , β 1 = 0 .
A difgerent Čech complex 7 / 31 A difgerent Čech complex on the 10 points: β 0 = 6 , β 1 = 0 .
A difgerent Čech complex 7 / 31 A difgerent Čech complex on the 10 points: β 0 = 6 , β 1 = 0 .
A difgerent Čech complex 7 / 31 A difgerent Čech complex on the 10 points: β 0 = 6 , β 1 = 0 .
Persistent homology Persistence diagram from Vietoris–Rips complex for the 10 points. 8 / 31 4 3 Death 2 1 component loop 0 0 1 2 3 4 Birth
So what’s wrong with this? What here is ‘noise’ and what is ‘signal’? How well do we represent the homology of the space of interest? 9 / 31
What do we want? Persistence diagram from Vietoris–Rips complex for the 10 points with the confidence band. 10 / 31 4 3 Death 2 1 component loop 0 0 1 2 3 4 Birth
Formal definition 11 / 31
X n following Statistical model and basic definitions Sample set S n X distribution P . P is concentrated on . Persistence diagram—multiset of points in plane. Denote by . 12 / 31 Space of interest: manifold M . Manifold M .
Statistical model and basic definitions distribution P . Persistence diagram—multiset of points in plane. Denote by . 12 / 31 Space of interest: manifold M . Sample set S n = { X 1 , . . . , X n } following P is concentrated on M . M with points sampled from P .
Statistical model and basic definitions distribution P . Persistence diagram—multiset of points in plane. Denote by . Sampled points S n . 12 / 31 Space of interest: manifold M . Sample set S n = { X 1 , . . . , X n } following P is concentrated on M .
Statistical model and basic definitions Persistence diagram—multiset of points in 12 / 31 distribution P . Space of interest: manifold M . 4 Sample set S n = { X 1 , . . . , X n } following 3 Death 2 P is concentrated on M . 1 component loop 0 0 1 2 3 4 Birth plane. Denote by P . Persistence diagram P for the example.
d A x y A y d S n x Čech filtrations; distance set filtration of distance function: Čech filtration. . increasing from 0 to for x i.e. the distance to the closest point in A . Čech filtration corresponds to lower level Čech filtration—sequence of Čech x inf D : Distance function for a set A radius. complexes with gradually increasing 13 / 31
d A x y A y d S n x Čech filtrations; distance set filtration of distance function: Čech filtration. . increasing from 0 to for x i.e. the distance to the closest point in A . Čech filtration corresponds to lower level Čech filtration—sequence of Čech x inf D : Distance function for a set A radius. complexes with gradually increasing 13 / 31
d S n x Čech filtrations; distance x Distance from p to A . p A . increasing from 0 to for set filtration of distance function: Čech filtration—sequence of Čech Čech filtration corresponds to lower level i.e. the distance to the closest point in A . radius. complexes with gradually increasing 13 / 31 Distance function for a set A ⊂ R D : d A ( x ) = inf y ∈ A ∥ y − x ∥ 2 ,
Čech filtrations; distance Čech filtration—sequence of Čech complexes with gradually increasing radius. i.e. the distance to the closest point in A . Čech filtration corresponds to lower level set filtration of distance function: 13 / 31 Distance function for a set A ⊂ R D : d A ( x ) = inf y ∈ A ∥ y − x ∥ 2 , Regions for increasing ε . { x : d S n ( x ) ≤ ε } , for ε increasing from 0 to ∞ .
Čech filtrations; distance Čech filtration—sequence of Čech complexes with gradually increasing radius. i.e. the distance to the closest point in A . Čech filtration corresponds to lower level set filtration of distance function: 13 / 31 Distance function for a set A ⊂ R D : d A ( x ) = inf y ∈ A ∥ y − x ∥ 2 , Regions for increasing ε . { x : d S n ( x ) ≤ ε } , for ε increasing from 0 to ∞ .
Čech filtrations; distance Čech filtration—sequence of Čech complexes with gradually increasing radius. i.e. the distance to the closest point in A . Čech filtration corresponds to lower level set filtration of distance function: 13 / 31 Distance function for a set A ⊂ R D : d A ( x ) = inf y ∈ A ∥ y − x ∥ 2 , Regions for increasing ε . { x : d S n ( x ) ≤ ε } , for ε increasing from 0 to ∞ .
What are we trying to get? Can we guarantee with high probability that they are close for all the points? We need to express ‘closeness’ somehow. 14 / 31 We can construct the persistence diagram of S n , ˆ P . We use it to estimate the persistence diagram of M , P .
What are we trying to get? Can we guarantee with high probability that they are close for all the points? We need to express ‘closeness’ somehow. 14 / 31 We can construct the persistence diagram of S n , ˆ P . We use it to estimate the persistence diagram of M , P .
M d Botuleneck distance max b a maximum pairwise cost. Botuleneck distance minimises the a b Perfect matching M of sets A and B a b M min A B W Botuleneck distance: (bijection). 15 / 31 Cost between points a ∈ A , b ∈ B : d ∞ ( a , b ) = max ( | a x − b x | , | a y − b y | ) . 2 3 d ∞ ( a , b ) = 3 .
Botuleneck distance M maximum pairwise cost. Perfect matching M of sets A and B max Botuleneck distance minimises the Botuleneck distance: (bijection). 15 / 31 (2 , 4) (2 . 8 , 4) Cost between points a ∈ A , b ∈ B : (4 . 8 , 2 . 8) d ∞ ( a , b ) = max ( | a x − b x | , | a y − b y | ) . (4 , 2) (0 , 0 . 8) W ∞ ( A , B ) = min ( a , b ) ∈ M d ∞ ( a , b ) . (0 , 0) W ∞ ( A , B ) =
Botuleneck distance M maximum pairwise cost. Perfect matching M of sets A and B max Botuleneck distance minimises the Botuleneck distance: (bijection). 15 / 31 (2 , 4) (2 . 8 , 4) Cost between points a ∈ A , b ∈ B : (4 . 8 , 2 . 8) d ∞ ( a , b ) = max ( | a x − b x | , | a y − b y | ) . (4 , 2) (0 , 0 . 8) W ∞ ( A , B ) = min ( a , b ) ∈ M d ∞ ( a , b ) . (0 , 0) W ∞ ( A , B ) =
Botuleneck distance M maximum pairwise cost. Perfect matching M of sets A and B max Botuleneck distance minimises the Botuleneck distance: (bijection). 15 / 31 (2 , 4) (2 . 8 , 4) Cost between points a ∈ A , b ∈ B : (4 . 8 , 2 . 8) d ∞ ( a , b ) = max ( | a x − b x | , | a y − b y | ) . (4 , 2) (0 , 0 . 8) W ∞ ( A , B ) = min ( a , b ) ∈ M d ∞ ( a , b ) . (0 , 0) W ∞ ( A , B ) = 0 . 8 .
, find c n such that is further than c n from the diagonal, then with probability at least What are we trying to get: take 2 Can we guarantee with high probability that they are close for all the points? Given confidence lim sup n W c n If the point in it is also not on the diagonal in , so it is significant . 16 / 31 We can construct the persistence diagram of S n , ˆ P . We use it to estimate the persistence diagram of M , P .
is further than c n from the diagonal, then with probability at least What are we trying to get: take 2 Can we guarantee with high probability that they are close for all the points? lim sup If the point in it is also not on the diagonal in , so it is significant . 16 / 31 We can construct the persistence diagram of S n , ˆ P . We use it to estimate the persistence diagram of M , P . Given confidence α ∈ (0 , 1) , find c n such that P ( W ∞ ( ˆ P , P ) > c n ) ≤ α . n →∞
What are we trying to get: take 2 Can we guarantee with high probability that they are close for all the points? lim sup 16 / 31 We can construct the persistence diagram of S n , ˆ P . We use it to estimate the persistence diagram of M , P . Given confidence α ∈ (0 , 1) , find c n such that P ( W ∞ ( ˆ P , P ) > c n ) ≤ α . n →∞ If the point in ˆ P is further than c n from the diagonal, then with probability at least 1 − α it is also not on the diagonal in P , so it is significant .
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