Statistical topological data analysis using. Persistence landscapes

Transcription

1 Motivation Persistence landscape Statistical topological data analysis using persistence landscapes Cleveland State University February 3, 2014 funded by AFOSR

2 Motivation Persistence landscape Topological data analysis (TDA) TDA attempts to recover topological and geometric information from sampled data. Data points in R n Geometric object Topological summary

3 Motivation Persistence landscape radius = 0

4 Motivation Persistence landscape radius = 1

5 Motivation Persistence landscape radius = 2

6 Motivation Persistence landscape radius = 3

7 Motivation Persistence landscape radius = 4

8 Motivation Persistence landscape radius = 5

9 Motivation Persistence landscape radius = 6

10 Motivation Persistence landscape radius = 7

11 Motivation Persistence landscape radius = 8

12 Motivation Persistence landscape radius = 9

13 Motivation Persistence landscape radius = 10

14 Motivation Persistence landscape radius = 11

15 Motivation Persistence landscape Degree 1 persistent homology: {(3, 9), (4, 6), (5, 11)}

16 Motivation Persistence landscape Mathematical viewpoint For each radius r, have a simplicial complex S r (X ) a vector space H(S r (X )) For r r, have the inclusion S r (X ) S r (X ) a linear map H(S r (X )) H(S r (X )) Persistent homology is the image of this map. This set of vector spaces and linear maps is called a persistence module. It has a complete discrete invariant: {(birth j, death j )}. There exist good algorithms. This summary is stable.

17 Motivation Persistence landscape Statistical viewpoint The topological summary as a random variable: Probability Space Summary Space TS (Ω, F, P) (SS, A, P ) X TS(X )

18 Challenges Motivation Persistence landscape Goal: use topological summaries to make inferences. We want to: construct summaries compare summaries average summaries use summaries for hypothesis testing and do so efficiently. My approach: the persistence landscape

19 Persistence landscape Recall that the persistence module consisted of linear maps H(S r (X )) H(S r (X )), for r r. The ranks of these maps gives us a function from R 2 to R.

20 Persistence landscape

21 Persistence landscape

22 Persistence landscape 2 λ 1 λ 2 λ

23 Persistence landscape

24 Properties Motivation Persistence landscape Definition Properties Mean Hypothesis testing Stability Lemma λ k (t) 0 λ k (t) λ k+1 (t) λ k is 1-Lipschitz 2 λ 1 λ 2 λ

25 Persistence landscape Consider λ 1, λ 2, λ 3,... as λ : N R R. Then λ = λ 1, and ( for 1 p <, λ p = k λ k p ) 1 p. 2 λ 1 λ 2 λ

26 Norms Motivation Persistence landscape Definition Properties Mean Hypothesis testing Stability For a persistence landscape λ, let (b j, d j ) be the corresponding birth-death pairs. Lemma 1 λ = 1 2 max j(d j b j ), and 2 λ 1 = 1 4 j (d j b j ) 2. 2 λ 1 λ 2 λ

27 Mean landscapes, λ (1),..., λ (n), have mean, λ = 1 n That is, λ k (t) = 1 n n i=1 λ (i) k (t) n λ (i). i=1 Interpretation: This is the average value of the largest h such that has rank at least k. H(S t h (X )) H(S t+h (X ))

28 Mean diagram vs mean landscape λ 2 λ λ 2 λ λ 2 λ

29 Linked annuli

30 Linked annuli

31 Linked annuli

32 Summary space ( Recall λ p = k λ k p ) 1 p. Let 1 p <. We assume λ := λ p <. That is, λ L p (N R). So λ is a random variable with values in a separable Banach space.

33 Law of Large Numbers λ L p (N R), λ is a real random variable. If E λ < then there exists E(λ) L p (N R) such that E(f (λ)) = f (E(λ)) for all continuous linear functionals f. For X 1,..., X n be an iid sample, and let λ (1),..., λ (n) be the corresponding persistence landscapes. Theorem (Strong Law of Large Numbers) λ (n) E(λ) almost surely if and only if E λ <.

34 Central limit theorems Theorem (Central Limit Theorem in L p (N R)) Assume p 2. If E λ < and E( λ 2 ) < then n[λ (n) E(λ)] converges weakly to a Gaussian random variable with the same covariance structure as λ. Corollary (Practical Central Limit Theorem) For any f L q (N R) with 1 p + 1 q = 1, let Y = f λ. (1) N R Then n[y n E(Y )] d N(0, Var(Y )). (2)

35 Mean landscapes for Gaussian Random Fields

36 Motivation Persistence landscape Definition Properties Mean Hypothesis testing Stability Mean landscapes for Gaussian Random Fields

37 Topological hypothesis testing

38 Topological hypothesis testing Points kernel density estimator filtered simplicial complex

39 Topological hypothesis testing

40 Topological hypothesis testing Null hypothesis: λ S 1 = λ T 1. Student s t-test: dim decision p-value 0 cannot reject 1 reject cannot reject

41 Topological hypothesis testing, noisy

42 Topological hypothesis testing, noisy

43 Topological hypothesis testing, noisy Null hypothesis: λ S λ T 2 = 0. Permutation test: dim decision p-value 0 reject reject reject

44 Topological hypothesis testing, noisy

45 Stability Motivation Persistence landscape Definition Properties Mean Hypothesis testing Stability Given f : X R, let λ(f ) the persistence landscape of sublevel sets of f. Theorem (Landscape stability theorem) Let f, g : X R. λ(f ) λ(g) f g. If X is nice and f and g are tame and Lipschitz then λ(f ) λ(g) p p C f g p k.