Persistent Local Homology: Theory, Applications, Algorithms

Transcription

1 Persistent Local Homology: Theory, Applications, Algorithms Paul Bendich Duke University Feb. 4, 2014

2 Basic Goal: stratification learning Q: Given a point cloud sampled from a stratified space, how do we cluster points that belong to the same stratum piece together (at a given scale)? A Points whose local structure glue together nicely belong to the same cluster. r 1 r 2

3 Teaser Image Left: points sampled from a stratified space. Right: Mapping into R 3 that makes local similarity apparent

4 Collaborators Bei Wang (SCI Institute, Utah) Sayan Mukherjee (Duke) John Harer (Duke) Dmitriy Morozov (LBNL) Herbert Edelsbrunner (IST Austria) David Cohen-Steiner (INRIA Sophia Antipolis) Bryan Jacobson (Duke) Ellen Gasparovic (Duke)

5 Talk Outline Applications Sliding-window Embedding and Motif-hunting for Signals (B., Gasparovic, Harer) Vizualization of High-Dimensional Data (Wang et. al.) Theory Stratifications and Local Homology Persistent Local Homology Inference Theorems Algorithms Computing PLH via Rips Complexes (Wang and Skraba 2013, Jacobson and B. 2014)

6 Sliding Window Embedding Persistence and Delay Embeddings: Sw1PerS (Perea and Harer, 2013) Application to wheeze signals (Emrani, Gentimis, Krim, 2013) Sliding window embedding Step/delay ; Window size Dimension window

7 Motif Repetition Repeated motifs in signal show up as singularities in SW-embedding. More interesting: similar motifs in two distinct signals: X shape means one repeated window, maybe at different times. Y shape means delayed synchronization for some time period. Detect with PLH rather than needing something like MDS. Computation of PLH (via Rips) not dependent (well... ) on ambient dimension.

8 Local parametrization: branching detection Apply circular-value coordinates (de Silva, Morozov, Vejdemo-Johansson 2009) to local cohomology.

9 Local parametrization: branching detection

10 Local parametrization: fake branching detection (b)-(c) Fake branching (e)-(g) True branching (a) (b) (c) (e) (f) (g)

11 Stratification: definition by picture X topological space. Decomposition of X into manifolds (strata) Two points in same connected component (piece) of same stratum are locally indistinguishable. =

12 Local Homology: definition Given point z in top. space X. LH of X at z is H k (X, X {z}). 1-dim LH of red point is rank one, of blue point is rank three.

13 Local Homology: alternate defnition Assuming X R D is (say) Whitney stratified. For each z, there exists a small enough radius r = r z such that H k (X, X {z}) = H k (X B r (z), X B r (z))

14 Local Homology: how about noise? What does small enough radius mean in noisy world? Simple idea: study H k (X B r (z), X B r (z)) at many radii.

15 Persistent LH Diagrams Fix a radius R and center point z R D Study module {H k (X α B R (z), X α B R (z)} Notation: let LH k (z, R) denote the diagram. death 10.5 α 1 7 α 2 α α 1 α 2 α birth

16 Persistent LH Diagrams Fix a radius R and center point z R D Study module {H k (X α B R (z), X α B R (z)} Notation: let LH k (z, R) denote the diagram. death 10.5 α 1 7 α 2 α3 3 α birth α 2 α 3

17 Persistent LH Diagrams Fix a radius R and center point z R D Study module {H k (X α B R (z), X α B R (z)} Notation: let LH k (z, R) denote the diagram. death 10.5 α 4 7 α 2 α α 2 α 3 α 4 birth

18 Persistent LH Diagrams Fix a radius R and center point z R D Study module {H k (X α B R (z), X α B R (z)} Notation: let LH k (z, R) denote the diagram. α 4 death 10.5 α 4 α 2 7 α 2 α birth α 3

19 Persistent LH Diagrams Fix a radius R and center point z R D Study module {H k (X α B R (z), X α B R (z)} Notation: let LH k (z, R) denote the diagram. α 3 death α birth

20 Persistent LH Diagrams Fix a radius R and center point z R D Study module {H k (X α B R (z), X α B R (z)} Notation: let LH k (z, R) denote the diagram. death α α 1 α 2 7 α 2 α 3 α birth

21 LH Vineyards Continuity in radius: W p (LH(z, R), LH(z, S)) S R. As R increases, dots in LH(z, R) move continuously and form vines. Union of vines is vineyard.

22 Local hom. feature size Define ρ k (z, R) to be smallest non-zero c.v of module {H k (X α B R (z), X α B R (z)} Here ρ 1 (x, R) = a and ρ 1 (y, R) = c. α 1 x a x x γ 1 y x b y β 1 c y y

23 LH Inference Theorem Suppose U is point sample from/near X, with d H (X, U) ɛ. Suppose ρ k (z, R) < ɛ 4 : rank of H k (X B r (z), X B r (z)) equals: Number of dots (u, v) LH k (z, R) with 0 u < ɛ and v > 3ɛ:

24 LH not nearly enough LH alone clearly not enough to differentiate local structure The rank of local homology groups of x, y agree in all dimensions. x = x y L x = = CL x y = CL y L y = CL y =

25 LH transfer map Partial solution: Study the following map: H(X B r (x), X B r (x)) f H(X B r (x) B r (y), X (B r (x) B r (y))) x y α 1 γ 1 β 1 x y f g α 1 γ 1 β 1

26 LH transfer map x y α 2 α 3 α 1 x γ 1 β1 y α 1 α 2 α 3 kerf f g γ 1 β 1

27 LH transfer map x y γ 1 x γ 2 β 2 α 1 β 1 y f g α 1 γ 1 β 1 γ 2 β 2 cokf

28 Clustering via transfer map Given points x and y such that B r (x) B r (y) : Put x and y into same piece at radius R if the two LH transfer maps are isos. Form transitive closure of this relation. death x y z #2 0 birth

29 What about noise? Transfer map below right not iso, but noisy sample will obscure that. Obvious fact: map is iso iff kernel and cokernel are zero. Idea: thicken space and study (co)kernel persistence. (a) (b)

30 Kernel persistent homology, formally H(X α B r (x), X α B r (x)) fα H(X α B r (x) B r (y), X α (B r (x) B r (y))) H(X β B r (x), X β B r (x)) f β H(X β B r (x) B r (y), X β (B r (x) B r (y))) ker f α ker f β cok f α cok f β

31 Kernel persistent homology for a space death 10.5 α 1 α 2 α 3 γ birth α 3 f γ1 α 1 α 2 α 1 α 2 kerf

32 Kernel persistent homology for a space α 1 death 10.5 α 1 α 2 7 α 2 α 3 γ birth α 3 f γ 1

33 Kernel persistent homology for a space death α 3 γ birth f α 3 α 3 kerf

34 Kernel persistent homology for a space α 3 death α birth f

35 Kernel persistent homology for a space α 3 death 10.5 α 1 α 2 α 3 γ 1 α 1 α birth

36 Kernel persistent homology for point cloud death 0 birth

37 Local hom. feature size Define ρ(x, y, R) = min{a, b, c}. α 1 x a x x γ 1 y x b y β 1 c y y

38 Topological Inference theorem Theorem (Topological Inference Theorem) Suppose that we have an ɛ-approximation U from X. Then for each pair of points x, y R N such that ρ(x, y, R) 4ɛ, x and y have same local stucture at radius R iff Dgm(ker f U )(ɛ, 3ɛ) Dgm(cok f U )(ɛ, 3ɛ) =. Empty death birth

39 Topological Inference theorem Theorem (Topological Inference Theorem) Suppose that we have an ɛ-approximation U from X. Then for each pair of points x, y R N such that ρ(x, y, R) 4ɛ, x and y have same local stucture at radius R iff Dgm(ker f U )(ɛ, 3ɛ) Dgm(cok f U )(ɛ, 3ɛ) =. p q death #2 (ɛ, 3ɛ) 0 birth p q

40 Algorithm Problems Actually computing LH diagrams is nasty. Naive Cech-like construction fails b/c intersection with boundary not convex! Correct power-cell algorithms (B. et al. 2007, B., Wang, Mukherjee 2012) scale like U 5 and 2 D. Z(α) p q P (α) Q(α)

41 Boundary Homology One approach: figure out appropriate sampling conditions and work with Rips directly (Wang and Skraba 2013). Or, make problem easier (Jacobson and B., 2014): Sub H(U α B R ) for H(U α B R, U B R ) Justifiable by LES of pair, or by appeal to historical error. death 10.5 α 1 7 α 2 α α 1 α 2 α birth

42 Boundary Cech Let U R D be a set of points, R > 0, D 3, z R D, B = B R (z). Theorem α < R, x 1,..., x k U, the intersection B ( k i=1 B α(x i )) is either empty or contractible. Appealing to the Persistent Nerve Lemma (Fraser 2012), this yields the persistence diagram of {H(U α B)} 0 α<r.

43 Boundary Rips V R B (U, 2ɛ) = {σ =< x 1,..., x n > U i, j B ɛ (x i ) B ɛ (x j ) B } Ĉ B (U, ɛ) = {σ =< x 1,..., x n > U ( n i=1 B ɛ(x i )) B }. By last theorem, the second complex computes the correct PLH. Simple interleaving result means we can use the first complex.

44 Planar Rips Theorem Let D > 2, x 1, x 2, z points R D, B = B R (z). Define P to be the plane in R D containing x 1, x 2, z. Then α < R, if B α (x 1 ) B α (x 2 ) B, then P (B α (x 1 ) B α (x 2 ) B). As a result, approximation of PLH amounts to: Constructing a distance matrix D entries in D obtained via planar computations feeding D to standard Rips software

45 The End THANK YOU FOR YOUR TIME AND ATTENTION!