Graph Induced Complex: A Data Sparsifier for Homology Inference

Transcription

1 Graph Induced Complex: A Data Sparsifier for Homology Inference Tamal K. Dey Fengtao Fan Yusu Wang Department of Computer Science and Engineering The Ohio State University October, 2013

2 Space, Sample and Complex Space (IMA) Graph Induced Complex October / 25

3 Space, Sample and Complex Space Samples (IMA) Graph Induced Complex October / 25

4 Space, Sample and Complex Space Samples Rips Complex (IMA) Graph Induced Complex October / 25

5 Complexes Delaunay complex difficult to compute in high dimensional spaces; (IMA) Graph Induced Complex October / 25

6 Complexes Delaunay complex difficult to compute in high dimensional spaces; Vietoris-Rips complex too large (5000 points in R 3 millions of simplicies); (IMA) Graph Induced Complex October / 25

7 Complexes Delaunay complex difficult to compute in high dimensional spaces; Vietoris-Rips complex too large (5000 points in R 3 millions of simplicies); Čech complex difficult to compute; also large; (IMA) Graph Induced Complex October / 25

8 Complexes Delaunay complex difficult to compute in high dimensional spaces; Vietoris-Rips complex too large (5000 points in R 3 millions of simplicies); Čech complex difficult to compute; also large; Witness complex manageable size; lack topological inference; (IMA) Graph Induced Complex October / 25

9 Subsamples (IMA) Graph Induced Complex October / 25

10 Subsamples Q (IMA) Graph Induced Complex October / 25

11 Subsamples R r (Q) (IMA) Graph Induced Complex October / 25

12 Subsamples R r (Q) Witness Complex (IMA) Graph Induced Complex October / 25

13 Subsamples R r (Q) Witness Complex (IMA) Graph Induced Complex October / 25

14 Our solution R r (P ) (IMA) Graph Induced Complex October / 25

15 Our solution R r (P ) G r (P, Q) (IMA) Graph Induced Complex October / 25

16 Input Assumptions P finite point set; (P, d) metric space; G(P ) be a graph; (IMA) Graph Induced Complex October / 25

17 Subsampling Q P a subset; ν(p) : the closest point of p P in Q; (IMA) Graph Induced Complex October / 25

18 Graph Induced Complex Graph induced complex G(P, Q, d) : {q 1,..., q k+1 } Q; a (k + 1)-clique in G(P ) with vertices p1,..., p k+1 ; ν(pi ) = q i ; (IMA) Graph Induced Complex October / 25

19 Graph Induced Complex Graph induced complex G(P, Q, d) : {q 1,..., q k+1 } Q; a (k + 1)-clique in G(P ) with vertices p1,..., p k+1 ; ν(pi ) = q i ; Remark: G(P, Q, d) depends on the metric d; Euclidean distance d E ; Graph based distance d G ; (IMA) Graph Induced Complex October / 25

20 Subsample Q of P δ-sample of P p P, q Q such that d(p, q) δ; (IMA) Graph Induced Complex October / 25

21 Subsample Q of P δ-sample of P p P, q Q such that d(p, q) δ; δ-sparse d(p, q) δ for any two distinct points p, q Q; (IMA) Graph Induced Complex October / 25

22 Subsample Q of P δ-sample of P p P, q Q such that d(p, q) δ; δ-sparse d(p, q) δ for any two distinct points p, q Q; Computing δ-sparse δ-sample Q iterative procedure; (IMA) Graph Induced Complex October / 25

23 Subsample Q of P δ-sample of P p P, q Q such that d(p, q) δ; δ-sparse d(p, q) δ for any two distinct points p, q Q; Computing δ-sparse δ-sample Q iterative procedure; δ Q i (IMA) Graph Induced Complex October / 25

24 Subsample Q of P δ-sample of P p P, q Q such that d(p, q) δ; δ-sparse d(p, q) δ for any two distinct points p, q Q; Computing δ-sparse δ-sample Q iterative procedure; δ δ q i+1 Q i Q i+1 (IMA) Graph Induced Complex October / 25

25 Subsample Q of P δ-sample of P p P, q Q such that d(p, q) δ; δ-sparse d(p, q) δ for any two distinct points p, q Q; Computing δ-sparse δ-sample Q iterative procedure; δ δ δ q i+1 Q i Q i+1 Q i+1 (IMA) Graph Induced Complex October / 25

26 Results H 1 inference in R n by d E and d G ; (IMA) Graph Induced Complex October / 25

27 Results H 1 inference in R n by d E and d G ; Surface reconstruction in R 3 ; (IMA) Graph Induced Complex October / 25

28 Results H 1 inference in R n by d E and d G ; Surface reconstruction in R 3 ; Improved H 1 inference in R n by d G from a lean subsample ; Complex size in log scale 14 Witness complex Rips complex 12 Graph induced complex Klein bottle in R 4 Dimension of H 1 14 Witness complex 12 Rips complex Graph induced complex δ δ (IMA) Graph Induced Complex October / 25

29 Results H 1 inference in R n by d E and d G ; Surface reconstruction in R 3 ; Improved H 1 inference in R n by d G from a lean subsample ; Complex size in log scale 14 Witness complex Rips complex 12 Graph induced complex Klein bottle in R 4 Dimension of H 1 14 Witness complex 12 Rips complex Graph induced complex δ δ Topological inference for compact sets in R n ; G α (P, Q, d) G 4(α+2δ) (P, Q, d) (IMA) Graph Induced Complex October / 25

30 Results H 1 inference in R n by d E and d G ; Surface reconstruction in R 3 ; Improved H 1 inference in R n by d G from a lean subsample ; Complex size in log scale 14 Witness complex Rips complex 12 Graph induced complex Klein bottle in R 4 Dimension of H 1 14 Witness complex 12 Rips complex Graph induced complex δ δ Topological inference for compact sets in R n ; G α (P, Q, d) G 4(α+2δ) (P, Q, d) (IMA) Graph Induced Complex October / 25

31 Simplicial map f : K L simplicial map for every simplex σ = {v 1, v 2,..., v k } K, f(σ) = {f(v 1 ), f(v 2 ),..., f(v k )} is a simplex in L (IMA) Graph Induced Complex October / 25

32 Simplicial map f : K L simplicial map for every simplex σ = {v 1, v 2,..., v k } K, f(σ) = {f(v 1 ), f(v 2 ),..., f(v k )} is a simplex in L w w u K v x f u L y x (IMA) Graph Induced Complex October / 25

33 Simplicial map f : K L simplicial map for every simplex σ = {v 1, v 2,..., v k } K, f(σ) = {f(v 1 ), f(v 2 ),..., f(v k )} is a simplex in L w w u K v x f u L y x [u] [u] [v] [u] [w] [w] [x] [x] [uv] [u] [uw] [uw] [vw] [uw] [vx] [vx] [uvw] [uw] (IMA) Graph Induced Complex October / 25

34 Contiguous maps f, g : K 1 K 2 two simplicial maps are contiguous for any simplex σ K 1, the simplices f(σ) and g(σ) are faces of a common simplex in K 2. (IMA) Graph Induced Complex October / 25

35 Contiguous maps f, g : K 1 K 2 two simplicial maps are contiguous for any simplex σ K 1, the simplices f(σ) and g(σ) are faces of a common simplex in K 2. u v x f g y z f([v]) = [y] g([v]) = [z], f([uv]) = [xy] g([uv]) = [xz] (IMA) Graph Induced Complex October / 25

36 Contiguous maps f, g : K 1 K 2 two simplicial maps are contiguous for any simplex σ K 1, the simplices f(σ) and g(σ) are faces of a common simplex in K 2. u v x f g y z f([v]) = [y] g([v]) = [z], f([uv]) = [xy] g([uv]) = [xz] Fact If f : K 1 K 2 and g : K 1 K 2 are contiguous, then the induced homomorphisms f : H n(k 1 ) H n(k 2 ) and g : H n(k 1 ) H n(k 2 ) are equal. (IMA) Graph Induced Complex October / 25

37 H 1 inference in R n Sample P from a space M; (IMA) Graph Induced Complex October / 25

38 H 1 inference in R n Sample P from a space M; Subsample Q : δ-sample, δ-sparse; (IMA) Graph Induced Complex October / 25

39 H 1 inference in R n Sample P from a space M; Subsample Q : δ-sample, δ-sparse; G α (P ) = 1-skeleton of R α (P ) (IMA) Graph Induced Complex October / 25

40 H 1 inference in R n Sample P from a space M; Subsample Q : δ-sample, δ-sparse; G α (P ) = 1-skeleton of R α (P ) The GIC G α (P, Q) Built on G α (P ) (IMA) Graph Induced Complex October / 25

41 H 1 inference in R n Sample P from a space M; Subsample Q : δ-sample, δ-sparse; G α (P ) = 1-skeleton of R α (P ) The GIC G α (P, Q) Built on G α (P ) h : R α (P ) G α (P, Q) simplicial map (IMA) Graph Induced Complex October / 25

42 H 1 inference in R n Sample P from a space M; Subsample Q : δ-sample, δ-sparse; G α (P ) = 1-skeleton of R α (P ) The GIC G α (P, Q) Built on G α (P ) h : R α (P ) G α (P, Q) simplicial map h : H(R α (P )) H(G α (P, Q)) isomorphism? (IMA) Graph Induced Complex October / 25

43 Contiguous maps between Rips and GIC R α (P ) h G α (P, Q, d) (IMA) Graph Induced Complex October / 25

44 Contiguous maps between Rips and GIC R α (P ) h R α+2δ (P ) G α (P, Q, d) (IMA) Graph Induced Complex October / 25

45 Contiguous maps between Rips and GIC i R α (P ) h R α+2δ (P ) G α (P, Q, d) (IMA) Graph Induced Complex October / 25

46 Contiguous maps between Rips and GIC R α (P ) h j R α+2δ (P ) G α (P, Q, d) (IMA) Graph Induced Complex October / 25

47 Contiguous maps between Rips and GIC i R α (P ) h j R α+2δ (P ) G α (P, Q, d) (IMA) Graph Induced Complex October / 25

48 Contiguous maps between Rips and GIC i R α (P ) h j R α+2δ (P ) G α (P, Q, d) j h contiguous to i; (IMA) Graph Induced Complex October / 25

49 Contiguous maps between Rips and GIC i R α (P ) h j R α+2δ (P ) j h contiguous to i; (j h) = i ; G α (P, Q, d) (j h), i : H(R α (P )) H(R α+2δ (P )) (IMA) Graph Induced Complex October / 25

50 Isomorphism of h Contiguous maps : j h = i ; (IMA) Graph Induced Complex October / 25

51 Isomorphism of h Contiguous maps : j h = i ; i isomorphism; i.e. P sampled from a manifold with positive reach; Prop 4.1 of [Dey and Wang 11] i : H 1 (R α (P )) H 1 (R β (P )) isomorphism; (IMA) Graph Induced Complex October / 25

52 Isomorphism of h Contiguous maps : j h = i ; i isomorphism; i.e. P sampled from a manifold with positive reach; Prop 4.1 of [Dey and Wang 11] i : H 1 (R α (P )) H 1 (R β (P )) isomorphism; h is injective; (IMA) Graph Induced Complex October / 25

53 Isomorphism of h Contiguous maps : j h = i ; i isomorphism; i.e. P sampled from a manifold with positive reach; Prop 4.1 of [Dey and Wang 11] i : H 1 (R α (P )) H 1 (R β (P )) isomorphism; h is injective; When h surjective? (IMA) Graph Induced Complex October / 25

54 Isomorphism of h Contiguous maps : j h = i ; i isomorphism; i.e. P sampled from a manifold with positive reach; Prop 4.1 of [Dey and Wang 11] i : H 1 (R α (P )) H 1 (R β (P )) isomorphism; h is injective; When h surjective? Higher dimensional homology (dim > 1), UNKNOWN; (IMA) Graph Induced Complex October / 25

55 Isomorphism of h Contiguous maps : j h = i ; i isomorphism; i.e. P sampled from a manifold with positive reach; Prop 4.1 of [Dey and Wang 11] i : H 1 (R α (P )) H 1 (R β (P )) isomorphism; h is injective; When h surjective? Higher dimensional homology (dim > 1), UNKNOWN; Positive answers for the 1-st dimensional homology : (IMA) Graph Induced Complex October / 25

56 Isomorphism of h Contiguous maps : j h = i ; i isomorphism; i.e. P sampled from a manifold with positive reach; Prop 4.1 of [Dey and Wang 11] i : H 1 (R α (P )) H 1 (R β (P )) isomorphism; h is injective; When h surjective? Higher dimensional homology (dim > 1), UNKNOWN; Positive answers for the 1-st dimensional homology : Euclidean distance de in R 3 ; (IMA) Graph Induced Complex October / 25

57 Isomorphism of h Contiguous maps : j h = i ; i isomorphism; i.e. P sampled from a manifold with positive reach; Prop 4.1 of [Dey and Wang 11] i : H 1 (R α (P )) H 1 (R β (P )) isomorphism; h is injective; When h surjective? Higher dimensional homology (dim > 1), UNKNOWN; Positive answers for the 1-st dimensional homology : Euclidean distance de in R 3 ; Graph distance dg in R n ; h : H 1 (R α (P )) H 1 (G α (P, Q)) isomorphism (IMA) Graph Induced Complex October / 25

58 Isomorphism of h Theorem P : an ɛ-sample of a surface with positive reach ρ in R 3 ; Q: a δ-sparse δ-sample of (P, d E ); ɛ ρ, 12ɛ α ρ, and 8ɛ δ ρ; h : H 1 (R α (P )) H 1 (G α (P, Q, d E )) isomorphism. (IMA) Graph Induced Complex October / 25

59 Isomorphism of h Theorem P : an ɛ-sample of a surface with positive reach ρ in R 3 ; Q: a δ-sparse δ-sample of (P, d E ); ɛ ρ, 12ɛ α ρ, and 8ɛ δ ρ; h : H 1 (R α (P )) H 1 (G α (P, Q, d E )) isomorphism. Theorem P : an ɛ-sample of manifold M with positive reach ρ; Q: a δ-sample of (P, d G ); 4ɛ α, δ 1 3 ρ, 3 5 h : H 1 (R α (P )) H 1 (G α (P, Q, d G )) isomorphism. (IMA) Graph Induced Complex October / 25

60 Improved H 1 Inference Homological loop feature size for simplicial complex K hlfs(k) = 1 inf{ c, c is non null-homologous 1-cycle in K}. 2 (IMA) Graph Induced Complex October / 25

61 Improved H 1 Inference Homological loop feature size for simplicial complex K hlfs(k) = 1 inf{ c, c is non null-homologous 1-cycle in K}. 2 ρ hlfs (IMA) Graph Induced Complex October / 25

62 Improved H 1 Inference Homological loop feature size for simplicial complex K hlfs(k) = 1 inf{ c, c is non null-homologous 1-cycle in K}. 2 ρ hlfs Theorem If Q is a δ-sample of (P, d G ) for δ < 1 2 hlfs(rα (P )) 1 α, then 2 h : H 1 (R α (P )) H 1 (G α (P, Q, d G )) is an isomorphism. (IMA) Graph Induced Complex October / 25

63 Experimental Result for Improved H 1 Inference Dimension of H Witness complex Rips complex Graph induced complex Complex size in log scale Witness complex Rips complex Graph induced complex δ (a) δ (b) Klein bottle in R 4 with P = 40, 000; GIC size : 154 (δ = 1.0) (IMA) Graph Induced Complex October / 25

64 Surface Reconstruction Crust [AB99] and Cocone [ACDL00]: Compute a subcomplex T Del P ; Argue T contains the restricted Delaunay triangulation Del M P ; Prune T to output a 2-manifold; (IMA) Graph Induced Complex October / 25

65 Surface Reconstruction by GIC in R 3 Restricted Delaunay triangulation Del M Q G(P, Q, d E ); (IMA) Graph Induced Complex October / 25

66 Surface Reconstruction by GIC in R 3 Restricted Delaunay triangulation Del M Q G(P, Q, d E ); Problem: GIC not embedded in R 3 ; t 2 t 1 (IMA) Graph Induced Complex October / 25

67 Surface Reconstruction by GIC in R 3 Restricted Delaunay triangulation Del M Q G(P, Q, d E ); Problem: GIC not embedded in R 3 ; t 2 t 1 Cleaning If V is the vertex set of t 1 and t 2 together, then at least one of t 1 and t 2 is not in Del V. The triangle which is not in Del V cannot be in Del Q as well. (IMA) Graph Induced Complex October / 25

68 Theorem P : ε-sample Q: δ-sparse, δ-sample of P 8ε δ 2 ρ, α 8ε 27 A triangulation T G α (P, Q, d E ) can be computed. FERTILITY BOTIJO mesh GIC mesh GIC 0-dim dim dim dim P = 1, 575, 055 for FERTILITY; P = 1, 049, 892 for BOTIJO; (IMA) Graph Induced Complex October / 25

69 GIC for Compact Sets Go beyond manifold and H 1 ; (IMA) Graph Induced Complex October / 25

70 GIC for Compact Sets Go beyond manifold and H 1 ; X R n compact set, and X λ offset; (IMA) Graph Induced Complex October / 25

71 GIC for Compact Sets Go beyond manifold and H 1 ; X R n compact set, and X λ offset; Homology of X λ captured by persistence of Rips complex on its samples P ; (IMA) Graph Induced Complex October / 25

72 GIC for Compact Sets Go beyond manifold and H 1 ; X R n compact set, and X λ offset; Homology of X λ captured by persistence of Rips complex on its samples P ; Interleave GIC with Rips complexes : persistence of GIC gives homology of X λ ; (IMA) Graph Induced Complex October / 25

73 GIC for Compact Sets Go beyond manifold and H 1 ; X R n compact set, and X λ offset; Homology of X λ captured by persistence of Rips complex on its samples P ; Interleave GIC with Rips complexes : persistence of GIC gives homology of X λ ; R α (P ) i 1 R h 1 α+2δ (P ) i 2 R 4β (P ) i 3 R 4β+2δ (P ) j 2 j 1 h 2 G α (P, Q, d) h G 4β (P, Q, d) Q δ-sparse δ-sample of P ; Q δ -sparse δ -sample of P with δ > δ; Denote β = α + 2δ; (IMA) Graph Induced Complex October / 25

74 GIC for Compact Sets Above diagram gives sequence H k (R α (P )) h 1 H k (G α (P, Q, d)) j 1 H k (R α+2δ (P )) i 2 H k (R 4β (P )) h 2 H k (G 4β (P, Q, d)) j 2 H k (R 4β+2δ (P )) (IMA) Graph Induced Complex October / 25

75 GIC for Compact Sets Above diagram gives sequence H k (R α (P )) h 1 H k (G α (P, Q, d)) j 1 H k (R α+2δ (P )) i 2 H k (R 4β (P )) h 2 H k (G 4β (P, Q, d)) j 2 H k (R 4β+2δ (P )) h = (h 2 i 2 j 1 ) simplicial map; Algorithms for persistence of simplicial maps [DFW]; (IMA) Graph Induced Complex October / 25

76 GIC for Compact Sets Above diagram gives sequence H k (R α (P )) h 1 H k (G α (P, Q, d)) j 1 H k (R α+2δ (P )) i 2 H k (R 4β (P )) h 2 H k (G 4β (P, Q, d)) j 2 H k (R 4β+2δ (P )) h = (h 2 i 2 j 1 ) simplicial map; Algorithms for persistence of simplicial maps [DFW]; Theorem P : an ɛ-sample of a compact set (X, d E ); Q: a δ-sparse δ-sample of (P, d) (d = d E or d G ); Q : a δ -sparse δ -sample of (P, d) (δ > δ); 0 < ɛ < 1 9 wfs(x), 2ɛ α 1 4 (wfs(x) ɛ) and (α + 2δ) δ 1 (wfs(x) ɛ), 4 im h = Hk (X λ ) (0 < λ < wfs(x)) (IMA) Graph Induced Complex October / 25

77 Conclusion Software for constructing GIC Available at authors webpages; (IMA) Graph Induced Complex October / 25

78 Conclusion Software for constructing GIC Available at authors webpages; Future work Go beyond of H 1 for h ; h : H n (R α (P )) H n (G α (P, Q)) (n 2) isomorphism? (IMA) Graph Induced Complex October / 25

79 Conclusion Software for constructing GIC Available at authors webpages; Future work Go beyond of H 1 for h ; h : H n (R α (P )) H n (G α (P, Q)) (n 2) isomorphism? Potential use in topological data analysis; (IMA) Graph Induced Complex October / 25

80 (IMA) Graph Induced Complex October / 25