A Multicover Nerve for Geometric Inference. Don Sheehy INRIA Saclay, France

Transcription

1 A Multicover Nerve for Geometric Inference Don Sheehy INRIA Saclay, France

2 Computational geometers use topology to certify geometric constructions.

3 Computational geometers use topology to certify geometric constructions. Surface Reconstruction - homeomorphic

4 Computational geometers use topology to certify geometric constructions. Surface Reconstruction - homeomorphic Medial Axis Approximation - homotopy equivalence

5 Computational geometers use topology to certify geometric constructions. Surface Reconstruction - homeomorphic Medial Axis Approximation - homotopy equivalence Topological Data Analysis - (persistent) homology

6 Computational geometers use topology to certify geometric constructions. Surface Reconstruction - homeomorphic Medial Axis Approximation - homotopy equivalence Topological Data Analysis - (persistent) homology

7 Topological Inference

8 Topological Inference Fixed Scale: Niyogi, Smale, Weinberger 2008

9 Topological Inference Fixed Scale: Niyogi, Smale, Weinberger 2008 Variable Scale: Chazal, Cohen-Steiner, Lieutier 2009

10 Topological Inference Fixed Scale: Niyogi, Smale, Weinberger 2008 Variable Scale: Chazal, Cohen-Steiner, Lieutier 2009 All Scales (Persistence): Edelsbrunner, Letscher, Zomorodian 2002

11 Topological Inference Fixed Scale: Niyogi, Smale, Weinberger 2008 Variable Scale: Chazal, Cohen-Steiner, Lieutier 2009 All Scales (Persistence): Edelsbrunner, Letscher, Zomorodian 2002 Guarantees: Chazal and Oudot, 2008

12 The Nerve Theorem

13 The Nerve Theorem Union of Shapes

14 The Nerve Theorem Union of Shapes Simplicial Complex

15 The Nerve Theorem Union of Shapes Simplicial Complex

16 The Nerve Theorem Union of Shapes Simplicial Complex

17 The Nerve Theorem Union of Shapes Simplicial Complex Key Fact: Preserves Topology as long as intersections are empty or contractible.

18 The Nerve Theorem Union of Shapes Simplicial Complex Cech Complex Key Fact: Preserves Topology as long as intersections are empty or contractible.

19 Geometric Persistent Homology

20 Geometric Persistent Homology Input: P R d

21 Geometric Persistent Homology Input: P R d P α = ball(p, α) p P

22 Geometric Persistent Homology Input: P R d P α = ball(p, α) p P

23 Geometric Persistent Homology Input: P R d P α = ball(p, α) p P

24 Geometric Persistent Homology Offsets Input: P R d P α = ball(p, α) p P

25 Geometric Persistent Homology Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

26 Geometric Persistent Homology Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

27 Geometric Persistent Homology Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

28 Geometric Persistent Homology Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

29 Geometric Persistent Homology Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

30 Geometric Persistent Homology Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

31 Geometric Persistent Homology Offsets Input: P R d P α = ball(p, α) p P Persistent Compute the Homology

32 k-covered regions

33 k-covered regions α

34 k-covered regions α Idea: Capture both mass and scale.

35 k-covered regions α Idea: Capture both mass and scale. Goal: Build a simplicial complex homotopy equivalent to the k-covered regions.

36 The k-nerve. k-nerve

37 The k-nerve. k-nerve

38 The k-nerve. k-nerve The k-nerve already gives the right topology, but...

39 The k-nerve. k-nerve The k-nerve already gives the right topology, but......there is no easy relationship between the complexes for different values of k.

40 Barycentric Decomposition

41 Barycentric Decomposition

42 Barycentric Decomposition

43 Barycentric Decomposition 0

44 Barycentric Decomposition 0 1

45 Barycentric Decomposition 0 1 2

46 Barycentric Decomposition

47 Barycentric Decomposition

48 The kth barycentric Cech complex is homotopy equivalent to the k-nerve. Cech Complex Barycentric Decomposition Filtered 2,α-offsets 2-nerve Barycentric Decomposition

49 The kth barycentric Cech complex is homotopy equivalent to the k-nerve. Cech Complex Barycentric Decomposition Filtered 2,α-offsets 2-nerve Barycentric Decomposition

50 The kth barycentric Cech complex is homotopy equivalent to the k-nerve. Cech Complex Barycentric Decomposition Filtered 2,α-offsets 2-nerve Barycentric Decomposition

51 The kth barycentric Cech complex is homotopy equivalent to the k-nerve. Cech Complex Barycentric Decomposition Filtered 2,α-offsets 2-nerve Barycentric Decomposition

52 The kth barycentric Cech complex is homotopy equivalent to the k-nerve. Cech Complex Barycentric Decomposition Filtered 2,α-offsets 2-nerve Barycentric Decomposition

53 The kth barycentric Cech complex is homotopy equivalent to the k-nerve. Cech Complex Barycentric Decomposition Filtered 2,α-offsets 2-nerve Barycentric Decomposition

54 The kth barycentric Cech complex is homotopy equivalent to the k-nerve. Cech Complex Barycentric Decomposition Filtered 2,α-offsets 2-nerve Barycentric Decomposition

55 A persistent version.

56 A persistent version. Input is a collection of filtrations, rather than a collection of sets.

57 A persistent version. Input is a collection of filtrations, rather than a collection of sets. The Result: Given a collection of convex filtrations, the persistent homology of the k-covered set is exactly that of the kth barycentric decomposition of the nerve of the filtrations.

58 What if we only have pairwise distances?

59 What if we only have pairwise distances? The Rips complex at scale r is the clique complex of the r-neighborhood graph. (the edges are the same as those in the Cech complex)

60 What if we only have pairwise distances? The Rips complex at scale r is the clique complex of the r-neighborhood graph. (the edges are the same as those in the Cech complex) New Result: Applying the same barycentric trick to the Rips complexes gives a 2-approximation to the persistent homology of k-covered region of balls.

61 Conclusion A filtered simplicial complex that captures the topology of the k-covered region of a collection of convex sets for all k. Guaranteed correct persistent homology. A guaranteed approximation via easier to compute Rips complexes.

62 Conclusion A filtered simplicial complex that captures the topology of the k-covered region of a collection of convex sets for all k. Guaranteed correct persistent homology. A guaranteed approximation via easier to compute Rips complexes. Thank you.

63

64 A 3-Step Process 1 Statistics De-noise and smooth the data. 2 Geometry Build a complex. 3 Topology (Algebra) Compute the persistent homology.

65 A 3-Step Process 1 Statistics De-noise and smooth the data. 2 Geometry Build a complex. 3 Topology (Algebra) Compute the persistent homology.

66 A 3-Step Process 1 Statistics De-noise and smooth the data. 2 Geometry Build a complex. 3 Topology (Algebra) Compute the persistent homology.

67 Goal: No more tuning parameters

68 Goal: No more tuning parameters i.e. Build a complex that works for every choice of de-noising parameters.

69 Capture both scale AND mass

70 Capture both scale AND mass See Also: [Chazal, Cohen-Steiner, Merigot, 2009]

71 Capture both scale AND mass See Also: [Chazal, Cohen-Steiner, Merigot, 2009] [Guibas, Merigot, Morozov, Yesterday]

72 k-nn distance.

73 k-nn distance. d P (x) =min p P x p P α = d 1 P [0,α]

74 k-nn distance. d P (x) =min p P x p P α = d 1 P [0,α] d k (x) = min max S ( P k) p S x p P α k = d 1 k [0,α]

75 k-nn distance. d P (x) =min p P x p P α = d 1 P [0,α] d k (x) = min max S ( P k) p S α x p P α k = d 1 k [0,α]

76 k-nn distance. d P (x) =min p P x p P α = d 1 P [0,α] d k (x) = min max S ( P k) p S α x p P α k = d 1 k [0,α] a multifiltration with parameters α and k.