We show that filtering the barycentric decomposition of a Cech complex by the cardinality of the vertices captures precisely the topology of k-covered regions among a collection of balls for all values of k. Moreover, we relate this result to the Vietoris-Rips complex to get an approximation in terms of the persistent homology.

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 ⊂ Rd

21. Geometric Persistent Homology
Input: P ⊂ Rd
α
P = ball(p, α)
p∈P

22. Geometric Persistent Homology
Input: P ⊂ Rd
α
P = ball(p, α)
p∈P

23. Geometric Persistent Homology
Input: P ⊂ Rd
α
P = ball(p, α)
p∈P

24. Geometric Persistent Homology
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P

25. Geometric Persistent Homology
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

26. Geometric Persistent Homology
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

27. Geometric Persistent Homology
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

28. Geometric Persistent Homology
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

29. Geometric Persistent Homology
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

30. Geometric Persistent Homology
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

31. Geometric Persistent Homology
Offsets Input: P ⊂ Rd
α
P = 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 0
1 2
2 1 0

47. Barycentric Decomposition 0
1 2
3
2 2
1 1
0

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

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

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

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

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

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

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

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

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

66. A 3-Step Process
Statistics
1 De-noise and smooth the data.
Geometry
2 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.
dP (x) = min |x − p| α
P = dP [0, α]
−1
p∈P

74. k-NN distance.
dP (x) = min |x − p| α
P = dP [0, α]
−1
p∈P
dk (x) = min max |x − p| α
Pk = dk [0, α]
−1
S∈(P ) p∈S
k

75. k-NN distance.
dP (x) = min |x − p| α
P = dP [0, α]
−1
p∈P
dk (x) = min max |x − p| α
Pk = dk [0, α]
−1
S∈(P ) p∈S
k
α

76. k-NN distance.
dP (x) = min |x − p| α
P = dP [0, α]
−1
p∈P
dk (x) = min max |x − p| α
Pk = dk [0, α]
−1
S∈(P ) p∈S
k
α
a multifiltration with parameters α and k.