1) The paper presents a method to approximate the Vietoris-Rips filtration using a zigzag filtration of linear size. 2) It embeds the zigzag filtration into an equivalent standard filtration and perturbs the metric so that the zigzag does not occur at the homology level. 3) This results in a zigzag filtration of size O(n) whose persistence diagram provides a (1+ε)-approximation of the persistence diagram for the Vietoris-Rips filtration.

1. Linear-Size Approximations to the
Vietoris-Rips Filtration
Don Sheehy
Geometrica Group
INRIA Saclay
This work appeared at SoCG 2012

2. The goal of topological data analysis
is to extract meaningful topological
information from data.

3. Use powerful ideas from computational
geometry to speed up persistent
homology computation when the data is
intrinsically low-dimensional.

12. A filtration is a growing sequence of spaces.

13. A filtration is a growing sequence of spaces.
A persistence diagram describes
the topological changes over time.

14. A filtration is a growing sequence of spaces.
Death
Birth
A persistence diagram describes
the topological changes over time.

15. The Vietoris-Rips Filtration encodes the topology of a
metric space when viewed at different scales.
Input: A finite metric space (P, d).
Output: A sequence of simplicial complexes {Rα }
such that σ ∈ Rα iff d(p, q) ≤ 2α for all p, q ∈ σ.

16. The Vietoris-Rips Filtration encodes the topology of a
metric space when viewed at different scales.
Input: A finite metric space (P, d).
Output: A sequence of simplicial complexes {Rα }
such that σ ∈ Rα iff d(p, q) ≤ 2α for all p, q ∈ σ.

17. The Vietoris-Rips Filtration encodes the topology of a
metric space when viewed at different scales.
Input: A finite metric space (P, d).
Output: A sequence of simplicial complexes {Rα }
such that σ ∈ Rα iff d(p, q) ≤ 2α for all p, q ∈ σ.

18. The Vietoris-Rips Filtration encodes the topology of a
metric space when viewed at different scales.
Input: A finite metric space (P, d).
Output: A sequence of simplicial complexes {Rα }
such that σ ∈ Rα iff d(p, q) ≤ 2α for all p, q ∈ σ.

19. The Vietoris-Rips Filtration encodes the topology of a
metric space when viewed at different scales.
Input: A finite metric space (P, d).
Output: A sequence of simplicial complexes {Rα }
such that σ ∈ Rα iff d(p, q) ≤ 2α for all p, q ∈ σ.

20. The Vietoris-Rips Filtration encodes the topology of a
metric space when viewed at different scales.
Input: A finite metric space (P, d).
Output: A sequence of simplicial complexes {Rα }
such that σ ∈ Rα iff d(p, q) ≤ 2α for all p, q ∈ σ.

21. The Vietoris-Rips Filtration encodes the topology of a
metric space when viewed at different scales.
Input: A finite metric space (P, d).
Output: A sequence of simplicial complexes {Rα }
such that σ ∈ Rα iff d(p, q) ≤ 2α for all p, q ∈ σ.
R∞ is the powerset 2P .

22. The Vietoris-Rips Filtration encodes the topology of a
metric space when viewed at different scales.
Input: A finite metric space (P, d).
Output: A sequence of simplicial complexes {Rα }
such that σ ∈ Rα iff d(p, q) ≤ 2α for all p, q ∈ σ.
R∞ is the powerset 2P .
This is too big!

23. Persistence Diagrams describe the changes in
topology corresponding to changes in scale.
Death
Birth

24. Persistence Diagrams describe the changes in
topology corresponding to changes in scale.
Death
Birth

25. Persistence Diagrams describe the changes in
topology corresponding to changes in scale.
Bottleneck Distance
d∞ = max |pi − qi |∞
B
i
Death
Birth

26. Persistence Diagrams describe the changes in
topology corresponding to changes in scale.
Bottleneck Distance
d∞ = max |pi − qi |∞
B
i
Death
In approximate persistence
diagrams, birth and death
times differ by at most a
constant factor.
Birth

27. Persistence Diagrams describe the changes in
topology corresponding to changes in scale.
Bottleneck Distance
d∞ = max |pi − qi |∞
B
i
Death
In approximate persistence
diagrams, birth and death
times differ by at most a
constant factor.
This is just the bottleneck
Birth distance of the log-scale
diagrams.

28. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

29. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

30. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

31. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

32. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

33. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

34. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

35. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

36. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

37. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

38. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

39. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

40. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

41. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

42. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

43. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

44. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

45. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

46. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

47. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

48. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

49. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

50. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

51. The Vietoris-Rips complex is a nerve of boxes.
Embed the input metric in Rn with the L∞ norm.
ˇ
In L∞ , the Rips complex is the same as the Cech complex

52. Some related work on sparse filtrations.

53. Some related work on sparse filtrations.
Previous Approaches at subsampling:

54. Some related work on sparse filtrations.
Previous Approaches at subsampling:
Witness Complexes [dSC04, BGO07]

55. Some related work on sparse filtrations.
Previous Approaches at subsampling:
Witness Complexes [dSC04, BGO07]
Persistence-based Reconstruction [CO08]

56. Some related work on sparse filtrations.
Previous Approaches at subsampling:
Witness Complexes [dSC04, BGO07]
Persistence-based Reconstruction [CO08]
Other methods:

57. Some related work on sparse filtrations.
Previous Approaches at subsampling:
Witness Complexes [dSC04, BGO07]
Persistence-based Reconstruction [CO08]
Other methods:
Meshes in Euclidean Space [HMSO10]

58. Some related work on sparse filtrations.
Previous Approaches at subsampling:
Witness Complexes [dSC04, BGO07]
Persistence-based Reconstruction [CO08]
Other methods:
Meshes in Euclidean Space [HMSO10]
Topological simplification [Z10, ALS11]

59. Key idea: Treat many close points as one point.

60. Key idea: Treat many close points as one point.
This idea is ubiquitous in computational geometry.

61. Key idea: Treat many close points as one point.
This idea is ubiquitous in computational geometry.
n-body simulation, approximate nearest neighbor search,
spanners, well-separated pair decomposition,...

62. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

63. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

64. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

65. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

66. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

67. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

68. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

69. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

70. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

71. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

72. Consider a 2-dimensional filtration parameterized
by both scale and sampling density.

73. Intuition: Remove points that are covered
by their neighbors.

74. Intuition: Remove points that are covered
by their neighbors.

75. Intuition: Remove points that are covered
by their neighbors.

76. Intuition: Remove points that are covered
by their neighbors.
No change
in topology

77. Intuition: Remove points that are covered
by their neighbors.
No change
in topology

78. Intuition: Remove points that are covered
by their neighbors.
No change
in topology

79. Intuition: Remove points that are covered
by their neighbors.
No change
in topology
Perturb the metric

81. Two tricks:

82. Two tricks:
1 Embed the zigzag in a topologically equivalent filtration.

83. Two tricks:
1 Embed the zigzag in a topologically equivalent filtration.
ˆ ˆ ˆ
Standard filtration → Rα → Rβ → Rγ →
→
→
→
Zigzag filtration ← Qα → Qβ ← Qγ →

84. Two tricks:
1 Embed the zigzag in a topologically equivalent filtration.
2 Perturb the metric so the persistence module does not zigzag.
ˆ ˆ ˆ
Standard filtration → Rα → Rβ → Rγ →
→
→
→
Zigzag filtration ← Qα → Qβ ← Qγ →

85. Two tricks:
1 Embed the zigzag in a topologically equivalent filtration.
2 Perturb the metric so the persistence module does not zigzag.
ˆ ˆ ˆ
Standard filtration → Rα → Rβ → Rγ →
→
→
→
Zigzag filtration ← Qα → Qβ ← Qγ →
← H(Qα ) → H(Qβ ) ← H(Qγ ) →

86. Two tricks:
1 Embed the zigzag in a topologically equivalent filtration.
2 Perturb the metric so the persistence module does not zigzag.
ˆ ˆ ˆ
Standard filtration → Rα → Rβ → Rγ →
→
→
→
Zigzag filtration ← Qα → Qβ ← Qγ →
∼
= ∼
=
← H(Qα ) → H(Qβ ) ← H(Qγ ) →

87. Two tricks:
1 Embed the zigzag in a topologically equivalent filtration.
2 Perturb the metric so the persistence module does not zigzag.
ˆ ˆ ˆ
Standard filtration → Rα → Rβ → Rγ →
→
→
→
Zigzag filtration ← Qα → Qβ ← Qγ →
∼
= ∼
=
→ H(Qα ) → H(Qβ ) → H(Qγ ) →

88. Two tricks:
1 Embed the zigzag in a topologically equivalent filtration.
2 Perturb the metric so the persistence module does not zigzag.
ˆ ˆ ˆ
Standard filtration → Rα → Rβ → Rγ →
→
→
→
Zigzag filtration ← Qα → Qβ ← Qγ →
∼
= ∼
=
→ H(Qα ) → H(Qβ ) → H(Qγ ) →
At the homology level, there is no zigzag.

89. The Result:

90. The Result:
Given an n point metric (P, d), there exists a zigzag filtration of
size O(n) whose persistence diagram (1+ ε)-approximates that of
the Rips filtration.

91. The Result:
Given an n point metric (P, d), there exists a zigzag filtration of
size O(n) whose persistence diagram (1+ ε)-approximates that of
the Rips filtration.
(Big-O hides doubling dimension and approximation factor, ε)

92. The Result:
Given an n point metric (P, d), there exists a zigzag filtration of
size O(n) whose persistence diagram (1+ ε)-approximates that of
the Rips filtration.
(Big-O hides doubling dimension and approximation factor, ε)

93. The Result:
Given an n point metric (P, d), there exists a zigzag filtration of
size O(n) whose persistence diagram (1+ ε)-approximates that of
the Rips filtration.
(Big-O hides doubling dimension and approximation factor, ε)

94. The Result:
Given an n point metric (P, d), there exists a zigzag filtration of
size O(n) whose persistence diagram (1+ ε)-approximates that of
the Rips filtration.
(Big-O hides doubling dimension and approximation factor, ε)
A metric with doubling dimension d is
one for which every ball of radius 2r
can be covered by 2d balls of radius r
for all r.

95. How to perturb the metric.

96. How to perturb the metric.
Let tp be the time when point p is removed.

97. How to perturb the metric.
Let tp be the time when point p is removed.

 0 if α ≤ (1 − ε)tp
wp (α) = α − (1 − ε)tp if (1 − ε)tp < α < tp
εα if tp ≤ α


98. How to perturb the metric.
Let tp be the time when point p is removed.

 0 if α ≤ (1 − ε)tp
wp (α) = α − (1 − ε)tp if (1 − ε)tp < α < tp
εα if tp ≤ α

0
0 (1 − ε)tp tp

99. How to perturb the metric.
Let tp be the time when point p is removed.

 0 if α ≤ (1 − ε)tp
wp (α) = α − (1 − ε)tp if (1 − ε)tp < α < tp
εα if tp ≤ α

0
0 (1 − ε)tp tp
ˆ
dα (p, q) = d(p, q) + wp (α) + wq (α)

100. How to perturb the metric.
Let tp be the time when point p is removed.

 0 if α ≤ (1 − ε)tp
wp (α) = α − (1 − ε)tp if (1 − ε)tp < α < tp
εα if tp ≤ α

0
0 (1 − ε)tp tp
ˆ
dα (p, q) = d(p, q) + wp (α) + wq (α)
Rips Complex: σ ∈ Rα ⇔ d(p, q) ≤ 2α for all p, q ∈ σ

101. How to perturb the metric.
Let tp be the time when point p is removed.

 0 if α ≤ (1 − ε)tp
wp (α) = α − (1 − ε)tp if (1 − ε)tp < α < tp
εα if tp ≤ α

0
0 (1 − ε)tp tp
ˆ
dα (p, q) = d(p, q) + wp (α) + wq (α)
Rips Complex: σ ∈ Rα ⇔ d(p, q) ≤ 2α for all p, q ∈ σ
ˆ ˆ
Relaxed Rips: σ ∈ Rα ⇔ dα (p, q) ≤ 2α for all p, q ∈ σ

102. How to perturb the metric.
Let tp be the time when point p is removed.

 0 if α ≤ (1 − ε)tp
wp (α) = α − (1 − ε)tp if (1 − ε)tp < α < tp
εα if tp ≤ α

0
0 (1 − ε)tp tp
ˆ
dα (p, q) = d(p, q) + wp (α) + wq (α)
Rips Complex: σ ∈ Rα ⇔ d(p, q) ≤ 2α for all p, q ∈ σ
ˆ ˆ
Relaxed Rips: σ ∈ Rα ⇔ dα (p, q) ≤ 2α for all p, q ∈ σ
ˆ
R 1+ε ⊆ Rα ⊆ Rα
α

103. Net-trees

104. Net-trees

105. Net-trees

106. Net-trees
A generalization of quadtrees to metric spaces.

107. Net-trees
A generalization of quadtrees to metric spaces.
One leaf per point of P.

108. Net-trees
A generalization of quadtrees to metric spaces.
One leaf per point of P.
Each node u has a representative rep(u) in P and a radius rad(p).

109. Net-trees
A generalization of quadtrees to metric spaces.
One leaf per point of P.
Each node u has a representative rep(u) in P and a radius rad(p).
Three properties:

110. Net-trees
A generalization of quadtrees to metric spaces.
One leaf per point of P.
Each node u has a representative rep(u) in P and a radius rad(p).
Three properties:
1 Inheritance: Every nonleaf u has a child with the same rep.

111. Net-trees
A generalization of quadtrees to metric spaces.
One leaf per point of P.
Each node u has a representative rep(u) in P and a radius rad(p).
Three properties:
1 Inheritance: Every nonleaf u has a child with the same rep.
2 Covering: Every heir is within ball(rep(u), rad(u))

112. Net-trees
A generalization of quadtrees to metric spaces.
One leaf per point of P.
Each node u has a representative rep(u) in P and a radius rad(p).
Three properties:
1 Inheritance: Every nonleaf u has a child with the same rep.
2 Covering: Every heir is within ball(rep(u), rad(u))
3 Packing: Any children v,w of u have d(rep(v),rep(w)) > K rad(u)

113. Net-trees
A generalization of quadtrees to metric spaces.
One leaf per point of P.
Each node u has a representative rep(u) in P and a radius rad(p).
Three properties:
1 Inheritance: Every nonleaf u has a child with the same rep.
2 Covering: Every heir is within ball(rep(u), rad(u))
3 Packing: Any children v,w of u have d(rep(v),rep(w)) > K rad(u)
Let up be the ancestor of all nodes represented by p.
1
Time to remove p: tp = ε(1−ε) rad(parent(up ))

114. Projection onto a net

115. Projection onto a net
Nα = {p ∈ P : tp ≥ α}

116. Projection onto a net
Nα = {p ∈ P : tp ≥ α}
ˆ
Qα is the subcomplex of Rα induced on Nα .

117. Projection onto a net
Nα = {p ∈ P : tp ≥ α}
ˆ
Qα is the subcomplex of Rα induced on Nα .
p if p ∈ Nα
πα (p) = ˆ
arg minq∈Nα d(p, q) otherwise

118. Projection onto a net
Nα = {p ∈ P : tp ≥ α}
ˆ
Qα is the subcomplex of Rα induced on Nα .
p if p ∈ Nα
πα (p) = ˆ
arg minq∈Nα d(p, q) otherwise
Nα is a Delone set.

119. Projection onto a net
Nα = {p ∈ P : tp ≥ α}
ˆ
Qα is the subcomplex of Rα induced on Nα .
p if p ∈ Nα
πα (p) = ˆ
arg minq∈Nα d(p, q) otherwise
Nα is a Delone set.
1 Covering: For all p ∈ P , there is a q ∈ Nα
such that d(p, q) ≤ ε(1 − ε)α.

120. Projection onto a net
Nα = {p ∈ P : tp ≥ α}
ˆ
Qα is the subcomplex of Rα induced on Nα .
p if p ∈ Nα
πα (p) = ˆ
arg minq∈Nα d(p, q) otherwise
Nα is a Delone set.
1 Covering: For all p ∈ P , there is a q ∈ Nα
such that d(p, q) ≤ ε(1 − ε)α.
2 Packing: For all distinct p, q ∈ Nα ,
d(p, q) ≥ Kpack ε(1 − ε)α.

121. Projection onto a net
Nα = {p ∈ P : tp ≥ α}
ˆ
Qα is the subcomplex of Rα induced on Nα .
p if p ∈ Nα
πα (p) = ˆ
arg minq∈Nα d(p, q) otherwise
Nα is a Delone set.
1 Covering: For all p ∈ P , there is a q ∈ Nα
such that d(p, q) ≤ ε(1 − ε)α.
2 Packing: For all distinct p, q ∈ Nα ,
d(p, q) ≥ Kpack ε(1 − ε)α.
Key Fact: ˆ ˆ
For all p, q ∈ P , d(πα (p), q) ≤ d(p, q).

122. Key Fact: ˆ ˆ
For all p, q ∈ P , d(πα (p), q) ≤ d(p, q).
This can be used to show that projection onto a net is a
homotopy equivalence.
Relaxed Rips: → Rα → Rβ → Rγ →
ˆ ˆ ˆ
→
→
→
Sparse Rips Zigzag: ← Qα → Qβ ← Qγ →

123. Key Fact: ˆ ˆ
For all p, q ∈ P , d(πα (p), q) ≤ d(p, q).
This can be used to show that projection onto a net is a
homotopy equivalence.
Relaxed Rips: → Rα → Rβ → Rγ →
ˆ ˆ ˆ
→
→
→
Sparse Rips Zigzag: ← Qα → Qβ ← Qγ →
← H(Qα ) → H(Qβ ) ← H(Qγ ) →

124. Key Fact: ˆ ˆ
For all p, q ∈ P , d(πα (p), q) ≤ d(p, q).
This can be used to show that projection onto a net is a
homotopy equivalence.
Relaxed Rips: → Rα → Rβ → Rγ →
ˆ ˆ ˆ
→
→
→
Sparse Rips Zigzag: ← Qα → Qβ ← Qγ →
∼
= ∼
=
← H(Qα ) → H(Qβ ) ← H(Qγ ) →

125. Key Fact: ˆ ˆ
For all p, q ∈ P , d(πα (p), q) ≤ d(p, q).
This can be used to show that projection onto a net is a
homotopy equivalence.
Relaxed Rips: → Rα → Rβ → Rγ →
ˆ ˆ ˆ
→
→
→
Sparse Rips Zigzag: ← Qα → Qβ ← Qγ →
∼
= ∼
=
→ H(Qα ) → H(Qβ ) → H(Qγ ) →

126. Why is the filtration only linear size?

127. Why is the filtration only linear size?
Standard trick from Euclidean geometry:

128. Why is the filtration only linear size?
Standard trick from Euclidean geometry:
1 Charge each simplex to its vertex with the earliest deletion time.

129. Why is the filtration only linear size?
Standard trick from Euclidean geometry:
1 Charge each simplex to its vertex with the earliest deletion time.
2 Apply a packing argument to the larger neighbors.

130. Why is the filtration only linear size?
Standard trick from Euclidean geometry:
1 Charge each simplex to its vertex with the earliest deletion time.
2 Apply a packing argument to the larger neighbors.
3 Conclude average O(1) simplices per vertex.

131. Why is the filtration only linear size?
Standard trick from Euclidean geometry:
1 Charge each simplex to its vertex with the earliest deletion time.
2 Apply a packing argument to the larger neighbors.
3 Conclude average O(1) simplices per vertex.
O(d2 )
1
ε

132. Really getting rid of the zigzags.

133. Really getting rid of the zigzags.
Xα = Qα
β≤α

134. Really getting rid of the zigzags.
Xα = Qα
β≤α
This is (almost) the clique complex of a hierarchical spanner!

135. Summary

136. Summary
Approximate the VR filtration with a Zigzag filtration.

137. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.

138. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.

139. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.
Use packing arguments to get linear size.

140. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.
Use packing arguments to get linear size.
Union trick eliminates zigzag.

141. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.
Use packing arguments to get linear size.
Union trick eliminates zigzag.
The Future

142. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.
Use packing arguments to get linear size.
Union trick eliminates zigzag.
The Future
Distance to a measure?

143. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.
Use packing arguments to get linear size.
Union trick eliminates zigzag.
The Future
Distance to a measure?
Other types of simplification.

144. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.
Use packing arguments to get linear size.
Union trick eliminates zigzag.
The Future
Distance to a measure?
Other types of simplification.
Construct the net-tree in O(n log n) time.

145. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.
Use packing arguments to get linear size.
Union trick eliminates zigzag.
The Future
Distance to a measure?
Other types of simplification.
Construct the net-tree in O(n log n) time.
An implementation.

146. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.
Use packing arguments to get linear size.
Union trick eliminates zigzag.
The Future
Distance to a measure?
Other types of simplification.
Construct the net-tree in O(n log n) time.
An implementation.
Thank You.

147. Summary
Approximate the VR filtration with a Zigzag filtration.
Remove points using a hierarchical net-tree.
Perturb the metric to straighten out the zigzags.
Use packing arguments to get linear size.
Union trick eliminates zigzag.
The Future
Distance to a measure?
Other types of simplification.
Construct the net-tree in O(n log n) time.
An implementation.
Thank You.