ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration
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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.
![Some related work on sparse filtrations.
Previous Approaches at subsampling:
Witness Complexes [dSC04, BGO07]](https://image.slidesharecdn.com/atmcs12linear-120930084840-phpapp01/85/ATMCS-Linear-Size-Approximations-to-the-Vietoris-Rips-Filtration-54-320.jpg)
![Some related work on sparse filtrations.
Previous Approaches at subsampling:
Witness Complexes [dSC04, BGO07]
Persistence-based Reconstruction [CO08]](https://image.slidesharecdn.com/atmcs12linear-120930084840-phpapp01/85/ATMCS-Linear-Size-Approximations-to-the-Vietoris-Rips-Filtration-55-320.jpg)
![Some related work on sparse filtrations.
Previous Approaches at subsampling:
Witness Complexes [dSC04, BGO07]
Persistence-based Reconstruction [CO08]
Other methods:](https://image.slidesharecdn.com/atmcs12linear-120930084840-phpapp01/85/ATMCS-Linear-Size-Approximations-to-the-Vietoris-Rips-Filtration-56-320.jpg)
![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]](https://image.slidesharecdn.com/atmcs12linear-120930084840-phpapp01/85/ATMCS-Linear-Size-Approximations-to-the-Vietoris-Rips-Filtration-57-320.jpg)
![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]](https://image.slidesharecdn.com/atmcs12linear-120930084840-phpapp01/85/ATMCS-Linear-Size-Approximations-to-the-Vietoris-Rips-Filtration-58-320.jpg)
ATMCS: Linear-Size Approximations to 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.