atmcs: linear-size approximations to the vietoris-rips filtration
DESCRIPTION
The Vietoris-Rips filtration is a versatile tool in topological data analysis. Unfortunately, it is often too large to construct in full. We show how to construct an $O(n)$-size filtered simplicial complex on an $n$-point metric space such that the persistence diagram is a good approximation to that of the Vietoris-Rips filtration. The filtration can be constructed in $O(n\log n)$ time. The constants depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. Our approach uses a hierarchical net-tree to sparsify the filtration. We can either sparsify the data by throwing out points at larger scales to give a zigzag filtration, or sparsify the underlying graph by throwing out edges at larger scales to give a standard filtration. Both methods yield the same guarantees.TRANSCRIPT
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Linear-Size Approximations to the Vietoris-Rips Filtration
Don SheehyGeometrica Group
INRIA Saclay
This work appeared at SoCG 2012
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The goal of topological data analysis is to extract meaningful topological
information from data.
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Use powerful ideas from computational geometry to speed up persistent
homology computation when the data is intrinsically low-dimensional.
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A filtration is a growing sequence of spaces.
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A filtration is a growing sequence of spaces.
A persistence diagram describes the topological changes over time.
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A filtration is a growing sequence of spaces.
Dea
thBirth
A persistence diagram describes the topological changes over time.
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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 {R!}such that ! ! R! i! d(p, q) " 2" for all p, q ! !.
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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 {R!}such that ! ! R! i! d(p, q) " 2" for all p, q ! !.
![Page 17: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/17.jpg)
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 {R!}such that ! ! R! i! d(p, q) " 2" for all p, q ! !.
![Page 18: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/18.jpg)
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 {R!}such that ! ! R! i! d(p, q) " 2" for all p, q ! !.
![Page 19: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/19.jpg)
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 {R!}such that ! ! R! i! d(p, q) " 2" for all p, q ! !.
![Page 20: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/20.jpg)
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 {R!}such that ! ! R! i! d(p, q) " 2" for all p, q ! !.
![Page 21: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/21.jpg)
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 {R!}such that ! ! R! i! d(p, q) " 2" for all p, q ! !.
R! is the powerset 2P .
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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 {R!}such that ! ! R! i! d(p, q) " 2" for all p, q ! !.
R! is the powerset 2P .
This is too big!
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Persistence Diagrams describe the changes in topology corresponding to changes in scale.
Birth
Dea
th
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Persistence Diagrams describe the changes in topology corresponding to changes in scale.
Birth
Dea
th
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Persistence Diagrams describe the changes in topology corresponding to changes in scale.
d!B = maxi
|pi ! qi|!
Bottleneck Distance
Birth
Dea
th
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Persistence Diagrams describe the changes in topology corresponding to changes in scale.
d!B = maxi
|pi ! qi|!
Bottleneck Distance
In approximate persistence diagrams, birth and death times differ by at most a constant factor.
Birth
Dea
th
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Persistence Diagrams describe the changes in topology corresponding to changes in scale.
d!B = maxi
|pi ! qi|!
Bottleneck Distance
This is just the bottleneck distance of the log-scale diagrams.
In approximate persistence diagrams, birth and death times differ by at most a constant factor.
Birth
Dea
th
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The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
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The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 30: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/30.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 31: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/31.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 32: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/32.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 33: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/33.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 34: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/34.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 35: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/35.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 36: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/36.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 37: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/37.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 38: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/38.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 39: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/39.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 40: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/40.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 41: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/41.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 42: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/42.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 43: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/43.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 44: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/44.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 45: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/45.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 46: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/46.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 47: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/47.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 48: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/48.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 49: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/49.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 50: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/50.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 51: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/51.jpg)
The Vietoris-Rips complex is a nerve of boxes.Embed the input metric in Rn with the L! norm.
In L!, the Rips complex is the same as the Cech complex
![Page 52: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/52.jpg)
Some related work on sparse filtrations.
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Previous Approaches at subsampling:
Some related work on sparse filtrations.
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Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]
Some related work on sparse filtrations.
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Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]Persistence-based Reconstruction [CO08]
Some related work on sparse filtrations.
![Page 56: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/56.jpg)
Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]Persistence-based Reconstruction [CO08]
Other methods:
Some related work on sparse filtrations.
![Page 57: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/57.jpg)
Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]Persistence-based Reconstruction [CO08]
Other methods:Meshes in Euclidean Space [HMSO10]
Some related work on sparse filtrations.
![Page 58: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/58.jpg)
Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]Persistence-based Reconstruction [CO08]
Other methods:Meshes in Euclidean Space [HMSO10]Topological simplification [Z10, ALS11]
Some related work on sparse filtrations.
![Page 59: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/59.jpg)
Key idea: Treat many close points as one point.
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Key idea: Treat many close points as one point.
This idea is ubiquitous in computational geometry.
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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,...
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Consider a 2-dimensional filtration parameterized by both scale and sampling density.
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Consider a 2-dimensional filtration parameterized by both scale and sampling density.
![Page 64: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/64.jpg)
Consider a 2-dimensional filtration parameterized by both scale and sampling density.
![Page 65: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/65.jpg)
Consider a 2-dimensional filtration parameterized by both scale and sampling density.
![Page 66: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/66.jpg)
Consider a 2-dimensional filtration parameterized by both scale and sampling density.
![Page 67: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/67.jpg)
Consider a 2-dimensional filtration parameterized by both scale and sampling density.
![Page 68: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/68.jpg)
Consider a 2-dimensional filtration parameterized by both scale and sampling density.
![Page 69: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/69.jpg)
Consider a 2-dimensional filtration parameterized by both scale and sampling density.
![Page 70: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/70.jpg)
Consider a 2-dimensional filtration parameterized by both scale and sampling density.
![Page 71: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/71.jpg)
Consider a 2-dimensional filtration parameterized by both scale and sampling density.
![Page 72: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/72.jpg)
Consider a 2-dimensional filtration parameterized by both scale and sampling density.
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Intuition: Remove points that are covered by their neighbors.
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Intuition: Remove points that are covered by their neighbors.
![Page 75: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/75.jpg)
Intuition: Remove points that are covered by their neighbors.
![Page 76: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/76.jpg)
Intuition: Remove points that are covered by their neighbors.
No change in topology
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Intuition: Remove points that are covered by their neighbors.
No change in topology
![Page 78: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/78.jpg)
Intuition: Remove points that are covered by their neighbors.
No change in topology
![Page 79: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/79.jpg)
Intuition: Remove points that are covered by their neighbors.
No change in topology
Perturb the metric
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Two tricks:
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Two tricks:1 Embed the zigzag in a topologically equivalent filtration.
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Two tricks:
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
1 Embed the zigzag in a topologically equivalent filtration.
Zigzag filtration
Standard filtration
![Page 84: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/84.jpg)
Two tricks:
2 Perturb the metric so the persistence module does not zigzag.
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
1 Embed the zigzag in a topologically equivalent filtration.
Zigzag filtration
Standard filtration
![Page 85: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/85.jpg)
Two tricks:
2 Perturb the metric so the persistence module does not zigzag.
! H(Q!) " H(Q") ! H(Q#) "
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
1 Embed the zigzag in a topologically equivalent filtration.
Zigzag filtration
Standard filtration
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!=
!=
Two tricks:
2 Perturb the metric so the persistence module does not zigzag.
! H(Q!) " H(Q") ! H(Q#) "
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
1 Embed the zigzag in a topologically equivalent filtration.
Zigzag filtration
Standard filtration
![Page 87: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/87.jpg)
!=
!=
Two tricks:
2 Perturb the metric so the persistence module does not zigzag.
! H(Q!) ! H(Q") ! H(Q#) !
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
1 Embed the zigzag in a topologically equivalent filtration.
Zigzag filtration
Standard filtration
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!=
!=
Two tricks:
2 Perturb the metric so the persistence module does not zigzag.
! H(Q!) ! H(Q") ! H(Q#) !
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
1 Embed the zigzag in a topologically equivalent filtration.
Zigzag filtration
Standard filtration
At the homology level, there is no zigzag.
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The Result:
![Page 90: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/90.jpg)
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.
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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)
![Page 92: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/92.jpg)
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)
![Page 93: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/93.jpg)
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)
![Page 94: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/94.jpg)
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.
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How to perturb the metric.
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How to perturb the metric.Let tp be the time when point p is removed.
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How to perturb the metric.
wp(!) =
!
"
#
0 if ! ! (1" ")tp!" (1" ")tp if (1" ")tp < ! < tp
"! if tp ! !
Let tp be the time when point p is removed.
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How to perturb the metric.
wp(!) =
!
"
#
0 if ! ! (1" ")tp!" (1" ")tp if (1" ")tp < ! < tp
"! if tp ! !
tp(1! !)tp00
Let tp be the time when point p is removed.
![Page 99: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/99.jpg)
How to perturb the metric.
d!(p, q) = d(p, q) + wp(!) + wq(!)
wp(!) =
!
"
#
0 if ! ! (1" ")tp!" (1" ")tp if (1" ")tp < ! < tp
"! if tp ! !
tp(1! !)tp00
Let tp be the time when point p is removed.
![Page 100: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/100.jpg)
How to perturb the metric.
d!(p, q) = d(p, q) + wp(!) + wq(!)
wp(!) =
!
"
#
0 if ! ! (1" ")tp!" (1" ")tp if (1" ")tp < ! < tp
"! if tp ! !
tp(1! !)tp00
Let tp be the time when point p is removed.
! ! R! " d(p, q) # 2" for all p, q ! !Rips Complex:
![Page 101: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/101.jpg)
How to perturb the metric.
d!(p, q) = d(p, q) + wp(!) + wq(!)
wp(!) =
!
"
#
0 if ! ! (1" ")tp!" (1" ")tp if (1" ")tp < ! < tp
"! if tp ! !
tp(1! !)tp00
Let tp be the time when point p is removed.
! ! R! " d(p, q) # 2" for all p, q ! !Rips Complex:
Relaxed Rips: ! ! R! " d!(p, q) # 2" for all p, q ! !
![Page 102: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/102.jpg)
How to perturb the metric.
d!(p, q) = d(p, q) + wp(!) + wq(!)
wp(!) =
!
"
#
0 if ! ! (1" ")tp!" (1" ")tp if (1" ")tp < ! < tp
"! if tp ! !
tp(1! !)tp00
Let tp be the time when point p is removed.
! ! R! " d(p, q) # 2" for all p, q ! !Rips Complex:
Relaxed Rips: ! ! R! " d!(p, q) # 2" for all p, q ! !
R !
1+"
! R! ! R!
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Net-trees
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Net-trees
![Page 105: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/105.jpg)
Net-trees
![Page 106: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/106.jpg)
Net-trees
A generalization of quadtrees to metric spaces.
![Page 107: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/107.jpg)
Net-trees
A generalization of quadtrees to metric spaces.One leaf per point of P.
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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).
![Page 109: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/109.jpg)
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:
![Page 110: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/110.jpg)
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.
![Page 111: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/111.jpg)
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))
![Page 112: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/112.jpg)
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)
![Page 113: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/113.jpg)
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.
Time to remove p: tp = 1!(1!!)rad(parent(up))
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Projection onto a net
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Projection onto a netN! = {p ! P : tp " !}
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Projection onto a netN! = {p ! P : tp " !}
Q! is the subcomplex of R! induced on N!.
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Projection onto a netN! = {p ! P : tp " !}
Q! is the subcomplex of R! induced on N!.
!!(p) =
!
p if p ! N!
argminq!N!d(p, q) otherwise
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N! is a Delone set.
Projection onto a netN! = {p ! P : tp " !}
Q! is the subcomplex of R! induced on N!.
!!(p) =
!
p if p ! N!
argminq!N!d(p, q) otherwise
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N! is a Delone set.
Projection onto a netN! = {p ! P : tp " !}
For all p ! P , there is a q ! N!
such that d(p, q) " !(1# !)".1 Covering:
Q! is the subcomplex of R! induced on N!.
!!(p) =
!
p if p ! N!
argminq!N!d(p, q) otherwise
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N! is a Delone set.
Projection onto a netN! = {p ! P : tp " !}
For all p ! P , there is a q ! N!
such that d(p, q) " !(1# !)".1 Covering:
For all distinct p, q ! N!,d(p, q) " Kpack!(1# !)".
2 Packing:
Q! is the subcomplex of R! induced on N!.
!!(p) =
!
p if p ! N!
argminq!N!d(p, q) otherwise
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N! is a Delone set.
Projection onto a netN! = {p ! P : tp " !}
For all p, q ! P , d(!!(p), q) " d(p, q).Key Fact:
For all p ! P , there is a q ! N!
such that d(p, q) " !(1# !)".1 Covering:
For all distinct p, q ! N!,d(p, q) " Kpack!(1# !)".
2 Packing:
Q! is the subcomplex of R! induced on N!.
!!(p) =
!
p if p ! N!
argminq!N!d(p, q) otherwise
![Page 122: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/122.jpg)
For all p, q ! P , d(!!(p), q) " d(p, q).Key Fact:
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
Relaxed Rips:
Sparse Rips Zigzag:
This can be used to show that projection onto a net is a homotopy equivalence.
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For all p, q ! P , d(!!(p), q) " d(p, q).Key Fact:
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
Relaxed Rips:
Sparse Rips Zigzag:
! H(Q!) " H(Q") ! H(Q#) "
This can be used to show that projection onto a net is a homotopy equivalence.
![Page 124: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/124.jpg)
For all p, q ! P , d(!!(p), q) " d(p, q).Key Fact:
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
Relaxed Rips:
Sparse Rips Zigzag:
!=
!=
! H(Q!) " H(Q") ! H(Q#) "
This can be used to show that projection onto a net is a homotopy equivalence.
![Page 125: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/125.jpg)
For all p, q ! P , d(!!(p), q) " d(p, q).Key Fact:
!! Q! "" Q" !! Q# ""
!! R! !! R" !! R# !!
!! !! !!
Relaxed Rips:
Sparse Rips Zigzag:
!=
!=
! H(Q!) ! H(Q") ! H(Q#) !
This can be used to show that projection onto a net is a homotopy equivalence.
![Page 126: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/126.jpg)
Why is the filtration only linear size?
![Page 127: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/127.jpg)
Why is the filtration only linear size?
Standard trick from Euclidean geometry:
![Page 128: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/128.jpg)
Why is the filtration only linear size?
Standard trick from Euclidean geometry:
1 Charge each simplex to its vertex with the earliest deletion time.
![Page 129: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/129.jpg)
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.
![Page 130: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/130.jpg)
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.
![Page 131: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/131.jpg)
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.
!
1
!
"O(d2)
![Page 132: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/132.jpg)
Really getting rid of the zigzags.
![Page 133: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/133.jpg)
Really getting rid of the zigzags.
X! =
!
"!!
Q!
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Really getting rid of the zigzags.
X! =
!
"!!
Q!
This is (almost) the clique complex of a hierarchical spanner!
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Summary
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SummaryApproximate the VR filtration with a Zigzag filtration.
![Page 137: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/137.jpg)
SummaryApproximate the VR filtration with a Zigzag filtration.Remove points using a hierarchical net-tree.
![Page 138: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/138.jpg)
SummaryApproximate the VR filtration with a Zigzag filtration.Remove points using a hierarchical net-tree.Perturb the metric to straighten out the zigzags.
![Page 139: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/139.jpg)
SummaryApproximate 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.
![Page 140: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/140.jpg)
SummaryApproximate 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.
![Page 141: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/141.jpg)
SummaryApproximate 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
![Page 142: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/142.jpg)
SummaryApproximate 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 FutureDistance to a measure?
![Page 143: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/143.jpg)
SummaryApproximate 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 FutureDistance to a measure?Other types of simplification.
![Page 144: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/144.jpg)
SummaryApproximate 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 FutureDistance to a measure?Other types of simplification.Construct the net-tree in O(n log n) time.
![Page 145: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/145.jpg)
SummaryApproximate 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 FutureDistance to a measure?Other types of simplification.Construct the net-tree in O(n log n) time.An implementation.
![Page 146: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/146.jpg)
SummaryApproximate 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 FutureDistance to a measure?Other types of simplification.Construct the net-tree in O(n log n) time.An implementation.
Thank You.
![Page 147: ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration](https://reader033.vdocument.in/reader033/viewer/2022060108/555096fab4c90595208b4673/html5/thumbnails/147.jpg)
SummaryApproximate 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 FutureDistance to a measure?Other types of simplification.Construct the net-tree in O(n log n) time.An implementation.
Thank You.Thank You.
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