Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

131

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 of Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Page 1: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster
Page 2: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Leopold Vietoris

Page 3: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Leopold VietorisEliyahu Rips

Page 4: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Linear-Size Approximations to the Vietoris-Rips Filtration

Don SheehyGeometrica Group

INRIA Saclay

This work will appear at SoCG 2012 in June.

Page 5: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

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Points, offsets, homology, and persistence.

Input: P ! Rd

Page 7: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! Rd

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Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! Rd

Page 9: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! Rd

Page 10: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! RdOffsets

Filtration

Page 11: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! RdOffsets

Filtration

Compute the Homology

Page 12: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! RdOffsets

Filtration

Compute the Homology

Page 13: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! RdOffsets

Filtration

Compute the Homology

Page 14: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! RdOffsets

Filtration

Compute the Homology

Page 15: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! RdOffsets

Filtration

Compute the Homology

Page 16: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! RdOffsets

Filtration

Compute the Homology

Page 17: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Points, offsets, homology, and persistence.

P! =!

p!P

ball(p,!)

Input: P ! RdOffsets

Filtration

Compute the Homology

Persistent

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Persistence Diagrams describe the changes in topology corresponding to changes in scale.

Birth

Dea

th

Page 19: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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

Page 21: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 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 24: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 25: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 26: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 27: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 28: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 29: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 .

Page 30: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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!

Page 31: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster
Page 32: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The ProblemWe want to compute the persistence diagram of the Rips

filtration of a finite metric space but its too big.

Page 33: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The ProblemWe want to compute the persistence diagram of the Rips

filtration of a finite metric space but its too big.

The SolutionBuild a linear size zigzag filtration that has a provably

close persistence diagram.Use that instead.

Page 34: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The ProblemWe want to compute the persistence diagram of the Rips

filtration of a finite metric space but its too big.

The SolutionBuild a linear size zigzag filtration that has a provably

close persistence diagram.Use that instead.

Obvious?

Page 35: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The ProblemWe want to compute the persistence diagram of the Rips

filtration of a finite metric space but its too big.

The SolutionBuild a linear size zigzag filtration that has a provably

close persistence diagram.Use that instead.

Obvious?

Page 36: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Page 37: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Page 38: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Page 39: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Page 40: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Page 41: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Previous Approaches at subsampling:

Page 42: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]

Page 43: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]Persistence-based Reconstruction [CO08]

Page 44: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]Persistence-based Reconstruction [CO08]

Other methods:

Page 45: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]Persistence-based Reconstruction [CO08]

Other methods:Meshes in Euclidean Space [HMSO10]

Page 46: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Intuition (why zigzag?)

Previous Approaches at subsampling:Witness Complexes [dSC04, BGO07]Persistence-based Reconstruction [CO08]

Other methods:Meshes in Euclidean Space [HMSO10]Topological simplification [Z10, ALS11]

Page 47: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster
Page 48: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Two tricks:

Page 49: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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

Page 50: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Two tricks:

!! Q! "" Q" !! Q# ""

!! R̂! !! R̂" !! R̂# !!

!! !! !!

1 Embed the zigzag in a topologically equivalent filtration.

Zigzag filtration

Standard filtration

Page 51: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Two tricks:

2 Perturb the metric so the persistence module does not zigzag.

!! Q! "" Q" !! Q# ""

!! R̂! !! R̂" !! R̂# !!

!! !! !!

1 Embed the zigzag in a topologically equivalent filtration.

Zigzag filtration

Standard filtration

Page 52: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Two tricks:

2 Perturb the metric so the persistence module does not zigzag.

! H(Q!) " H(Q") ! H(Q#) "

!! Q! "" Q" !! Q# ""

!! R̂! !! R̂" !! R̂# !!

!! !! !!

1 Embed the zigzag in a topologically equivalent filtration.

Zigzag filtration

Standard filtration

Page 53: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

!=

!=

Two tricks:

2 Perturb the metric so the persistence module does not zigzag.

! H(Q!) " H(Q") ! H(Q#) "

!! Q! "" Q" !! Q# ""

!! R̂! !! R̂" !! R̂# !!

!! !! !!

1 Embed the zigzag in a topologically equivalent filtration.

Zigzag filtration

Standard filtration

Page 54: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

!=

!=

Two tricks:

2 Perturb the metric so the persistence module does not zigzag.

! H(Q!) ! H(Q") ! H(Q#) !

!! Q! "" Q" !! Q# ""

!! R̂! !! R̂" !! R̂# !!

!! !! !!

1 Embed the zigzag in a topologically equivalent filtration.

Zigzag filtration

Standard filtration

Page 55: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

!=

!=

Two tricks:

2 Perturb the metric so the persistence module does not zigzag.

! H(Q!) ! H(Q") ! H(Q#) !

!! Q! "" Q" !! Q# ""

!! R̂! !! R̂" !! R̂# !!

!! !! !!

1 Embed the zigzag in a topologically equivalent filtration.

Zigzag filtration

Standard filtration

At the homology level, there is no zigzag.

Page 56: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

The Result:

Page 57: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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.

Page 58: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 59: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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)

Doubling Constant: ! = maxmetric balls B

min

!

"

#

|S| : S ! P,B !$

q!S

ball(q,1

2radius(B)

%

&

'

Page 60: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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)

Doubling Constant: ! = maxmetric balls B

min

!

"

#

|S| : S ! P,B !$

q!S

ball(q,1

2radius(B)

%

&

'

Page 61: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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)

Doubling Constant: ! = maxmetric balls B

min

!

"

#

|S| : S ! P,B !$

q!S

ball(q,1

2radius(B)

%

&

'

Page 62: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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)

Doubling Constant: ! = maxmetric balls B

min

!

"

#

|S| : S ! P,B !$

q!S

ball(q,1

2radius(B)

%

&

'

Doubling Dimension: log2(!)

Page 63: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

How to perturb the metric (and why).

Page 64: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

How to perturb the metric (and why).Let tp be the time when point p is removed.

Page 65: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

How to perturb the metric (and why).

wp(!) =

!

"

#

0 if ! ! (1" ")tp!" (1" ")tp if (1" ")tp < ! < tp

"! if tp ! !

Let tp be the time when point p is removed.

Page 66: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

How to perturb the metric (and why).

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 67: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

How to perturb the metric (and why).

d̂!(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 68: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

How to perturb the metric (and why).

d̂!(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 69: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

How to perturb the metric (and why).

d̂!(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: ! ! R̂! " d̂!(p, q) # 2" for all p, q ! !

Page 70: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

How to perturb the metric (and why).

d̂!(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: ! ! R̂! " d̂!(p, q) # 2" for all p, q ! !

R !

1+"

! R̂! ! R!

Page 71: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

Page 72: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

Page 73: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

Page 74: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

One leaf per point of P.

Page 75: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

One leaf per point of P.Each node u has a representative rep(u) in P and a radius rad(p).

Page 76: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

One leaf per point of P.Each node u has a representative rep(u) in P and a radius rad(p).Three properties:

Page 77: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

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 78: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

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 79: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

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 80: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Net-trees

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))

Page 81: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Projection onto a net

Page 82: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Projection onto a net

N! = {p ! P : tp " !}

Page 83: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Projection onto a net

N! = {p ! P : tp " !}

Q! is the subcomplex of R̂! induced on N!.

Page 84: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Projection onto a net

N! = {p ! P : tp " !}

!!(p) = arg minq!N!

d̂(p, q)

Q! is the subcomplex of R̂! induced on N!.

Page 85: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

N! is a Delone set.

Projection onto a net

N! = {p ! P : tp " !}

!!(p) = arg minq!N!

d̂(p, q)

Q! is the subcomplex of R̂! induced on N!.

Page 86: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

N! is a Delone set.

Projection onto a net

N! = {p ! P : tp " !}

!!(p) = arg minq!N!

d̂(p, q)

For all p ! P , there is a q ! N!

such that d(p, q) " !(1# !)".1 Covering:

Q! is the subcomplex of R̂! induced on N!.

Page 87: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

N! is a Delone set.

Projection onto a net

N! = {p ! P : tp " !}

!!(p) = arg minq!N!

d̂(p, q)

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 R̂! induced on N!.

Page 88: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

N! is a Delone set.

Projection onto a net

N! = {p ! P : tp " !}

!!(p) = arg minq!N!

d̂(p, q)

For all p, q ! P , d̂(!!(p), q) " d̂(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 R̂! induced on N!.

Page 89: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Page 90: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Page 91: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

Page 92: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

A Simplicial Map is a map between vertex setsthat maps simplices to simplices.

Page 93: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

A Simplicial Map is a map between vertex setsthat maps simplices to simplices.

Two simplicial maps f, g : X ! Y are contiguousif for all simplices ! " X, f(!) # g(!) " Y .

Page 94: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

A Simplicial Map is a map between vertex setsthat maps simplices to simplices.

Two simplicial maps f, g : X ! Y are contiguousif for all simplices ! " X, f(!) # g(!) " Y .

Fact: If f, g : X ! Y are contiguous simplicial mapsthen the homomorphisms f!, g! : H(X) ! H(Y )are identical.

Page 95: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

A Simplicial Map is a map between vertex setsthat maps simplices to simplices.

Two simplicial maps f, g : X ! Y are contiguousif for all simplices ! " X, f(!) # g(!) " Y .

We will show that i ! !! is contiguous with theidentity map on R̂!.

Fact: If f, g : X ! Y are contiguous simplicial mapsthen the homomorphisms f!, g! : H(X) ! H(Y )are identical.

Page 96: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

Page 97: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

Claim: i ! !! is contiguous to the identity map on R̂!.

Page 98: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Claim: i ! !! is contiguous to the identity map on R̂!.

Page 99: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Claim: i ! !! is contiguous to the identity map on R̂!.

Page 100: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Page 101: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

Page 102: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

p

q

Page 103: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

Page 104: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

!!(q)

Page 105: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

!!(q)

Page 106: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

!!(q)

Page 107: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

!!(q)

Page 108: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

!!(q)

Page 109: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

!!(q)

Page 110: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

!!(q)

Page 111: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

!!(q)

Page 112: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

For all p, q ! P , d̂(!!(p), q) " d̂(p, q).Key Fact:

Let i : Q! !! R̂! be the canonical inclusion.

Goal: Show that i! : H(Q") ! H(R̂") is an isomorphism.

First we need to show that it’s a simplicial map, i.e. thatfor all ! ! R̂!, (i " "!)(!) ! R̂!.

Su!ces to show d̂!(p, q) ! 2! implies d̂("!(p),"!(q)) ! 2!.

Follows from fact above.

Claim: i ! !! is contiguous to the identity map on R̂!.

Second we, need to prove contiguity, i.e.{p, q} ! R̂! implies {p, q,!!(p),!!(q)} ! R̂!.

!!(p)

p

q

!!(q)

Page 113: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Why is the filtration only linear size?

Page 114: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Why is the filtration only linear size?

Standard trick from Euclidean geometry:

Page 115: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 116: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 117: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 118: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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.

2O(d2)

Page 119: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

Summary

Page 120: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

SummaryApproximate the VR filtration with a Zigzag filtration.

Page 121: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

SummaryApproximate the VR filtration with a Zigzag filtration.

Remove points using a hierarchical net-tree.

Page 122: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

SummaryApproximate the VR filtration with a Zigzag filtration.

Remove points using a hierarchical net-tree.

Perturb the metric to straighten out the zigzags.

Page 123: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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 124: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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.

The Future

Page 125: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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.

The FutureDistance to a measure?

Page 126: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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.

The FutureDistance to a measure?Other types of simplification.

Page 127: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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.

The FutureDistance to a measure?Other types of simplification.Construct the net-tree in O(n log n) time.

Page 128: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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.

The FutureDistance to a measure?Other types of simplification.Construct the net-tree in O(n log n) time.An implementation.

Page 129: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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.

The FutureDistance to a measure?Other types of simplification.Construct the net-tree in O(n log n) time.An implementation.

Thank You.

Page 130: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster

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.

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.

Page 131: Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at University of Muenster