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Vietoris–Rips Thickenings of the Circle
and Centrally–Symmetric Orbitopes
Advisor: Dr. Henry Adams
Committee: Dr. Amit Patel, Dr. Gloria Luong
Johnathan Bush
Master’s thesis defense – October 19th, 2018
Background
The Vietoris–Rips Complex
In[69]:= Demo[data1, 0, .41]
Out[69]=
Cech simplicialcomplex
Appearance
draw one simplices
draw Cech complex
draw Rips complex
Filtrationparameter
t 0
CechRips.nb 3
Definition
Let X be a metric space and r > 0 a scale parameter. The
Vietoris–Rips complex of X, denoted VR(X; r), has vertex set
X and a simplex for every finite subset σ ⊆ X such that
diam(σ) ≤ r.2
The Vietoris–Rips Complex
In[70]:= Demo[data1, 0, .41]
Out[70]=
Cech simplicialcomplex
Appearance
draw one simplices
draw Cech complex
draw Rips complex
Filtrationparameter
t 0.14
4 CechRips.nb
Definition
Let X be a metric space and r > 0 a scale parameter. The
Vietoris–Rips complex of X, denoted VR(X; r), has vertex set
X and a simplex for every finite subset σ ⊆ X such that
diam(σ) ≤ r.2
The Vietoris–Rips Complex
In[71]:= Demo[data1, 0, .41]
Out[71]=
Cech simplicialcomplex
Appearance
draw one simplices
draw Cech complex
draw Rips complex
Filtrationparameter
t 0.192
CechRips.nb 5
Definition
Let X be a metric space and r > 0 a scale parameter. The
Vietoris–Rips complex of X, denoted VR(X; r), has vertex set
X and a simplex for every finite subset σ ⊆ X such that
diam(σ) ≤ r.2
The Vietoris–Rips Complex
In[72]:= Demo[data1, 0, .41]
Out[72]=
Cech simplicialcomplex
Appearance
draw one simplices
draw Cech complex
draw Rips complex
Filtrationparameter
t 0.267
6 CechRips.nb
Definition
Let X be a metric space and r > 0 a scale parameter. The
Vietoris–Rips complex of X, denoted VR(X; r), has vertex set
X and a simplex for every finite subset σ ⊆ X such that
diam(σ) ≤ r.2
The Vietoris–Rips Complex
In[73]:= Demo[data1, 0, .41]
Out[73]=
Cech simplicialcomplex
Appearance
draw one simplices
draw Cech complex
draw Rips complex
Filtrationparameter
t 0.338
CechRips.nb 7
Definition
Let X be a metric space and r > 0 a scale parameter. The
Vietoris–Rips complex of X, denoted VR(X; r), has vertex set
X and a simplex for every finite subset σ ⊆ X such that
diam(σ) ≤ r.2
The Vietoris–Rips Complex
In[74]:= Demo[data1, 0, .41]
Out[74]=
Cech simplicialcomplex
Appearance
draw one simplices
draw Cech complex
draw Rips complex
Filtrationparameter
t 0.41
8 CechRips.nb
Definition
Let X be a metric space and r > 0 a scale parameter. The
Vietoris–Rips complex of X, denoted VR(X; r), has vertex set
X and a simplex for every finite subset σ ⊆ X such that
diam(σ) ≤ r.2
The Vietoris–Rips Complex
In applications of persistent homology, we consider
Vietoris–Rips complexes at all scale parameters.
In[71]:= Demo[data1, 0, .41]
Out[71]=
Cech simplicialcomplex
Appearance
draw one simplices
draw Cech complex
draw Rips complex
Filtrationparameter
t 0.192
CechRips.nb 5
In[74]:= Demo[data1, 0, .41]
Out[74]=
Cech simplicialcomplex
Appearance
draw one simplices
draw Cech complex
draw Rips complex
Filtrationparameter
t 0.41
8 CechRips.nb
3
Hausmann’s Theorem
Theorem
Let M be a compact Riemannian manifold and r > 0 be
sufficiently small. Then, VR(M ; r) 'M . [4]
• Downsides of the proof:
� Hausmann’s map VR(M ; r)→M depends upon a
total order of all points in M .
� VR(M ; r) does not inherit the metric of M . In
particular, the inclusion M ↪→ VR(M ; r) is not
continuous.
4
Hausmann’s Theorem
Theorem
Let M be a compact Riemannian manifold and r > 0 be
sufficiently small. Then, VR(M ; r) 'M . [4]
• Downsides of the proof:
� Hausmann’s map VR(M ; r)→M depends upon a
total order of all points in M .
� VR(M ; r) does not inherit the metric of M . In
particular, the inclusion M ↪→ VR(M ; r) is not
continuous.
4
Hausmann’s Theorem
Theorem
Let M be a compact Riemannian manifold and r > 0 be
sufficiently small. Then, VR(M ; r) 'M . [4]
• Downsides of the proof:
� Hausmann’s map VR(M ; r)→M depends upon a
total order of all points in M .
� VR(M ; r) does not inherit the metric of M . In
particular, the inclusion M ↪→ VR(M ; r) is not
continuous. 4
Latschev’s Theorem
Theorem
Let M be a closed Riemannian manifold and X a metric space
δ–close to M in the Gromov-Hausdorff distance. Then,
VR(X; r) 'M for r > 0 sufficiently small. [5]
• Generalization of Hausmann’s theorem. Applies, in
particular, to samplings X ⊆M .
• Again, the value of “sufficiently small” r > 0 depends on
the curvature of M .
• The value of δ depends on r.
• Same downsides as Hausmann’s proof.
5
Latschev’s Theorem
Theorem
Let M be a closed Riemannian manifold and X a metric space
δ–close to M in the Gromov-Hausdorff distance. Then,
VR(X; r) 'M for r > 0 sufficiently small. [5]
• Generalization of Hausmann’s theorem. Applies, in
particular, to samplings X ⊆M .
• Again, the value of “sufficiently small” r > 0 depends on
the curvature of M .
• The value of δ depends on r.
• Same downsides as Hausmann’s proof.
5
Latschev’s Theorem
Theorem
Let M be a closed Riemannian manifold and X a metric space
δ–close to M in the Gromov-Hausdorff distance. Then,
VR(X; r) 'M for r > 0 sufficiently small. [5]
• Generalization of Hausmann’s theorem. Applies, in
particular, to samplings X ⊆M .
• Again, the value of “sufficiently small” r > 0 depends on
the curvature of M .
• The value of δ depends on r.
• Same downsides as Hausmann’s proof.
5
Latschev’s Theorem
Theorem
Let M be a closed Riemannian manifold and X a metric space
δ–close to M in the Gromov-Hausdorff distance. Then,
VR(X; r) 'M for r > 0 sufficiently small. [5]
• Generalization of Hausmann’s theorem. Applies, in
particular, to samplings X ⊆M .
• Again, the value of “sufficiently small” r > 0 depends on
the curvature of M .
• The value of δ depends on r.
• Same downsides as Hausmann’s proof.
5
VR(X; r) for large scale parameters
Theorem (Adamszek, Adams)
Let S1 be the circle of unit circumference. Then,
VR(S1; r) '
S2`+1 if `2`+1 < r < `+1
2`+3 for ` ∈ N. [1]
0
S1 S3 S5 S7
13
25
3749
r =1
3 r =2
5r =
3
7
6
VR(X; r) for large scale parameters
Theorem (Adamszek, Adams)
Let S1 be the circle of unit circumference. Then,
VR(S1; r) '
S2`+1 if `2`+1 < r < `+1
2`+3∨∞ S2` if r = `2`+1
for ` ∈ N. [1]
0
S1 S3 S5 S7
13
25
3749
r =1
3 r =2
5r =
3
7
6
VR(X; r) for large scale parameters
Theorem (Adamszek, Adams)
Let S1 be the circle of unit circumference. Then,
VR(S1; r) '
S2`+1 if `2`+1 < r < `+1
2`+3∨∞ S2` if r = `2`+1
for ` ∈ N. [1]
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r =1
3 r =2
5r =
3
76
Metric Vietoris–Rips Thickenings
Definition (Adamaszek, Adams, Frick)
For a metric space X and r ≥ 0, the Vietoris–Rips thickeningVRm(X; r) is the set
VRm(X; r) =
{k∑
i=0
λixi | k ∈ N, xi ∈ X, and diam({x0, . . . , xk}) ≤ r
}
equipped with the 1-Wasserstein metric. [2]
1/3 1/3 1/61/6
1/31/3 1/3
7
Metric Vietoris–Rips Thickenings
• An analogue of Hausmann’s theorem holds for the
Vietoris–Rips metric thickening VRm(M ; r).
• Notably, this theorem admits a nicer proof:
� The homotopy equivalence VRm(M ; r)→M is
canonically defined.
� The inclusion M ↪→ VRm(M ; r) is continuous.
8
Results
Main Theorem
Theorem (Adams, B.)
Let S1 be the circle of unit circumference, and let r = 13 . Then,
the Vietoris–Rips metric thickening thickening VRm(S1; r) is
homotopy equivalent to S3.
• 1/3 is the side length of an inscribed equilateral triangle.
• Recall that VR(S1; 13) '
∨∞ S2
9
Main Theorem
Theorem (Adams, B.)
Let S1 be the circle of unit circumference, and let r = 13 . Then,
the Vietoris–Rips metric thickening thickening VRm(S1; r) is
homotopy equivalent to S3.
• 1/3 is the side length of an inscribed equilateral triangle.
• Recall that VR(S1; 13) '
∨∞ S2
9
Main Theorem (proof)
We construct a homotopy equivalence via
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
where SM4 : S1 → R4 is defined by
SM4(θ) = (cos(θ), sin(θ), cos(3θ), sin(3θ)) ,
p is radial projection, B4 = conv(SM4(S1)) ⊆ R4 is the (second)
Barvinok–Novik orbitope, and ι is inclusion.
10
Main Theorem (proof)
We construct a homotopy equivalence via
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
where SM4 : S1 → R4 is defined by
SM4(θ) = (cos(θ), sin(θ), cos(3θ), sin(3θ)) ,
p is radial projection, B4 = conv(SM4(S1)) ⊆ R4 is the (second)
Barvinok–Novik orbitope, and ι is inclusion.
10
Main Theorem (proof)
We construct a homotopy equivalence via
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
where SM4 : S1 → R4 is defined by
SM4(θ) = (cos(θ), sin(θ), cos(3θ), sin(3θ)) ,
p is radial projection,
B4 = conv(SM4(S1)) ⊆ R4 is the (second)
Barvinok–Novik orbitope, and ι is inclusion.
10
Main Theorem (proof)
We construct a homotopy equivalence via
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
where SM4 : S1 → R4 is defined by
SM4(θ) = (cos(θ), sin(θ), cos(3θ), sin(3θ)) ,
p is radial projection, B4 = conv(SM4(S1)) ⊆ R4 is the (second)
Barvinok–Novik orbitope,
and ι is inclusion.
10
Main Theorem (proof)
We construct a homotopy equivalence via
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
where SM4 : S1 → R4 is defined by
SM4(θ) = (cos(θ), sin(θ), cos(3θ), sin(3θ)) ,
p is radial projection, B4 = conv(SM4(S1)) ⊆ R4 is the (second)
Barvinok–Novik orbitope, and ι is inclusion.
10
Barvinok–Novik Orbitopes
Fix k ≥ 1 and define the symmetric moment curve
SM2k : R/2πZ→ R2k
θ 7→ (cos(θ), sin(θ), cos(3θ), sin(3θ), . . . , cos((2k − 1)θ), sin((2k − 1)θ)).
Define the k-th Barvinok–Novik orbitope by
B2k = conv(SM2k(S1)).
The precise facial struc-
ture of B2k is unknown
for k > 2.
Our proof technique in-
volves the faces of B4.
11
Barvinok–Novik Orbitopes
Fix k ≥ 1 and define the symmetric moment curve
SM2k : R/2πZ→ R2k
θ 7→ (cos(θ), sin(θ), cos(3θ), sin(3θ), . . . , cos((2k − 1)θ), sin((2k − 1)θ)).
Define the k-th Barvinok–Novik orbitope by
B2k = conv(SM2k(S1)).
The precise facial struc-
ture of B2k is unknown
for k > 2.
Our proof technique in-
volves the faces of B4.
11
Barvinok–Novik Orbitopes
Fix k ≥ 1 and define the symmetric moment curve
SM2k : R/2πZ→ R2k
θ 7→ (cos(θ), sin(θ), cos(3θ), sin(3θ), . . . , cos((2k − 1)θ), sin((2k − 1)θ)).
Define the k-th Barvinok–Novik orbitope by
B2k = conv(SM2k(S1)).
The precise facial struc-
ture of B2k is unknown
for k > 2.
Our proof technique in-
volves the faces of B4.11
Barvinok–Novik Orbitopes
Theorem (4.1 of [3])
The proper faces and subfaces of B4 are
• 0-dimensional faces, SM4(t) for t ∈ S1,
• 1-dimensional faces, conv(SM4({t1, t2})) where t1 6= t2 are
the edges of an arc of S1 of length less than or equal to 13 ,
• 2-dimensional faces, conv(SM4({t, t+ 13 , t+ 2
3})) for t ∈ S1.
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2 Torus.nb
By contrast, the simplicies of VRm(S1; 13) are
• Vertices, t ∈ S1,
• All simplices of the form conv({t1, . . . , tl}),where t1, . . . , tm belong to an arc of S1 of
length less than or equal to 13,
• 2-simplices, conv({t, t+ 13, t+ 2
3}) for t ∈ S1.
12
Barvinok–Novik Orbitopes
Theorem (4.1 of [3])
The proper faces and subfaces of B4 are
• 0-dimensional faces, SM4(t) for t ∈ S1,
• 1-dimensional faces, conv(SM4({t1, t2})) where t1 6= t2 are
the edges of an arc of S1 of length less than or equal to 13 ,
• 2-dimensional faces, conv(SM4({t, t+ 13 , t+ 2
3})) for t ∈ S1.
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��
��
��
��
��
2 Torus.nb
By contrast, the simplicies of VRm(S1; 13) are
• Vertices, t ∈ S1,
• All simplices of the form conv({t1, . . . , tl}),where t1, . . . , tm belong to an arc of S1 of
length less than or equal to 13,
• 2-simplices, conv({t, t+ 13, t+ 2
3}) for t ∈ S1.
12
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Steps:
(1) Prove ι is well-defined and continuous.
(2) Extend SM4 to VRm(S1; r).
(3) Prove SM4
(VRm(S1; r)
)misses the origin.
(4) Prove p ◦ SM4 and ι are homotopy inverses.
13
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Steps:
(1) Prove ι is well-defined and continuous.
(2) Extend SM4 to VRm(S1; r).
(3) Prove SM4
(VRm(S1; r)
)misses the origin.
(4) Prove p ◦ SM4 and ι are homotopy inverses.
13
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Steps:
(1) Prove ι is well-defined and continuous.
(2) Extend SM4 to VRm(S1; r).
(3) Prove SM4
(VRm(S1; r)
)misses the origin.
(4) Prove p ◦ SM4 and ι are homotopy inverses.
13
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Steps:
(1) Prove ι is well-defined and continuous.
(2) Extend SM4 to VRm(S1; r).
(3) Prove SM4
(VRm(S1; r)
)misses the origin.
(4) Prove p ◦ SM4 and ι are homotopy inverses.
13
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Steps:
(1) Prove ι is well-defined and continuous.
(2) Extend SM4 to VRm(S1; r).
(3) Prove SM4
(VRm(S1; r)
)misses the origin.
(4) Prove p ◦ SM4 and ι are homotopy inverses.
14
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Steps:
(1) Prove ι is well-defined and continuous.
(2) Extend SM4 to VRm(S1; r).
(3) Prove SM4
(VRm(S1; r)
)misses the origin.
(4) Prove p ◦ SM4 and ι are homotopy inverses.
15
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Steps:
(1) Prove ι is well-defined and continuous.
(2) Extend SM4 to VRm(S1; r).∑
i λiθi 7→∑
i λiSM2k(θi)
(3) Prove SM4
(VRm(S1; r)
)misses the origin.
(4) Prove p ◦ SM4 and ι are homotopy inverses.
15
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Steps:
(1) Prove ι is well-defined and continuous.
(2) Extend SM4 to VRm(S1; r).∑
i λiθi 7→∑
i λiSM2k(θi)
(3) Prove SM4
(VRm(S1; r)
)misses the origin.
(4) Prove p ◦ SM4 and ι are homotopy inverses.
16
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Steps:
(1) Prove ι is well-defined and continuous.
(2) Extend SM4 to VRm(S1; r).∑
i λiθi 7→∑
i λiSM2k(θi)
(3) Prove SM4
(VRm(S1; r)
)misses the origin.
(4) Prove p ◦ SM4 and ι are homotopy inverses.
17
Main Theorem (proof) – Prove p◦SM4 and ι are h. inverses
• (p ◦ SM4) ◦ ι = id∂B4 .
• To show ι ◦ (p ◦ SM4) ' idVRm(S1;r) we use a linear
homotopy H.
• H is well-defined only if ι ◦ (p ◦ SM4) does not “increase
diameter” too much.
18
Main Theorem (proof) – Prove p◦SM4 and ι are h. inverses
• (p ◦ SM4) ◦ ι = id∂B4 .
• To show ι ◦ (p ◦ SM4) ' idVRm(S1;r) we use a linear
homotopy H.
• H is well-defined only if ι ◦ (p ◦ SM4) does not “increase
diameter” too much.
18
Main Theorem (proof) – Prove p◦SM4 and ι are h. inverses
• In general, this “non-diameter increasing” property of p
depends on the facial structure of B2k.
• We use Farkas’ Lemma to exclude certain cases.
• Continuity of H follows from the fact that VRm(S1; r) is a
metric r-thickening.
19
Main Theorem (proof) – Prove p◦SM4 and ι are h. inverses
• In general, this “non-diameter increasing” property of p
depends on the facial structure of B2k.• We use Farkas’ Lemma to exclude certain cases.
• Continuity of H follows from the fact that VRm(S1; r) is a
metric r-thickening.
19
Main Theorem (proof) – Prove p◦SM4 and ι are h. inverses
• In general, this “non-diameter increasing” property of p
depends on the facial structure of B2k.• We use Farkas’ Lemma to exclude certain cases.
• Continuity of H follows from the fact that VRm(S1; r) is a
metric r-thickening.
19
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Thus, ι ◦ (p ◦SM4) ' idVRm(S1;r) and VRm(S1; 1/3) ' ∂B4 ∼= S3.
• Geometric proof of the homotopy type of a Vietoris–Rips
metric thickening.
• Conjecture: for k−12k−1 ≤ r <
k2k+1 ,
p2k ◦ SM2k : VRm≤ (S1; r)→ ∂B2k
is a homotopy equivalence.
20
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Thus, ι ◦ (p ◦SM4) ' idVRm(S1;r) and VRm(S1; 1/3) ' ∂B4 ∼= S3.
• Geometric proof of the homotopy type of a Vietoris–Rips
metric thickening.
• Conjecture: for k−12k−1 ≤ r <
k2k+1 ,
p2k ◦ SM2k : VRm≤ (S1; r)→ ∂B2k
is a homotopy equivalence.
20
Main Theorem (proof)
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
Thus, ι ◦ (p ◦SM4) ' idVRm(S1;r) and VRm(S1; 1/3) ' ∂B4 ∼= S3.
• Geometric proof of the homotopy type of a Vietoris–Rips
metric thickening.
• Conjecture: for k−12k−1 ≤ r <
k2k+1 ,
p2k ◦ SM2k : VRm≤ (S1; r)→ ∂B2k
is a homotopy equivalence.
20
References
[1] M. Adamaszek and H. Adams, The vietoris–rips complexes of a circle, Pacific Journal of
Mathematics, 290 (2017), pp. 1–40.
[2] M. Adamaszek, H. Adams, and F. Frick, Metric reconstruction via optimal transport, To appear
in the SIAM Journal on Applied Algebra and Geometry, (2018).
[3] A. Barvinok and I. Novik, A centrally symmetric version of the cyclic polytope, Discrete &
Computational Geometry, 39 (2008), pp. 76–99.
[4] J.-C. Hausmann, On the vietoris-rips complexes and a cohomology theory for metric spaces,
Ann. Math. Studies, 138 (1995), pp. 175–188.
[5] J. Latschev, Vietoris–Rips complexes of metric spaces near a closed Riemannian manifold,
Archiv der Mathematik, 77 (2001), pp. 522–528.
Thank you!
21
