Henry Adams (Colorado State University), Micha l ...adams/talks/Metric...Metric reconstruction via...

Metric reconstruction via optimal transport

Henry Adams (Colorado State University), Micha l Adamaszek(MOSEK ApS), Florian Frick (Carnegie Mellon)

Thanks to my graduate students: Johnathan Bush, BrittanyCarr, Mark Heim, Lara Kassab, Joshua Mirth, Alex Williams

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

For X a metric space and scale r > 0, theVietoris–Rips simplicial complex VR(X ; r) has

vertex set X

simplex {x0, . . . , xk} when diam({x0, . . . , xk}) ≤ r .

In[70]:= Demo[data1, 0, .41]

Out[70]=


Appearance

draw one simplices

draw Cech complex

draw Rips complex

Filtrationparameter

t 0.14

4 CechRips.nb

Definition


vertex set X


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Out[71]=


Appearance

draw one simplices

draw Cech complex

draw Rips complex

Filtrationparameter

t 0.192

CechRips.nb 5

Definition


vertex set X


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Out[72]=


Appearance

draw one simplices

draw Cech complex

draw Rips complex

Filtrationparameter

t 0.267

6 CechRips.nb

Definition


vertex set X


In[73]:= Demo[data1, 0, .41]

Out[73]=


Appearance

draw one simplices

draw Cech complex

draw Rips complex

Filtrationparameter

t 0.338

CechRips.nb 7

Definition


vertex set X


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Out[74]=


Appearance

draw one simplices

draw Cech complex

draw Rips complex

Filtrationparameter

t 0.41

8 CechRips.nb

Definition


vertex set X


Definition


vertex set X


Theorem (Hausmann, 1995)

For M a compact Riemannian manifold and r small,VR(M; r) ' M.



Theorem (Latschev, 2001)

For r small and dGH(X ,M) small (depending on r),

VR(X ; r) ' M.



Theorem (Latschev, 2001)

For r small and dGH(X ,M) small (depending on r),

VR(X ; r) ' M.

Theorem (Chazal, de Silva, Oudot, 2013)

For X and Y totally bounded metric spaces,

d(PH(VR(X ;−)),PH(VR(Y ;−))) ≤ 2dGH(X ,Y ).



Downsides of the proof:

map VR(M; r)→ M depends on the choice of a totalordering of all points in M, and

the inclusion M ↪→ VR(M; r) is not continuous.

Let S1 be the circle of unit circumference.

Theorem (Adamaszek, A)

VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3 for some k ∈ N.

0

S1 S3 S5

S7

13

25

3749

r =1

3 r =2

5r =

3

7



VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3∨∞ S2k if r = k2k+1

for some k ∈ N.

0

S1 S3 S5

S7

13

25

3749

r =1

3 r =2

5r =

3

7



VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3∨∞ S2k if r = k2k+1

for some k ∈ N.

Intuition

VR(6 points, 13) ' S2 VR(9 points, 13) '∨2 S2



VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3∨∞ S2k if r = k2k+1

for some k ∈ N.

Intuition

��

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��

Torus.nb 7



VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3∨∞ S2k if r = k2k+1

for some k ∈ N.

Intuition

��

��

��

��

��

��

6 Torus.nb



VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3∨∞ S2k if r = k2k+1

for some k ∈ N.

Intuition

��

��

��

��

��

��

Torus.nb 5



VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3∨∞ S2k if r = k2k+1

for some k ∈ N.

Intuition

��

��

��

��

��

��

4 Torus.nb



VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3∨∞ S2k if r = k2k+1

for some k ∈ N.

Intuition

��

��

��

��

��

��

Torus.nb 3



VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3∨∞ S2k if r = k2k+1

for some k ∈ N.

Intuition

��

��

��

��

��

��

2 Torus.nb



VR(S1; r) '{S2k+1 if k2k+1 < r <

k+12k+3∨∞ S2k if r = k2k+1

for some k ∈ N.

0

S1 S3 S5

S7

13

25

3749

r =1

3 r =2

5r =

3

7

Čech/nerve version is analogous! By contrast, VRm(S1; r) ' S3.

The Vietoris–Rips simplicial complex may not be metrizable!

Metric space M X ⊆ M VR(X ; r)

Definition

For X a metric space and r > 0, the Vietoris–Rips metricthickening is

VRm(X ; r) ={

k∑

i=0

λixi

∣∣∣∣∣ λi ≥ 0,∑

i

λi = 1, xi ∈ X , diam({x0, . . . , xk}) ≤ r},

equipped with the 1-Wasserstein metric.

1/3 1/3 1/61/6

1/31/3 1/3

Note VRm(X ; r) is a metric r -thickening of X .

Definition

For X a metric space and r > 0, the Vietoris–Rips metricthickening is

VRm(X ; r) ={

k∑

i=0

λiδxi

∣∣∣∣∣ λi ≥ 0,∑

i

λi = 1, xi ∈ X , diam({x0, . . . , xk}) ≤ r},

equipped with the 1-Wasserstein metric.

1/3 1/3 1/61/6

1/31/3 1/3

Note VRm(X ; r) is a metric r -thickening of X .

Theorem (Adamaszek, A, Frick)

If M is a Riemannian manifold and r is sufficiently small, then

VRm(M; r) ' M.

1/3 1/61/6

1/3

Proof.

VRm(M; r)

f

))M

ιkk

• f : ∑i λiδxi 7→ Fréchet mean.• f ◦ ι = idM , and ι ◦ f ' idVRm(M;r) via a linear homotopy.

We can now say something about the n-sphere Sn

Theorem (Adamaszek, A, Frick)

VRm(Sn; r) '{Sn r < rc

Σn+1 SO(n+1)An+2 r = rc .

SO(n + 1) = group of orientation-preserving isometries of Rn+1.An+2 = group of orientation-preserving symmetries of ∆

n+1.rc = diameter of inscribed regular (n + 1)-simplex.

Theorem (Unpublished)

Čm

(Sn; r) '{Sn r < 14Σn+1 RPn r = 14 .

Open questions

1 VRm(Sn; r) and Čm

(Sn; r) for larger r? Lovász’ stronglyself-dual polytopes

2 Other manifolds M?

3 VR

References

Micha l Adamaszek, Henry Adams, Florian Frick, Metricreconstruction via optimal transport, SIAM Journal onApplied Algebra and Geometry 2 (2018).

Micha l Adamaszek and Henry Adams, The Vietoris–Ripscomplexes of a circle, Pacific Journal of Mathematics 290(2017), 1–40.

Henry Adams, Johnathan Bush, and Florian Frick, Metricthickenings, Borsuk–Ulam theorems, and orbitopes (2019), inpreparation.

Jean-Claude Hausmann, On the Vietoris–Rips complexes anda cohomology theory for metric spaces, Annals ofMathematics Studies 138 (1995), 175–188.

Thank you!

Theorem (Hausman, 1995)

Let M be a Riemannian manifold with r(M) > 0.If 0 < r ≤ r(M), then VR(M; r) ' M.

Definition

Let r(M) be the largest satisfying:(a) If d(x , y) < 2r(M), then ∃! shortest geodesic between x and y .

u

x

y 0 if M has positive injectivity radius and boundedsectional curvature (in particular if M compact).

We can now say something about VRm(Sn; r) and Čm

(Sn,Sn; r)

VRm(S1; 13) ' VRm(S1; 13 − �) ∪ (D2 × S1)' (S1 × D2) ∪S1×S1 (D2 × S1)= S1 ∗ S1

= S3




(Sn,Sn; r)


= S3


= S3




(Sn,Sn; r)


= S3

VRm(S1; 13) ' VRm(S1; 13 − �) ∪ (C (S1)× S1)' (S1 × C (S1)) ∪S1×S1 (C (S1)× S1)= S1 ∗ S1

= S3




(Sn,Sn; r)


= S3

VRm(S1; 13) ' VRm(S1; 13 − �) ∪ (C (S1)× S1)' (S1 × C (S1)) ∪S1×S1 (C (S1)× S1)= S1 ∗ S1

= Σ2 S1




(Sn,Sn; r)


= S3

VRm(S1; 13) ' VRm(S1; 13 − �) ∪ (C (S1)×SO(2)A3

)

' (S1 × C (SO(2)A3 )) ∪ (C (S1)× SO(2)A3 )

= S1 ∗ SO(2)A3= Σ2 SO(2)A3




(Sn,Sn; r)


= S3

VRm(S1; rc) ' VRm(S1; rc − �) ∪ (C (S1)× SO(2)A3 )' (S1 × C (SO(2)A3 )) ∪ (C (S

1)× SO(2)A3 )= S1 ∗ SO(2)A3= Σ2 SO(2)A3




(Sn,Sn; r)


= S3


1)× SO(2)A3 )= S1 ∗ SO(2)A3= Σ2 SO(2)A3


(Sn,Sn; r)


= S3

VRm(Sn; rc) ' VRm(Sn; rc − �) ∪ (C (Sn)× SO(n+1)An+2 )' (Sn × C (SO(n+1)An+2 )) ∪ (C (S

n)× SO(n+1)An+2 )= Sn ∗ SO(n+1)An+2= Σn+1 SO(n+1)An+2




(Sn,Sn; r)


= S3

Čm

(Sn,Sn; 14) ' Čm

(Sn, Sn; 14 − �) ∪ (C (Sn)× RPn)' (Sn × C (RPn)) ∪ (C (Sn)× RPn)= Sn ∗ RPn

= Σn+1 RPn



Metric thickenings, Borsuk–Ulam theorems, andorbitopes

Henry Adams (Colorado State University)Johnathan Bush (Colorado State University)

Florian Frick (Carnegie Mellon)

Paper in preparation

Theorem (Borsuk–Ulam)

For f : Sn → Rn, there exists a point x ∈ Sn with f (x) = f (−x).

22 2. The Borsuk–Ulam Theorem

more demanding.) Next, in Section 2.4, we prove Tucker’s lemma differently,introducing some of the most elementary notions of simplicial homology.

As for (3), we will examine various generalizations and strengtheningslater; much more can be found in Steinlein’s surveys [Ste85], [Ste93] and inthe sources he quotes.

Finally, as for applications (4), just wait and see.

2.1 The Borsuk–Ulam Theorem in Various Guises

One of the versions of the Borsuk–Ulam theorem, the one that is perhaps theeasiest to remember, states that for every continuous mapping f :Sn → Rn,there exists a point x ∈ Sn such that f(x) = f(−x). Here is an illustrationfor n = 2. Take a rubber ball, deflate and crumple it, and lay it flat:

Then there are two points on the surface of the ball that were diametricallyopposite (antipodal) and now are lying on top of one another!

Another popular interpretation, found in almost every textbook, says thatat any given time there are two antipodal places on Earth that have the sametemperature and, at the same time, identical air pressure (here n = 2).2

It is instructive to compare this with the Brouwer fixed point theorem,which says that every continuous mapping f :Bn → Bn has a fixed point:f(x) = x for some x ∈ Bn. The statement of the Borsuk–Ulam theoremsounds similar (and actually, it easily implies the Brouwer theorem; see be-low). But it involves an extra ingredient besides the topology of the considered

2 Although anyone who has ever touched a griddle-hot stove knows that the tem-perature need not be continuous.











Figure credit: Jǐŕı Matoušek, Using the Borsuk–Ulam theorem



Theorem (Gromov’s “waist” theorem)

For f : Sn → Rn, there exists some y ∈ Rn withvol0(f

−1(y)) ≥ vol0(S0 ⊆ Sn).















For f : Sn → Rk with k ≤ n, there exists some y ∈ Rn withvoln−k(f −1(y)) ≥ voln−k(Sn−k ⊆ Sn).
















Izydorek–Rybicki [IR92], Mramor-Kosta [MK95], Volovikov [Vol96], Koikara–Mukerjee [KM96],Živaljević [Živ99], de Mattos–dos Santos [dMdS07], Crabb–Jaworowski [CJ09], Schick–Simon–Spiecż–Toruńczyk [SSST11], Blagojević–M.–Ziegler [BMZ11] and [Mat11], and Singh [Sin11].

Main application. Gromov proved in [Gro03] the following version of the Borsuk–Ulamtheorem.

Theorem 1.2 (Gromov’s waist of the sphere theorem). Let f : Sn ! Rk be a continuousmap where n � k � 0.Then there exists a point z 2 Rk such that for any " > 0,

voln

⇣U"

�f�1(z)

�⌘� voln

⇣U"

�Sn�k

�⌘.

Here, voln denotes the standard measure on Sn, U"(X) denotes the "-neighborhood of a set

X ✓ Sn with respect to the standard metric on Sn, and Sn�k is the (n � k)-dimensionalequator of Sn.

Memarian [Mem09] gave a more detailed proof of Gromov’s theorem. Karasev andVolovikov [KV11] generalized it to maps f : Sn ! M of even degree from the n-sphereto arbitrary k-manifolds M , see figure 1.

z

Sn

Mk

f

f�1(z)

Figure 1: Example of the Gromov–Memarian–Karasev–Volovikov theorem for n = 2 andM = S1. In this example, f�1(z) is not a large preimage.

The main application of this paper is a parametrized version of this Gromov–Memarian–Karasev–Volovikov theorem.

Theorem 1.3 (Parametrized Gromov–Memarian–Karasev–Volovikov waist of the sphere the-orem). Let f : B ⇥ Sn ! E be a bundle map over B, where Sn ,! B ⇥ Sn ! B is the trivialSn bundle over B and M ,! E p�! B is a fiber bundle over B whose fiber is a k-manifold M .If n = k then we further assume that the fiber maps fb : S

n ! M have even degree at everybase point b 2 B.Then there exist a subset Z ✓ E such that for all z 2 Z and " > 0,

voln

⇣U"

�f�1(z)

�⌘� voln

⇣U"

�Sn�k

�⌘, (1)

and such that Z is the set of limit points of a sequence of subsets Zi ✓ E with

(pE |Z)⇤ : H⇤(B; F2) ! H⇤(Zi; F2)

being injective. Here, voln is the standard measure on the fiber Sn over pE(z), U"(.) denotes

the "-neighborhood in that fiber, and H⇤ denotes Čech cohomology.

2

Figure credit: Benjamin Matschke, Journal of Topology & Analysis





What about f : Sn → Rk with k ≥ n?













Theorem (A, Bush, Frick)

For f : S1 → R2k+1, there exists a set {x0, . . . , x2k+1} of diameterat most k2k+1 such that

∑λi f (xi ) =

∑λi f (−xi ).





∑λi f (xi ) =

∑λi f (−xi ).

Proof: S2k+1 ' VRm(S1; k2k+1)f−→ R2k+1

Sharpness: f = (cos θ, sin θ, cos 3θ, sin 3θ, cos 5θ, sin 5θ, . . .)





∑λi f (xi ) =

∑λi f (−xi ).


0

S1 S3 S5

S7

13

25

3749

r =1

3 r =2

5r =

3

7






∑λi f (xi ) =

∑λi f (−xi ).


��

��

��

��

��

��

Torus.nb 7






∑λi f (xi ) =

∑λi f (−xi ).


��

��

��

��

��

��

6 Torus.nb






∑λi f (xi ) =

∑λi f (−xi ).


��

��

��

��

��

��

Torus.nb 5






∑λi f (xi ) =

∑λi f (−xi ).


��

��

��

��

��

��

4 Torus.nb






∑λi f (xi ) =

∑λi f (−xi ).


��

��

��

��

��

��

Torus.nb 3






∑λi f (xi ) =

∑λi f (−xi ).


��

��

��

��

��

��

2 Torus.nb





For f : Sn → Rn+2, there exists a set {x0, . . . , xn+2} of diameterat most rn such that

∑λi f (xi ) =

∑λi f (−xi ).

Proof: Sn+2 “ ⊆ ” VRm(Sn; rn) f−→ Rn+2

rn is the side-length of an inscribed simplex


For f : Sn → Rn+2, there exists a point ∑λixi of diameter at mostthat of an inscribed simplex such that f (

∑λixi ) = f (

∑λi (−xi )).

Henry Adams (Colorado State University), Micha l ...adams/talks/Metric...Metric reconstruction via...

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