The many paths to discrete curvaturemath.sjtu.edu.cn/conference/9shcc/ppt/Prasad Tetali.pdf · The many paths to discrete curvature Prasad Tetali Georgia Institute of Technology Shanghai

The many paths to discrete curvature

Prasad Tetali

Georgia Institute of Technology

Shanghai Conference on Combinatorics

Prasad Tetali The many paths to discrete curvature

Collaboration Acknowledgment

N. Gozlan, C. Roberto, P-M. Samson, P. Tetali, “Dispalcement Convexity inGraphs and Related Inequalities,” Probab. Th. Rel. Fields, 2013.

M. Erbar, J. Maas, P. Tetali, “Discrete Ricci Curvature bounds forBernoulli-Laplace and the Random Transposition Models,” Annales Fac.Sci. Toulouse, 2015.

R. Che, W. Huang, Y. Li, P. Tetali, “Convergence to Global Equilibrium inFokker-Planck Equations on a Graph and Talagrand-type inequalities,” J.Diff. Equations, 2016.

B. Klartag, G. Kozma, P. Ralli, P. Tetali, “Discrete Curvature and AbelianGroups,” Canadian J. Math., 2016.

N. Gozlan, C. Roberto, P-M. Samson, P. Tetali, “Kantorovich Duality forgeneral Transport Costs and Applications,” J. Functional Analysis, accepted.

M. Erbar, C. Henderson, G. Menz, P. Tetali, “Ricci Curvature bounds forWeakly Interacting Markov Chains,” Electronic J. Probab., 2017.


























Classical Brunn-Minkowski

Theorem

For all bounded Borel measurable sets A ,B in Rn, the following are equivalent:1 For A + B = {a + b : a 2 A , b 2 B} is the Minkowski sum of A and B:

voln(A + B)1/n � voln(A)1/n + voln(B)1/n

2 In the multiplicative form, for every t 2 [0, 1],

voln((1 � t)A + tB) � voln(A)1�t voln(B)t .

In other words, for A ,B ⇢ Rn, the t-midpoint set Mt(A ,B), for 0 t 1,satisfies:

log |Mt | � (1 � t) log |A |+ t log |B | ,where Mt := {(1 � t)a + tb : a 2 A , b 2 B} , and | · | is the Leb. meas.


Strengthened Brunn-Minkowski under positive Ricci

Theorem (Cordero-Erasquin, McCann, Schmuckenschlager, ...) 2006,...

If Rn replaced by a smooth Riemannian manifold X with Ricci curvature bdd.below by K > 0, then:

log |Mt | � (1 � t) log |A |+ t log |B |+ K2

t(1 � t)d(A ,B)2 ,

with d: Haussdorff distance, Mt := {�(t) : � a geodesic s.t. �(0) 2 A , �(1) 2 B} .

“When the space is positively curved, there exist many more midpointsbetween sets of points.”


Discrete Brunn-Minkowski type Inequality?

Definition. Graph G = (V ,E); for A ,B ⇢ V , for 0 t 1, letMt(A ,B) be the t-midpoint set of A and B. i.e., for a 2 A , b 2 B,

Mt(a, b) := {z 2 V(G) : d(a, z) ⇡ td(a, b), d(z, b) ⇡ (1 � t)d(a, b)} ,

E.g.,M1/2(a, b) := {m : d(a,m)+d(m, b) = d(a, b); d(a,m) = d(m, b)+ {�1, 0,+1}}and

Mt(A ,B) = [a2A ,b2B Mt(a, b) .

Ollivier-Villani (2012): Brunn-Minkowski type inequality on the discretehypercube ⌦n = {0, 1}n equipped with the Ham. distance d(x, y) =

Pni=1 1Ixi,yi .

|M1/2(A ,B)| � |A |1/2|B |1/2e1

16n d2(A ,B), 8A ,B ⇢ ⌦n

Above is a consequence of the following: 8⌫0, ⌫1, 9⌫1/2 such that

H(⌫1/2|µ) 12

H(⌫0|µ) +12

H(⌫1|µ) �1

16nW2

1 (⌫0, ⌫1),

where µ is the uniform measure on ⌦n.






Mt(A ,B) = [a2A ,b2B Mt(a, b) .


Pni=1 1Ixi,yi .

|M1/2(A ,B)| � |A |1/2|B |1/2e1

16n d2(A ,B), 8A ,B ⇢ ⌦n


H(⌫1/2|µ) 12

H(⌫0|µ) +12

H(⌫1|µ) �1

16nW2

1 (⌫0, ⌫1),







Mt(A ,B) = [a2A ,b2B Mt(a, b) .


Pni=1 1Ixi,yi .

|M1/2(A ,B)| � |A |1/2|B |1/2e1

16n d2(A ,B), 8A ,B ⇢ ⌦n


H(⌫1/2|µ) 12

H(⌫0|µ) +12

H(⌫1|µ) �1

16nW2

1 (⌫0, ⌫1),



Kantorovich metric and Relative EntropyDefinitions. (⌦, d, µ): metric space w/ probab. measure µ.Let ⌫ be another probability measure on ⌦. Then

1.W1(⌫, µ) := inf

�!(⌫,µ)

X

x,y2⌦d(x, y)�(x, y) ,

where � is a coupling (joint distribution) of ⌫ and µ:X

y

�(x, y) = µ(x), 8x, andX

x

�(x, y) = ⌫(y), 8y .

* Kantorovich, Wasserstein, KROV distance, Earth mover distance, ...

* If d 2 0, 1, then W1(⌫, µ) = k⌫ � µktv := supA⇢⌦⇣⌫(A) � µ(A)

⌘.

2.

H(⌫|µ) :=X

x2⌦⌫(x) log

⌫(x)µ(x)

.

* Kullback-Liebler, ...



1.W1(⌫, µ) := inf

�!(⌫,µ)

X

x,y2⌦d(x, y)�(x, y) ,


y

�(x, y) = µ(x), 8x, andX

x

�(x, y) = ⌫(y), 8y .



⌘.

2.

H(⌫|µ) :=X

x2⌦⌫(x) log

⌫(x)µ(x)

.




1.W1(⌫, µ) := inf

�!(⌫,µ)

X

x,y2⌦d(x, y)�(x, y) ,


y

�(x, y) = µ(x), 8x, andX

x

�(x, y) = ⌫(y), 8y .



⌘.

2.

H(⌫|µ) :=X

x2⌦⌫(x) log

⌫(x)µ(x)

.




1.W1(⌫, µ) := inf

�!(⌫,µ)

X

x,y2⌦d(x, y)�(x, y) ,


y

�(x, y) = µ(x), 8x, andX

x

�(x, y) = ⌫(y), 8y .



⌘.

2.

H(⌫|µ) :=X

x2⌦⌫(x) log

⌫(x)µ(x)

.




1.W1(⌫, µ) := inf

�!(⌫,µ)

X

x,y2⌦d(x, y)�(x, y) ,


y

�(x, y) = µ(x), 8x, andX

x

�(x, y) = ⌫(y), 8y .



⌘.

2.

H(⌫|µ) :=X

x2⌦⌫(x) log

⌫(x)µ(x)

.



Specific Open Questions

Definition. Graph G = (V ,E); for A ,B ⇢ V , for 0 t 1, letMt(A ,B) be the t-midpoint set of A and B.

1. G = Qn, the discrete n-cube, for n � 1. Then, is there a c � 0:

|Mt(A ,B)| � |A |1�t |B |t e(c/n) t(1�t) d2(A ,B) ??

2. G = Sn, the symmetric group with the Hamming (or, say, thetransposition) distance defining adjacency. Then, is there a c � 0:


* “Curvature” term ...


















Ollivier-Villani

“ In positive curvature, balls are closer than their centers.”

Coarse Ricci: Take two small balls and compute the transportationdistance between them. If it is smaller than the distance between thecenters of the balls, then coarse Ricci is positive.

W1(µx , µy) =: (1 � (x, y))d(x, y) ,Examples.1. n-cube: µx uniform on the n+1 neighbors of x (including itself). Forx, y, neighbors, (x, y) = 2/(n + 1).

2. Sn with transpositions: For �, ⌧ differing in a transposition,(x, y) = 1/(n

2).


(Coarse) Ricci of Hypercube

Proposition (folklore)

The n-cube has coarse Ricci = 2/(n + 1).

Proof sketch.(i) Consider the lazy random walk on the n-cube.

(ii) Use a simple (path) coupling argument to show that two copies of thechain started at neighboring vertices x, y 2 {0, 1}n can be “coupled” withprobability at least 2/(n + 1) in one step.


(Coarse) Ricci of Transposition Graph

Proposition

Sn with transpositions has coarse Ricci = 1/(n2).

Proof sketch.Lower bound:(i) Consider the lazy random transposition chain on Sn.(ii) Use a simple (path) coupling argument to show that two copies of thechain started at �, ⌧ 2 Sn with d(�, ⌧) = 1, can be coupled withprobability at least 1/(n

2) in one step.

Upper bound: Follows by showing that there is no better coupling – use(Kantorovich’s) dual formulation of W1:

W1(⌫, µ) = supf : 1�Lip

✓E⌫f � Eµf

◆.


A general problem

(X, d) a complete separable metric space.

Definition [Relative entropy]

Let µ, ⌫ be two Borel probability measures on X

H(⌫|µ) :=Z

logd⌫dµ

d⌫

if ⌫ is absolutely continuous with respect to µ (otherwise we set H(⌫|µ) = 1).

The map ⌫ 7! H(⌫|µ) is always convex in the usual sense:

H((1 � t)⌫0 + t⌫1|µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ), 8t 2 [0, 1].

Problem: Examine the convexity of H along other types of paths (⌫t)t2[0,1]interpolating between ⌫0 and ⌫1.

Convexity along geodesics of the Wasserstein W2 distance hasconnections with curvature of the underlying space and functional inequalities(Log-Sobolev, Talagrand) or Brunn-Minkowski type inequalities.


A general problem




H(⌫|µ) :=Z

logd⌫dµ

d⌫



H((1 � t)⌫0 + t⌫1|µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ), 8t 2 [0, 1].




A general problem




H(⌫|µ) :=Z

logd⌫dµ

d⌫



H((1 � t)⌫0 + t⌫1|µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ), 8t 2 [0, 1].




A general problem




H(⌫|µ) :=Z

logd⌫dµ

d⌫



H((1 � t)⌫0 + t⌫1|µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ), 8t 2 [0, 1].




A general problem




H(⌫|µ) :=Z

logd⌫dµ

d⌫



H((1 � t)⌫0 + t⌫1|µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ), 8t 2 [0, 1].




I. Displacement convexity of theentropy in a continuous setting.


Geodesic spaces

A metric space (E, d) is said geodesic if for all x0, x1 2 E there is at least one path� : [0, 1]! E such that �(0) = x0 and �(1) = x1 and

d(�(s), �(t)) = |t � s|d(x0, x1), 8t , s 2 [0, 1].

Such a path is called a constant speed geodesic between x0 and x1.


Geodesics in the Wasserstein space

Let Pp(X), p � 1, be the set of Borel probability measures having a finite p-thmoment.

Definition [Lp-Wasserstein distance]

Let ⌫0, ⌫1 2 Pp(X);

Wpp (⌫0, ⌫1) = inf

⇡2P(⌫0 ,⌫1)

"

dp(x0, x1) ⇡(dx0dx1),

where P(⌫0, ⌫1) is the set of couplings of ⌫0 and ⌫1.

Theorem

If p > 1, the space (Pp(X),Wp) is geodesic if and only if (X, d) is geodesic.


Geodesics in the Wasserstein space

Let Pp(X), p � 1, be the set of Borel probability measures having a finite p-thmoment.

Definition [Lp-Wasserstein distance]

Let ⌫0, ⌫1 2 Pp(X);

Wpp (⌫0, ⌫1) = inf

⇡2P(⌫0 ,⌫1)

"

dp(x0, x1) ⇡(dx0dx1),

where P(⌫0, ⌫1) is the set of couplings of ⌫0 and ⌫1.

Theorem

If p > 1, the space (Pp(X),Wp) is geodesic if and only if (X, d) is geodesic.


Displacement Convexity of Entropy

Theorem

Let (M, g) be a complete connected Riemannian manifold and suppose thatµ 2 P(M) is absolutely continuous with µ(dx) = e�V(x) dx. The following areequivalent:

1 µ verifies the CD(K ,1) condition, for some K 2 R:

Ric + Hess V � Kg

2 The relative entropy functional is K -displacement convex with respect to theW2 metric: for all ⌫0, ⌫1 2 P2(M), absolutely continuous with respect to µ,there is a W2-geodesic (⌫t)t2[0,1] connecting ⌫0 to ⌫1 such that

H(⌫t |µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ) � Kt(1 � t)

2W2

2 (⌫0, ⌫1), 8t 2 [0, 1]

McCann, Cordero-McCann-Schmuckenschlager, Sturm-Von Renesse,Lott-Villani, Sturm.


Implications: Bakry-Emery etc.

1. This equivalence sets the ground for a possible definition of the conditionCD(K ,1) on geodesic spaces (Lott-Villani; Sturm).

2. Brunn-Minkowski type inequalities. Suppose K � 0, then

µ([A ,B]t) � µ(A)1�tµ(B)t eKt(1�t)

2 d2(A ,B), 8t 2 [0, 1],

where [A ,B]t = {x 2 M;9(a, b) 2 A ⇥ B , d(x, a) =(1 � t)d(a, b) and d(x, b) = td(a, b)}.

3. Talagrand’s inequality. Suppose K > 0, then

W22 (⌫0, µ)

2K

H(⌫0|µ), 8⌫0 2 P2(M).

4. HWI inequality. Otto-Villani, 2000.


Implications: Bakry-Emery etc.

1. This equivalence sets the ground for a possible definition of the conditionCD(K ,1) on geodesic spaces (Lott-Villani; Sturm).

2. Brunn-Minkowski type inequalities. Suppose K � 0, then

µ([A ,B]t) � µ(A)1�tµ(B)t eKt(1�t)

2 d2(A ,B), 8t 2 [0, 1],

where [A ,B]t = {x 2 M;9(a, b) 2 A ⇥ B , d(x, a) =(1 � t)d(a, b) and d(x, b) = td(a, b)}.

3. Talagrand’s inequality. Suppose K > 0, then

W22 (⌫0, µ)

2K

H(⌫0|µ), 8⌫0 2 P2(M).

4. HWI inequality. Otto-Villani, 2000.


5. Log-Sobolev inequality. Suppose K > 0, then

H(⌫0|µ) 2K

I(⌫0|µ), 8⌫0.

6. Prekopa-Leindler type inequalities. Suppose K 2 R, and fix t 2 (0, 1).If f , g, h : M ! R are such that

h(z) � (1 � t)f(x) + tg(y) � Kt(1 � t)2

d2(x, y), 8x, y 2 M, z 2 [x, y]t

then Zeh(z) µ(dz) �

Zef(x) µ(dx)

!1�t Zeg(y) µ(dy)

!t


II. Displacement convexity of theentropy in a discrete setting.


Extension to discrete settings

Question: Is it possible to extend this theory to discrete settings, for examplefinite graphs ?

Two obstructions:

Talagrand’s inequality

W22 (⌫0, µ) Const. H(⌫0|µ), 8⌫0

does not hold in discrete setting unless µ is a Dirac.

W2-geodesics do not exist in discrete setting; even on a two-point space!

W2 is not adapted neither for defining the path ⌫t nor for measuring theconvexity defect/excess of the entropy.


Extension to discrete settings

Question: Is it possible to extend this theory to discrete settings, for examplefinite graphs ?

Two obstructions:

Talagrand’s inequality

W22 (⌫0, µ) Const. H(⌫0|µ), 8⌫0

does not hold in discrete setting unless µ is a Dirac.

W2-geodesics do not exist in discrete setting; even on a two-point space!

W2 is not adapted neither for defining the path ⌫t nor for measuring theconvexity defect/excess of the entropy.


..., Schmuckenschlager, Chung-Yau, Sammer-T., Ollivier, Ollivier-Villani,Bonciocat-Sturm, Maas, Chow-Huang-Li-Zhou, Erbar-Maas,Gozlan-Roberto-Samson-T., Mielke, Hillion, Leonard, Lehec, Joulin, Veysseire,Bauer et al, Klartag et al, Munch, . . .


The various paths to curvature...

Dobrushin, Sammer, ..., Ollivier: W1 contraction approach

M. Erbar, J. Maas; S-N. Chow, W. Huang, Y. Li, H-M. Zhou: Displacementconvexity of entropy using interpolation along newW2 geodesic, a PDEapproach, where the geodesics minimize certain kinetic energy intransporting one density (of molecules) to another.

N. Gozlan, C. Roberto, P-M. Samson, P.T.: Displacement convexity ofentropy using interpolation along W1 geodesics, a probabilistic approach.

Schmuckenschlager, Klartag-Kozma-Ralli-P.T.:Discrete versions of the Bochner formula, the so-called �2 criterion,comparing the 2nd derivative (of the variance/entropy of the heat flow) withthe first derivative...

Bauer-Horn-Lin-Lippner-Mangoubi-S.T.Yau:“Li-Yau Inequality on Graphs” : a local notion of curvature-dimensioninequality with some consequences























Interpolation along W1-geodesics : Gozlan,Roberto, Samson, and T.


GRST approach

similar approach developed independently by E. Hillion

Definition [L1-Wasserstein distance]

For all ⌫0, ⌫1 2 P(V), W1(⌫0, ⌫1) = inf⇡2P(⌫0 ,⌫1)

!

d(x0, x1) ⇡(dx0dx1).

We seek to prove convexity of the entropy along (pseudo)-W1-geodesics ⌫t

connecting ⌫0 to ⌫1 :

There is some K � 0 such that for all ⌫0, ⌫1 there is some pseudo W1-geodesic ⌫t

connecting ⌫0 to ⌫1 such that for all t 2 [0, 1]

H(⌫t |µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ) �Kt(1 � t)

2⇥ · · · .

A good candidate for the transport term in the right hand side is W21 (⌫0, ⌫1) .


GRST approach



For all ⌫0, ⌫1 2 P(V), W1(⌫0, ⌫1) = inf⇡2P(⌫0 ,⌫1)

!

d(x0, x1) ⇡(dx0dx1).





H(⌫t |µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ) �Kt(1 � t)

2⇥ · · · .



GRST approach



For all ⌫0, ⌫1 2 P(V), W1(⌫0, ⌫1) = inf⇡2P(⌫0 ,⌫1)

!

d(x0, x1) ⇡(dx0dx1).





H(⌫t |µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ) �Kt(1 � t)

2⇥ · · · .



GRST approach



For all ⌫0, ⌫1 2 P(V), W1(⌫0, ⌫1) = inf⇡2P(⌫0 ,⌫1)

!

d(x0, x1) ⇡(dx0dx1).





H(⌫t |µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ) �Kt(1 � t)

2⇥ · · · .



Our main example - the discrete hypercube ⌦n

Theorem [GRST]

Let µ be a probability on {0, 1} and denote by µn its n-fold tensor product. Thefollowing displacement convexity properties of the entropy hold true.

For all ⌫0, ⌫1 2 P(⌦n), there is a coupling ⇡ 2 P(⌫0, ⌫1) such that for all t 2 [0, 1], itholds

H(⌫⇡t |µn) (1 � t)H(⌫0|µn) + tH(⌫1|µn) � 2t(1 � t)n

W21 (⌫0, ⌫1)

where⌫⇡t =

X

x0 ,x12⌦n

⇡(x0, x1)X

z2Jx0 ,x1Ktd(x0 ,z)(1 � t)d(z,x1)�z ,

where z 2 Jx0, x1K means that z belongs to some geodesic joining x0 to x1.



Theorem [GRST]




W21 (⌫0, ⌫1)

where⌫⇡t =

X

x0 ,x12⌦n

⇡(x0, x1)X

z2Jx0 ,x1Ktd(x0 ,z)(1 � t)d(z,x1)�z ,




Theorem [GRST]




W21 (⌫0, ⌫1)

where⌫⇡t =

X

x0 ,x12⌦n

⇡(x0, x1)X

z2Jx0 ,x1Ktd(x0 ,z)(1 � t)d(z,x1)�z ,



W1-geodesics


For all ⌫0, ⌫1 2 P(V), W1(⌫0, ⌫1) = inf⇡2P(⌫0 ,⌫1)

!

d(x0, x1) ⇡(dx0dx1).

Important example:Let V = {0, 1, . . . , n}; consider the Binomial distribution B(n, t) defined by

B(n, t) =nX

k=0

nk

!tk (1 � t)n�k�k 2 P(V).

Then (B(n, t))t2[0,1] is a W1 constant-speed geodesic connecting �0 to �n.


Implications (Discrete Setting)

Displacement convexityw + u

Prekopa-Leindler + HWIu + w

Modified log-Sob Weak transport+

log-Sob for the Gaussian


Bochner formula (iterated gradient �2 criterion) :application: discrete Buser ...


Bakry-Emery, ... ,Schmuckenschlager,..., Klartag-KozmaG = (V ,E), and Graph Laplacian � = �(D � A)

Definition [Bochner w/ parameter K]

Curvature of G is at least K , if 8f : V ! R, and 8x 2 V ,

��(f , f)(x) � 2�(f ,�f)(x) � K �(f , f)(x) ,

where�f(x) =

X

y:(x,y)2E(f(y) � f(x)) .

And given f , g : V ! R, also define:

�(f , g)(x) =12

X

y:(x,y)2E(f(x) � f(y))(g(x) � g(y)) .

Note: When f = g, we get the more commonly denoted (square of the l2-type)discrete gradient: for each x 2 V ,

�(f)(x) := �(f , f)(x) =12

X

y:(x,y)2E(f(x) � f(y))2 =: |rf(x)|2 .





��(f , f)(x) � 2�(f ,�f)(x) � K �(f , f)(x) ,

where�f(x) =

X

y:(x,y)2E(f(y) � f(x)) .


�(f , g)(x) =12

X

y:(x,y)2E(f(x) � f(y))(g(x) � g(y)) .


�(f)(x) := �(f , f)(x) =12

X

y:(x,y)2E(f(x) � f(y))2 =: |rf(x)|2 .





��(f , f)(x) � 2�(f ,�f)(x) � K �(f , f)(x) ,

where�f(x) =

X

y:(x,y)2E(f(y) � f(x)) .


�(f , g)(x) =12

X

y:(x,y)2E(f(x) � f(y))(g(x) � g(y)) .


�(f)(x) := �(f , f)(x) =12

X

y:(x,y)2E(f(x) � f(y))2 =: |rf(x)|2 .





��(f , f)(x) � 2�(f ,�f)(x) � K �(f , f)(x) ,

where�f(x) =

X

y:(x,y)2E(f(y) � f(x)) .


�(f , g)(x) =12

X

y:(x,y)2E(f(x) � f(y))(g(x) � g(y)) .


�(f)(x) := �(f , f)(x) =12

X

y:(x,y)2E(f(x) � f(y))2 =: |rf(x)|2 .


�2 Calculus

Equivalently ...


�2(f)(x) � K �(f)(x) ,

where�2(f) := �2(f , f) =

12��(f) � �(f ,�f) .

Basically, using the iterated gradient:

2�2(f , g) = ��(f , g) � �(f ,�g) � �(�f , g) .


�2 Calculus

Equivalently ...


�2(f)(x) � K �(f)(x) ,

where�2(f) := �2(f , f) =

12��(f) � �(f ,�f) .


2�2(f , g) = ��(f , g) � �(f ,�g) � �(�f , g) .


�2 Calculus

Equivalently ...


�2(f)(x) � K �(f)(x) ,

where�2(f) := �2(f , f) =

12��(f) � �(f ,�f) .


2�2(f , g) = ��(f , g) � �(f ,�g) � �(�f , g) .


�2 Calculus (contd.)

The iterated gamma can be written out:

2�2(f)(x) = ��(f)(x) � 2�(f ,�f)(x)

=X

v⇠x

�(f)(v) � d(x)�(f)(x) �X

v⇠x

f(v)⇣�f(v) ��f(x)

⌘

=✓X

v⇠x

f(v)◆2� d(x)

2

X

v⇠x

f2(v) +X

u⇠v⇠x

f2(u) � 4f(u)f(v) + 3f2(v)2

=✓X

v⇠x

f(v)◆2�

X

v⇠x

d(x) + d(v)2

f2(v) +12

X

u⇠v⇠x

⇣(f(u) � 2f(v)

⌘2. (1)

For d-regular, further simplifies to:

2�2(f)(x) = dX

v⇠x

f2(v) +✓X

v⇠x

f(v)◆2

+X

v⇠x

X

u⇠v

✓ f2(u)2� 2f(u)f(v)

◆. (2)




2�2(f)(x) = ��(f)(x) � 2�(f ,�f)(x)

=X

v⇠x

�(f)(v) � d(x)�(f)(x) �X

v⇠x

f(v)⇣�f(v) ��f(x)

⌘

=✓X

v⇠x

f(v)◆2� d(x)

2

X

v⇠x

f2(v) +X

u⇠v⇠x

f2(u) � 4f(u)f(v) + 3f2(v)2

=✓X

v⇠x

f(v)◆2�

X

v⇠x

d(x) + d(v)2

f2(v) +12

X

u⇠v⇠x

⇣(f(u) � 2f(v)

⌘2. (1)


2�2(f)(x) = dX

v⇠x

f2(v) +✓X

v⇠x

f(v)◆2

+X

v⇠x

X

u⇠v

✓ f2(u)2� 2f(u)f(v)

◆. (2)




2�2(f)(x) = ��(f)(x) � 2�(f ,�f)(x)

=X

v⇠x

�(f)(v) � d(x)�(f)(x) �X

v⇠x

f(v)⇣�f(v) ��f(x)

⌘

=✓X

v⇠x

f(v)◆2� d(x)

2

X

v⇠x

f2(v) +X

u⇠v⇠x

f2(u) � 4f(u)f(v) + 3f2(v)2

=✓X

v⇠x

f(v)◆2�

X

v⇠x

d(x) + d(v)2

f2(v) +12

X

u⇠v⇠x

⇣(f(u) � 2f(v)

⌘2. (1)


2�2(f)(x) = dX

v⇠x

f2(v) +✓X

v⇠x

f(v)◆2

+X

v⇠x

X

u⇠v

✓ f2(u)2� 2f(u)f(v)

◆. (2)


Cheeger is tight under non-negative curvature!h := h(G) := minA⇢V |@A |/|A |, the Cheeger const, of G; � > 0 spectralgap of G.

Theorem (A-M, L-S, J-S 80’s)

h2

c1 maxdeg(G) � c2h .

Theorem (Klartag-Kozma-Ralli-T. ’14)

1. Bochner w/ parameter K implies: for any subset A ⇢ V,

|@A | � 12

min⇢p

�,�p2|K |

�|A |

✓1 � |A ||V |

◆.

2. Bochner w/ parameter K � 0 implies � � K, and hence:

� 16h2 .

3. Abelian Cayley graphs satisfy Bochner w/ parameter K � 0.


CD(K, 1) with K � 0: Examples

1. Complete graph Kn: Curvature = 1 + n/2.

2. Discrete n-cube Qn: Curvature = 2.

3. Symmetric group S(n) with the transposition metric: Curvature = 2

4. General Proposition. If T is the maximum number of trianglescontaining any edge, then K 2 + T/2.

5. Abelian necessary! On Sn, the (left) Cayley graph generated by(12), (12...n) ± 1, the Cheeger constant is c1n�2, while the spectral gap isc2n�3, with c1, c2 > 0, independent of n.


Cheeger, CD(K, 1) with K � 0

Qn. 1 Can we characterize the graphs or Markov kernels which satisfynon-negative curvature?

Qn. 2 Can we characterize the graphs for which the Cheeger inequality istight?

Qn. 3 More examples? Non-crossing partition lattice NC(n)?


Erbar-Maas:

X : finite set;K : X ⇥ X! R+ : irreducible Markov kernel;⇡ : the unique invariant probab. measure for K .

Let

P(X) :=⇢⇢ : X! R+ |

X

x2X⇡(x)⇢(x) = 1

�

and let P+(X) ⇢ P(X) be the subset consisting of strictly positive probabilitydensities on X.

Given ⇢ 2 P(X), recall the logarithmic mean of ⇢(x) and ⇢(y):

• ⇢(x, y) := ✓⇣⇢(x), ⇢(y)

⌘:=

Z 1

0⇢(x)1�p⇢(y)pdp =

⇢(x) � ⇢(y)log ⇢(x) � log ⇢(y)

.

• H(⇢) =P

x2X ⇡(x)⇢(x) log ⇢(x) .


Displacement Convexity and Ricci a la Maas

Definition (Maas)

Now consider the metricW defined for ⇢0, ⇢1 2 P(X) by

W(⇢0, ⇢1)2 := inf

⇢,

(12

Z 1

0

X

x,y2X( t(x) � t(y))2⇢t(x, y)K(x, y)⇡(x)dt

),

where the infimum runs over all sufficiently regular curves ⇢ : [0, 1]! P(X) and : [0, 1]! RX satisfying the continuity equation

8>>>><>>>>:

ddt⇢t(x) +

X

y2X( t(y) � t(x))⇢t(x, y)K(x, y) = 0 8x 2 X ,

⇢(0) = ⇢0 , ⇢(1) = ⇢1 .

(3)

Definition (Maas)

Markov kernel K has Ricci curvature bounded from below by 2 R, if for anyconstant-speed geodesic {⇢t }t2[0,1] in (P(X),W), we have

H(⇢t) (1 � t)H(⇢0) + tH(⇢1) �

2t(1 � t)W(⇢0, ⇢1)

2 .


The PDE approach

Maas-Erbar: Displacement convexity of the entropy on the discrete hypercube⌦n = {0, 1}n.

H(⌫t |µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ) �t(1 � t)

nW2

2(⌫0, ⌫1)

The (finite dimensional) space P(⌦n) is equipped with a Riemannian metric.

[Their metric construction is general and makes sense for every reversibleMarkov kernel on a finite graph].

The pseudo Wasserstein distanceW2 corresponds to the geodesic distance onP(⌦n) and ⌫t is a geodesic connecting ⌫0 to ⌫1.

The functionW22 is not a transport cost.

This implies some versions of LSI and HWI, and a Talagrand typeinequality for theW2 and W1 distances with sharp constants.


The PDE approach


H(⌫t |µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ) �t(1 � t)

nW2

2(⌫0, ⌫1)







The PDE approach


H(⌫t |µ) (1 � t)H(⌫0|µ) + tH(⌫1|µ) �t(1 � t)

nW2

2(⌫0, ⌫1)







Ricci vs Coarse Ricci Curvature

Theorem (M.Erbar-J.Maas-P.T.)1. k -slice of the n-cube (with 1-0 transpositions) has Ricci =⇥(n/[k(n � k)]) ;

2. Sn with transpositions has Ricci = ⌦(1/n2) .

Conjecture

Sn with transpositions has Ricci = ⌦(1/n).

Remark. It can not be better than (1/n), since both the spectral gap andthe modified Log-Sobolev (entropy) constant are ⇥(1/n), and these areimplied by (the stronger notions of) Ricci – e.g., those of Erbar-Maas andGozlan-Roberto-Samson-T.










Thank you!


Back to the Cube: Kruskal-Katona consequence

Let A ⇢ {0, 1}n, B := 0 (singleton); let d = 1|A |

Pa2A d(0, a) . Then

Proposition

|Mt(0,A)|2 � t2|A |t et log(1/t) d2

.

Proof idea. We may use a theorem of Lovasz:If |A | = (x

m), for some x 2 R, then its shadow has size, |@A | � ( xm�1) .

Idea: write A as the (disjoint) union of its intersection with the level-sets,and use the observation: Mj/m(0,Am) = @m�jAm , for 0 j m.


Zero Curvature

Theorem (B-M on Z)Let 0 t 1. Then 8A ,B ⇢ Z, we have:

|Mt(A ,B)| � |A |1�t |B |t ,

where Mt(A ,B) = {b(1 � t)a + tbc, d(1 � t)a + tbe; a 2 A , b 2 B} .

Proof sketch. Prove directly using Brunn-Minkowski on R; or derivePrekopa-Leindler on Z using that on R and then (as usual) restrict toindicator functions:

f = 1A , g = 1B , h = 1Mt (A ,B) .


Zero Curvature (contd.)

Theorem (P-L on Z)

Let 0 t 1. If f , g, h : Z! R+, s.t. we have:

h(m) � f(a)1�t g(b)t , 8a, b 2 Z , and m 2 Mt(a, b) ,

then we also have:X

m2Zh(m) �

✓X

a2Af(a)

◆1�t✓X

b2Bg(b)

◆t.

Proof sketch. Apply the classical continuous Prekopa-Leindler to thefunctions:

f(x) = f(bx + 1/2c), g(x) = g(bx + 1/2c), h(x) = h(bx + 1/2c), x 2 R .

The hypothesis can be verified for these functions by checking that:

m := b(1 � t)x + ty + 1/2c 2 Mt(bx + 1/2c, by + 1/2c) .


Continuing with the Integers

1. (J. Melbourne). Previous result can be extended to Z mod n.

2. (K. Costello, P.T.): The t = 1/2 case of Z (and Z mod p) can behandled with the Cauchy-Davenport sumset inequality!


Sn : midpoint injection

Theorem (K. Costello, P.T.)Consider the transposition graph on Sn, with d being the transpositiondistance. Then, for all A ,B ⇢ Sn, we have

|M1/2(A ,B)|2 � |A ||B | .

Proof sketch. Consider a 2 A and b 2 B be so that d(a, b) = d.

Let M1(a, b) be the set of perms. of distance bd/2c from a and distancedd/2e from b, and

M2(a, b) be the set of perms. of distance dd/2e from a and distancebd/2c from b.

define canonical paths between a 2 A and b 2 B and exhibit an injectionfrom A ⇥ B to M1 ⇥M2.


Sn : a canonical path construction

Towards an injection from A ⇥ B to M1 ⇥M2:

• Factorize ba�1 = td td�1 . . . t1 as a product of d transpositions (from rightto left):

i) If ba�1 is a single cycle, write it as (a1a2 · · · ak ), where ak < ai for alli , k . Then factorize the cycle as (a1a2)(a2a3) . . . (ak�1ak ).

ii) If it consists of several cycles, sort the cycles in decreasing order bythe smallest element in each cycle, then concatenate the factorizationsfor each cycle.

• Define x = tbd/2c · · · t1 and y = td · · · t1+bd/2c. Then define

m1 := xa and m2 := x�1yxa = x�1b .

Thus given m1 and m2, we know x�1y = m2m�11 .


Sn : path argument (contd.)

• Define x = tbd/2c · · · t1 and y = td · · · t1+bd/2c. Then define

m1 := xa and m2 := x�1yxa = x�1b .

Thus given m1 and m2, we know x�1y = m2m�11 .

Claim (near commutativity of x and y). Any two elements are in the samecycle in x�1y iff they are in the same cycle in yx.


Sn : injection completed

Claim (near commutativity of x and y). Any two elements are in the samecycle in x�1y iff they are in the same cycle in yx.

• Using the claim, knowing the cycles of x�1y, we will know the cycles ofyx, and hence the distance d.• By the (canonical) path construction, we may know which cycles arepart of x and which are part of y, and which cycle (if any) is split!Furthermore, we know where in the (shared) cycle the split occurred,since we know d and which cycles are from x.

Thus we may recover x and y and hence also a and b, given m1 and m2.


Documents

The many paths to discrete curvaturemath.sjtu.edu.cn/conference/9shcc/ppt/Prasad Tetali.pdf · The many paths to discrete curvature Prasad Tetali Georgia Institute of Technology Shanghai