# MLSI: Theory, Examples and Consequences Lecture 3

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## MLSI: Theory, Examples and Consequences

Lecture 3

Georgia Institute of Technology and Indian Institute of Science

ICTS, Bengaluru, August 5-17, 2019

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

1 Discrete curvature and Conjectures

2 SLC and Matroid Bases Exchange

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Bochner-Bakry- ´ Emery Criterion

The following 2nd derivative criterion is also useful, and inspires a notion ofdiscrete curvature, mimicking the Ricci curvature.

Proposition (Bochner’46, Lichn´erowicz’58) inff

E(f,f)

Varπf =:λ= inf

f

E(−Lf,f)

E(f,f) =:µP =:µ . Proof. λµ: Use Cauchy-Schwartz. Indeed, w/Eπf = 0,

E(f,f) =E(f(−Lf))

Varπf1/2

E(−Lf)21/2

1

λ(E(f,f))1/2

E(−Lf)21/2

= 1

λ(E(f,f))1/2

E(−Lf,f)1/2

.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Bochner-Bakry- ´ Emery Criterion (contd.)

Other direction,λµ: 2nd derivative and integration. Starting with d

dtVar(Htf) =−2E(Htf,Htf) and d

dtE(Htf,Htf) =−2E(Htf,−LHtf). Observe that ast→ ∞, Htf Eπf, and soE(Htf,Htf)0. Hence,

E(f,f) = Z

0

d

dtE(Htf,Htf)dt= 2 Z

0

E(Htf,−L(Htf))dt

= 2

Z

0

E(−L(Htf),Htf)dt P

Z

0

E(Htf,Htf)dt

= −µP

Z

0

d

dtVar(Htf)dt=µPVarf, implying thatλP =λPµP.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## MLSI and Bakry- ´ Emery Criterion

What about repeating the above for the entropy constant?

Proposition (Bakry- ´Emery’85) inf

f>0

E(f,logf)

Entπf =:αinf

f>0

u(f) E(f,logf), where u(f) =E(−Lf,logf) +E(f,(−Lf)/f).

Some relevant references include:

[Boudou, Caputo, Dai Pra, Posta, 2005]. “Spectral gap estimates for interacting particle systems via a Bochner type identity,” J. Funct.

Analysis.

[M. Erbar, J. Maas, P.T. 2015].“Discrete Curvature Bounds for

Bernoulli-Laplace and Random Transposition Models,” Annales Fac. Sci.

Toulouse.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Bochner formula (iterated gradient Γ

2

criterion) :

application: discrete Buser ...

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Bakry- ´ Emery, ... , Schmuckenschl¨ ager,...

G= (V,E), and Graph Laplacian ∆ =−(DA)

Definition [Bochner w/ parameter K]

Curvature ofG is at leastK, if∀f :V R, and∀x V,

∆Γ(f,f)(x)2Γ(f,∆f)(x)K Γ(f,f)(x), where

∆f(x) = X

y:(x,y)∈E

(f(y)f(x)). And givenf,g :V R, also define:

Γ(f,g)(x) = 1 2

X

y:(x,y)∈E

(f(x)f(y))(g(x)g(y)).

Note: Γ(f)(x) := Γ(f,f)(x) = 12P

y:(x,y)∈E(f(x)f(y))2=:|∇f(x)|2.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Bakry- ´ Emery, ... , Schmuckenschl¨ ager,...

G= (V,E), and Graph Laplacian ∆ =−(DA) Definition [Bochner w/ parameter K]

Curvature ofG is at leastK, if∀f :V R, and∀x V,

∆Γ(f,f)(x)2Γ(f,∆f)(x)K Γ(f,f)(x),

where

∆f(x) = X

y:(x,y)∈E

(f(y)f(x)). And givenf,g :V R, also define:

Γ(f,g)(x) = 1 2

X

y:(x,y)∈E

(f(x)f(y))(g(x)g(y)).

Note: Γ(f)(x) := Γ(f,f)(x) = 12P

y:(x,y)∈E(f(x)f(y))2=:|∇f(x)|2.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Bakry- ´ Emery, ... , Schmuckenschl¨ ager,...

G= (V,E), and Graph Laplacian ∆ =−(DA) Definition [Bochner w/ parameter K]

Curvature ofG is at leastK, if∀f :V R, and∀x V,

∆Γ(f,f)(x)2Γ(f,∆f)(x)K Γ(f,f)(x), where

∆f(x) = X

y:(x,y)∈E

(f(y)f(x)). And givenf,g :V R, also define:

Γ(f,g)(x) = 1 2

X

y:(x,y)∈E

(f(x)f(y))(g(x)g(y)).

Note: Γ(f)(x) := Γ(f,f)(x) = 12P

y:(x,y)∈E(f(x)f(y))2=:|∇f(x)|2.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Bakry- ´ Emery, ... , Schmuckenschl¨ ager,...

G= (V,E), and Graph Laplacian ∆ =−(DA) Definition [Bochner w/ parameter K]

Curvature ofG is at leastK, if∀f :V R, and∀x V,

∆Γ(f,f)(x)2Γ(f,∆f)(x)K Γ(f,f)(x), where

∆f(x) = X

y:(x,y)∈E

(f(y)f(x)). And givenf,g :V R, also define:

Γ(f,g)(x) = 1 2

X

y:(x,y)∈E

(f(x)f(y))(g(x)g(y)).

Note: Γ(f)(x) := Γ(f,f)(x) = 12P

y:(x,y)∈E(f(x)f(y))2=:|∇f(x)|2.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Γ

2

Calculus

Equivalently ...

Curvature ofG is at least K, if∀f :V →R, and ∀x∈V, Γ2(f)(x)≥K Γ(f)(x),

where

Γ2(f) := Γ2(f,f) = 1

2∆Γ(f)−Γ(f,∆f).

2(f,g) = ∆Γ(f,g)−Γ(f,∆g)−Γ(∆f,g).

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Γ

2

## Calculus

Equivalently ...

Curvature ofG is at least K, if∀f :V →R, and ∀x∈V, Γ2(f)(x)≥K Γ(f)(x),

where

Γ2(f) := Γ2(f,f) = 1

2∆Γ(f)−Γ(f,∆f).

2(f,g) = ∆Γ(f,g)−Γ(f,∆g)−Γ(∆f,g).

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Γ

2

Calculus

Equivalently ...

Curvature ofG is at least K, if∀f :V →R, and ∀x∈V, Γ2(f)(x)≥K Γ(f)(x),

where

Γ2(f) := Γ2(f,f) = 1

2∆Γ(f)−Γ(f,∆f). Basically, using the iterated gradient:

2(f,g) = ∆Γ(f,g)−Γ(f,∆g)−Γ(∆f,g).

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Γ

2

## Calculus (contd.)

The iterated gamma can be written out:

2(f)(x) = ∆Γ(f)(x)2Γ(f,∆f)(x)

=X

v∼x

Γ(f)(v)d(x)Γ(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) +1 2

X

u∼v∼x

(f(u)2f(v)2

. (2.1) Ford-regular, further simplifies to:

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)

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Γ

2

Calculus (contd.)

The iterated gamma can be written out:

2(f)(x) = ∆Γ(f)(x)2Γ(f,∆f)(x)

=X

v∼x

Γ(f)(v)d(x)Γ(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) +1 2

X

u∼v∼x

(f(u)2f(v)2

. (2.1)

Ford-regular, further simplifies to: 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)

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Γ

2

## Calculus (contd.)

The iterated gamma can be written out:

2(f)(x) = ∆Γ(f)(x)2Γ(f,∆f)(x)

=X

v∼x

Γ(f)(v)d(x)Γ(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) +1 2

X

u∼v∼x

(f(u)2f(v)2

. (2.1) Ford-regular, further simplifies to:

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)

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Cheeger is tight if curvature ≥ 0

h:=h(G) := minA⊂V|∂A|/|A|, the Cheeger const, ofG;λ >0: gap(G).

Theorem (A-M, L-S, J-S 80’s) h2

c1maxdeg(G) λc2h. Theorem (Klartag-Kozma-Ralli-T. ’14)

1. Bochner w/ parameter K implies: for any subset AV ,

|∂A| ≥ 1 2minn√

λ, λ

p2|K| o|A|

1 |A|

|V|

.

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

λ16h2.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## CD(K, ∞ ) 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 triangles containing any edge, then K ≤2 +T/2.

5. Abelian necessary! OnSn, the (left) Cayley graph generated by (12),(12...n)±1, the Cheeger constant is c1n−2, while the spectral gap is c2n−3, with c1,c2>0, independent of n.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Cheeger, CD(K, ∞ ) with K ≥ 0

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

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

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

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Ollivier-Villani

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

Coarse Ricci: Take two small balls and compute the transportation distance between them. If it is smaller than the distance between the centers of the balls, then coarse Ricci ispositive.

W1x, µy) =: (1κ(x,y))d(x,y), Examples.

1. n-cube: µx uniform on the n+1 neighbors of x (including itself). For x,y, neighbors,κ(x,y) = 2/(n+ 1).

2. Snwith transpositions: Forσ, τ differing in a transposition, κ(x,y) = 1/ n2

.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## (Coarse) Ricci of Hypercube

Proposition (folklore)

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

Proof sketch.

(i) Consider thelazyrandom walk on then-cube.

(ii) Use a simple (path) coupling argument to show that two copies of the chain started atneighboring verticesx,y ∈ {0,1}n can be “coupled”

with probability at least 2/(n+ 1) in one step.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## (Coarse) Ricci of Transposition Graph

Proposition

Snwith transpositions has coarse Ricci=1/ n2 .

Proof sketch.

Lower bound:

(i) Consider thelazyrandom transposition chain onSn.

(ii) Use a simple (path) coupling argument to show that two copies of the chain started atσ, τ Sn withd(σ, τ) = 1, can be coupled with probability at least 1/ n2

in one step.

Upper bound: Follows by showing that there is no better coupling – use (Kantorovich’s) dual formulation ofW1:

W1(ν, µ) = sup

f: 1−Lip

Eνf Eµf .

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Peres-Tetali “Conjecture”!

Doescoarse Ricci at least κ >0 imply the MLSI constantα≥κ?

Something weaker is known: namely that aW1 transport-Entropy inequality follows, thanks to(Marton??, Eldan, Lehec, Lee’16)

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

MLSI for Bases Exchange Walk

I am thankful to Heng Guo for letting me borrow some of his slides in this section.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Matroid

A matroidM= (E,I) consists of a finite ground setE and a collectionI of subsets ofE (independent sets) such that:

∅ ∈ I;

if S ∈ I,T ⊆S, then T ∈ I (downward closed);

if S,T ∈ I and |S|>|T|, then there exists an element i ∈S\T such thatT ∪ {i} ∈ I (augment axiom).

Maximum independent sets are thebases.

For any two bases, there is a sequence of exchanges of ground set elements that take one basis to the other.

Letn=|E|andr be the rank, namely the size of any basis.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Bases-exchange walk

The following Markov chainPBX,π converges to a “homogeneous SLC”π:

1 remove an element uniformly at random from the current basis (call the resulting set S);

2 addi 6∈S with probability proportional to π(S∪ {i}).

The implementation of the second step may be non-trivial.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## MLSI for Matroid Basis Exchange

Theorem (Mary Cryan-Heng Guo-Giorgos Mousa) For any f : Ω→R≥0,

EPBX,π(f,logf)≥ 1

r ·Entπ(f), where r is the rank of the matroid.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Many open problems remain!

Sampling spanning trees ofG with no broken circuit- the same as evaluation TG(1,0) of the Tutte polynomial, leading coefficient |a1G(λ)]|of the chromatic polynomial (up to sign), number of maximumG-parking functions, sampling acyclic orientations with a unique sink...

Sampling all acyclic orientations of G Negative correlation for random forests ofG

(More generally) characterization of matroids with negative correlation for random bases.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Strongly log-concave polynomials

Log-concave polynomial

A polynomialp ∈R≥0[x1, . . . ,xn] is log-concave(atx) if the Hessian∇2logp(x) is negative semi-definite.

⇒ ∇2p(x) has at most one positive eigenvalue.

Strongly log-concave polynomial

A polynomialp ∈R≥0[x1, . . . ,xn] is strongly log-concaveif for any index setI ⊆[n],∂Ip is log-concave at 1.

Originally introduced byGurvits(2009), equivalent to:

Completely log-concave (Anari, Oveis Gharan, and Vinzant, 2018);

Lorentzian polynomials (Br¨and´en and Huh, 2019+).

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Strongly log-concave distributions

A distributionπ: 2[n]→R≥0 is strongly log-concaveif so is its generating polynomial

gπ(x) = X

S⊆[n]

π(S)Y

i∈S

xi.

An important example of homogeneous strongly log-concave distributions is the uniform distribution over bases of a matroid (Anari, Oveis Gharan, and Vinzant 2018; Br¨and´en and Huh 2019+).

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Alternative characterization for SLC

Br¨and´en and Huh(2019+): Anr-homogeneous multiaffine

polynomialp with non-negative coefficients isstrongly log-concave if and only if:

the support of p is the bases of some matroid;

after taking r−2 partial derivatives, the quadratic is real stableor 0.

Real stable: p(x)6= 0 if=(xi)>0 for alli.

Real stable polynomials (and strongly Rayleigh distributions) capture only “balanced” matroids, whereas SLC polynomials capture all matroids.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Bases-exchange walk

The following Markov chainPBX,π converges to a homogeneous SLCπ:

1 remove an element uniformly at random from the current basis (call the resulting set S);

2 addi 6∈S with probability proportional to π(S∪ {i}).

The implementation of the second step may be non-trivial.

Recall the mixing time defn.

tmix(P, ε) := min

t

t | kPt(x0,·)−πkTV≤ε .

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Theorem (Cryan-Guo-Mousa 2019)

For any r -homogeneous strongly log-concave distributionπ, tmix(PBX,π, ε)≤r

log log 1 πmin

+ log 1 2ε2

,

whereπmin= minx∈Ωπ(x).

Previously,Anari, Liu, Oveis Gharan, and Vinzant(2019):

tmix(PBX,π, ε)≤r

log 1 πmin

+ log1 ε

E.g. for the uniform distribution over bases of matroids (withn elements and rankr), the new bound is O(r(logr+ log logn)), whereas the previous bound isO(r2logn).

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Levels of independent sets

The set of all independent sets of a matroidMisdownward closed.

LetM(k) be the set of independent sets of size k. Thus,M(r) is the set of all bases.

LetMi denote the matroid Mafter contractingi, which is another matroid itself.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Weights for independent sets

EquipMwith the following inductively defined weight function:

w(I) :=

(π(I)Zr if|I|=r, P

I0⊃I,|I0|=|I|+1w(I0) if|I|<r, for some normalization constantZr >0.

For example, we may choosew(B) = 1 for allB ∈ B andZr =|B|, which corresponds to the uniform distribution overB.

Letπk be the distribution such thatπk(I)∝w(I), and Zk be the corresponding normalizing constant.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Different views

Polynomial: ∂x∂p

i;settingxi = 0; (r−k)!∂x

Ip(1)

Matroid: contraction over i;deletion of i;w(I)

Distribution: conditioning on having i;conditioning on not having i; proportional toπk(I)

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Random walk between levels

There are two natural random walks converging toπk. The “down-up” random walk Pk:

→ 1. remove an element of I ∈ M(k) uniformly at random to get I0 ∈ M(k−1);

2. move toJ such thatJ ∈ M(k),J ⊃I0 with probability w(Iw(J)0). The bases-exchange walkPBX,π=Pr.

The “up-down” walk Pk is defined similarly.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Random walk between levels

There are two natural random walks converging toπk. The “down-up” random walk Pk:

1. remove an element of I ∈ M(k) uniformly at random to get I0 ∈ M(k−1);

→ 2. move toJ such thatJ ∈ M(k),J ⊃I0 with probability w(Iw(J)0). The bases-exchange walkPBX,π=Pr.

The “up-down” walk Pk is defined similarly.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Decomposing the walks

LetAk be the matrix whose rows are indexed by M(k) and columns byM(k+ 1) such thatAk(I,J) = 1 if and only ifI ⊂J.

Letwk ={w(I)}I∈M(k), and Pk+1 := 1

k+ 1·ATk;

Pk := diag(wk)−1Akdiag(wk+1). We have

Pk+1=Pk+1 Pk; Pk =PkPk+1 .

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Decomposing the walks

LetAk be the matrix whose rows are indexed by M(k) and columns byM(k+ 1) such thatAk(I,J) = 1 if and only ifI ⊂J.

Letwk ={w(I)}I∈M(k), and Pk+1 := 1

k+ 1·ATk;

Pk := diag(wk)−1Akdiag(wk+1).

We have

Pk+1=Pk+1 Pk; Pk =PkPk+1 .

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Decomposing the walks

LetAk be the matrix whose rows are indexed by M(k) and columns byM(k+ 1) such thatAk(I,J) = 1 if and only ifI ⊂J.

Letwk ={w(I)}I∈M(k), and Pk+1 := 1

k+ 1·ATk;

Pk := diag(wk)−1Akdiag(wk+1).

We have

Pk+1=Pk+1 Pk; Pk =PkPk+1 .

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Decomposing the walks

LetAk be the matrix whose rows are indexed by M(k) and columns byM(k+ 1) such thatAk(I,J) = 1 if and only ifI ⊂J.

Letwk ={w(I)}I∈M(k), and Pk+1 := 1

k+ 1·ATk;

Pk := diag(wk)−1Akdiag(wk+1).

We have

Pk+1=Pk+1 Pk; Pk =PkPk+1 .

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Key lemma

Lemma

For any k≥2and f :M(k)→R≥0, Entπk(f)

k ≥ Entπk−1(Pk−1 f) k−1 .

ApplyingPk−1 to the left corresponds to the random walkPk. The lemma asserts that Pk contracts the relative entropy by at least (1−1/k):

Entπk−1(Pk−1 f)≤(1−1/k)Entπk(f).

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Key lemma

Lemma

For any k≥2and f :M(k)→R≥0, Entπk(f)

k ≥ Entπk−1(Pk−1 f) k−1 .

ApplyingPk−1 to the left corresponds to the random walkPk. The lemma asserts that Pk contracts the relative entropy by at least (1−1/k):

Entπk−1(Pk−1 f)≤(1−1/k)Entπk(f).

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Base case

For the base case, we want to show that

Entπ2(f)−2Entπ1(P1f)≥0.

Usingalogab ≥a−b for a,b>0, we can get Entπ2(f)−2Entπ1(P1f)≥1− 1

2Z2 ·hTWh, whereWij =w({i,j}) andh=P1f.

SinceW = (r−2)!Zr2gπ(1), it has at most one positive eigenvalue. The quadratic form is maximized ath =P1f =1, which helps prove the base case...

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Base case

For the base case, we want to show that

Entπ2(f)−2Entπ1(P1f)≥0.

Usingalogab ≥a−b for a,b>0, we can get Entπ2(f)−2Entπ1(P1f)≥1− 1

2Z2 ·hTWh, whereWij =w({i,j}) andh=P1f.

SinceW = (r−2)!Zr2gπ(1), it has at most one positive eigenvalue. The quadratic form is maximized ath =P1f =1, which helps prove the base case...

(47)

Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Base case

For the base case, we want to show that

Entπ2(f)−2Entπ1(P1f)≥0.

Usingalogab ≥a−b for a,b>0, we can get Entπ2(f)−2Entπ1(P1f)≥1− 1

2Z2 ·hTWh, whereWij =w({i,j}) andh=P1f.

SinceW = (r−2)!Zr2gπ(1), it has at most one positive eigenvalue. The quadratic form is maximized ath=P1f =1, which helps prove the base case...

(48)

Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Bound the mixing time directly

For a distributionτonM(k), the relative entropyD(τkπk) = Entπk(Dk−1τ) where Dk= diag(πk). Moreover, after one step ofPk, the distribution is

TPk)T= (Pk)Tτ. SincePkis reversible,Dk−1(Pk)T=PkDk−1. D (Pk)Tτkπk

= Entπk(Dk−1(Pk)Tτ)

= Entπk(PkD−1k τ)

= Entπk(PkPk−1 Dk−1τ)

Entπk−1(Pk−1 Dk−1τ) (Jensen’s inequality)

11 k

Entπk(Dk−1τ) (entropy contraction)

=

11 k

D kπk).

The mixing time bound follows from Pinsker’s inequality 2σk2TVD(τkσ).

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

## Many open problems remain!

Sampling spanning trees ofG with no broken circuit- the same as evaluation TG(1,0) of the Tutte polynomial, leading coefficient |a1G(λ)]|of the chromatic polynomial (up to sign), number of maximumG-parking functions, sampling acyclic orientations with a unique sink...

Sampling all acyclic orientations of G Negative correlation for random forests ofG

(More generally) characterization of matroids with negative correlation for random bases.

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Outline Discrete curvature and Conjectures SLC and Matroid Bases Exchange

Fin

## Thank you!

References

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