Introduction to the Correlation Decay Method Yitong Yin Nanjing - - PowerPoint PPT Presentation

Introduction to the Correlation Decay Method

Yitong Yin Nanjing University

Introduction to Partition Functions W

• rkshop, July 20, 2018

Computational Aspects of Partition Functions, CIB@EPFL

• July 20: Introduction to the correlation

decay method (Yitong)

• July 23: Correlation decay for distributed

counting (Yitong)

• July 25: Beyond bounded degree graphs

(Piyush)

• July 26: Correlation decay, zeros of

polynomials, and the Lovász local lemma (Piyush)

Counting Independent Set

hardcore model:

• λ < λc(Δ) ⇒ FPTAS
• λ > λc(Δ) ⇒ no FPRAS unless NP=RP

∀ I ⊆ V:

uniqueness threshold: Computing ZG(λ) in graphs with constant max-degree ≤ ∆

[Galanis Štefankovič Vigoda 12] [Sly Sun 12] [Weitz 06]

undirected graph G(V,E) fugacity λ > 0

partition function:

Z = ZG(λ) = X

I⊆V

w(I)

w(I) = ( λ|I| I is an independent set in G

• therwise

λc(∆) = (∆ − 1)∆−1 (∆ − 2)∆

⟸ strong spatial mixing

≈ e ∆ − 2

Spin System

undirected graph G = (V, E) finite integer q ≥ 2 σ ∈ [q]V configuration w(σ) = Y

{u,v}∈E

A(σu, σv) Y

v∈V

b(σv) weight: A : [q] × [q] → R≥0 b : [q] → R≥0 symmetric q×q matrix q-vector (symmetric binary constraint) partition function:

ZG = X

σ∈[q]V

w(σ)

Gibbs distribution:

µG(σ) = w(σ) ZG

(unary constraint)

• 2-spin system: q =2,
• hardcore model:
• Ising model:
• multi-spin system: general q ≥ 2
• Potts model:
• q-coloring: β=0

undirected graph G = (V, E) σ ∈ [q]V configuration w(σ) = Y

{u,v}∈E

A(σu, σv) Y

v∈V

b(σv) weight:

σ ∈ {0, 1}V

1 1

     β β ... β     

A =

   1 . . . 1   

b =

A =  a00 a01 a10 a11

• b =

 b0 b1

• A =

0 1 1 1

• b =

 λ 1

• A =

β 1 1 β

• symmetric

finite integer q ≥ 2

b = λ 1

Marginal Probability

Gibbs distribution µG over all configurations in [q]V undirected graph G = (V, E) finite integer q ≥ 2

specified on an arbitrary subset Λ⊂V

∀ possible boundary condition σ ∈ [q]Λ marginal distribution at vertex v ∈ V : µσ

v

∀x ∈ [q] : µσ

v(x) =

Pr

X∼µG[Xv = x | XΛ = σ]

w(σ) ZG = µ(σ) =

n

Y

i=1

µσ1,...,σi−1

vi

(σi)

Marginal Probability

Gibbs distribution µG over all configurations in [q]V undirected graph G = (V, E) finite integer q ≥ 2

specified on an arbitrary subset Λ⊂V

∀ possible boundary condition σ ∈ [q]Λ marginal distribution at vertex v ∈ V : µσ

v

∀x ∈ [q] : µσ

v(x) =

Pr

X∼µG[Xv = x | XΛ = σ]

approximately computing µσ

v

approximately computing ZG

• approx. inference
• approx. counting

Spatial Mixing (Decay of Correlation)

G v

dist(v,Λ)

weak spatial mixing (WSM) at rate δ( ): µσ

v : marginal distribution at vertex v conditioning on σ

boundary conditions

• n infinite graphs:

WSM

uniqueness of infinite- volume Gibbs measure

σ

µσ

v

WSM of hardcore model

• n infinite Δ-regular tree

Λ ∀σ, τ ∈ [q]Λ : kµσ

v µτ vkT V  δ(distG(v, Λ))

λ ≤ λc(∆)

Spatial Mixing (Decay of Correlation)

strong spatial mixing (SSM) at rate δ( ): weak spatial mixing (WSM) at rate δ( ): µσ

v : marginal distribution at vertex v conditioning on σ

SSM

marginal probabilities are well approximated by the local information

G v

dist(v,Δ)

Δ

Λ\Δ

weak spatial mixing (WSM) at rate δ( ): ∀σ, τ ∈ [q]Λ : kµσ

v µτ vkT V  δ(distG(v, Λ))

kµσ

v µτ vkT V  δ(distG(v, ∆))

∀σ, τ ∈ [q]Λ that differ on Δ:

pT , Pr

I∼µT[v is unoccupied by I ]

Ti

v ui

T hardcore model:

independent set I in T

µT (I) ∝ λ|I|

Tree Recurrence

pTi , Pr

I∼µTi

[ui is unoccupied by I ]

u1 ud

T1 Td

pT , Pr

I∼µT[v is unoccupied by I ]

Ti

v ui

T

= ZT (v is unoccupied) ZT (v is unoccupied) + ZT (v is occupied)

ZT (event A) , X

I: A holds

w(I)

where

pT , Pr

I∼µT[v is unoccupied by I ]

Ti

v ui

T

= ZT (v is unoccupied) ZT (v is unoccupied) + ZT (v is occupied)

ZT (v is unoccupied) =

d

Y

i=1

ZTi

Product rule:

= 1 1 + λ Qd

i=1 pTi

ZT (v is occupied) = λ

d

Y

i=1

ZTi(ui is unoccupied)

= Qd

i=1 ZTi

Qd

I=1 ZTi + λ Qd i=1 ZTi(ui is unoccupied)

Ti

v ui

T

Product rule: σi

, Pr

I∼µT[v is unoccupied by I | σ ]

pσ

T

= ZT (v is unoccupied ∧ σ) ZT (v is unoccupied ∧ σ) + ZT (v is occupied ∧ σ)

ZT (v is unoccupied ∧ σ) =

d

Y

i=1

ZTi(σi)

= 1 1 + λ Qd

i=1 pσi Ti

ZT (v is occupied ∧ σ) = λ

d

Y

i=1

ZTi(ui is unoccupied ∧ σi)

= Qd

i=1 ZTi(σi)

Qd

I=1 ZTi + λ Qd i=1 ZTi(σi)(ui is unoccupied ∧ σi)

Ti

v ui

T hardcore model:

independent set I in T

µT (I) ∝ λ|I|

Tree Recurrence

σi

, Pr

I∼µT[v is unoccupied by I | σ ]

pσ

T

= 1 1 + λ Qd

i=1 pσi Ti

pσ

T

Occupancy ratio:

Rσ

T ,

PrT [v is occupied | σ] PrT [v is unoccupied | σ]

= (1 − pσ

T )/pσ T

Rσ

T = λ d

Y

i=1

1 Rσi

Ti + 1

Ti

v vi

T 2-spin system:

Occupancy ratio:

Tree Recurrence

σi

Rσ

T = b0

b1

d

Y

i=1

a00Rσi

Ti + a01

a10Rσi

Ti + a11

Rσ

T , PrX∼µT [Xv = 0 | σ]

PrX∼µT [Xv = 1 | σ]

A = a00 a01 a10 a11

• b =

 b0 b1

• µT (σ) ∝

Y

uv∈E

aσ(u),σ(v) Y

v∈V

bσ(v)

a Möbius transformation

Ti

v vi

T hardcore model:

Tree Recurrence

σi

Rσ

T = λ d

Y

i=1

1 Rσi

Ti + 1

A = 0 1 1 1

• b =

λ 1

• Occupancy ratio:

Rσ

T , PrX∼µT [Xv = 0 | σ]

PrX∼µT [Xv = 1 | σ]

v ui

G

Rv = PrI∼µG[v is occupied by I] PrI∼µG[v is unoccupied by I] =

ZG(v is occupied) ZG(v is unoccupied)

vi ui

Rv = PrI∼µG[v is occupied by I] PrI∼µG[v is unoccupied by I] =

ZG(v is occupied) ZG(v is unoccupied)

v1 vd

G’

λ1/d λ1/d λ1/d

= ZG0(v1, . . . , vd = • · · · •) ZG0(v1, . . . , vd = · · · )

vi ui

Rv = PrI∼µG[v is occupied by I] PrI∼µG[v is unoccupied by I] =

ZG(v is occupied) ZG(v is unoccupied)

τi : v1, . . . , vi−1 = · · · vi+1, . . . , vd = • · · · •

G’

λ1/d

: occupied : unoccupied = ZG0(v1, . . . , vd = • · · · •) ZG0(v1, . . . , vd = · · · )

=

d

Y

i=1

ZG0(v1, . . . , vd =

i−1

z }| { · · · •

d−i

z }| {

• · · · •)

ZG0(v1, . . . , vd = · · · | {z }

i−1

• · · · •

| {z }

d−i

)

=

d

Y

i=1

Rτi

G0,vi

Rv = PrI∼µG[v is occupied by I] PrI∼µG[v is unoccupied by I] =

ZG(v is occupied) ZG(v is unoccupied) Gi

ui

: occupied : unoccupied

τi : v1, . . . , vi−1 = · · · vi+1, . . . , vd = • · · · •

= ZG0(v1, . . . , vd = • · · · •) ZG0(v1, . . . , vd = · · · )

=

d

Y

i=1

ZG0(v1, . . . , vd =

i−1

z }| { · · · •

d−i

z }| {

• · · · •)

ZG0(v1, . . . , vd = · · · | {z }

i−1

• · · · •

| {z }

d−i

)

=

d

Y

i=1

Rτi

G0,vi

=

d

Y

i=1

λ1/d Rτi

Gi,ui + 1 = λ d

Y

i=1

1 Rτi

Gi,ui + 1

v v v

Tree Recurrence

µ(I) ∝ λ|I| hardcore model:

independent set I in G

v

v1 v2 v3

: occupied : unoccupied

Rσ

v =

Pr[v is occupied | σ] Pr[v is unoccupied | σ]

Rσ1

1 =

Pr[v1 is occupied | σ1] Pr[v1 is unoccupied | σ1]

Rσ2

2 =

Pr[v2 is occupied | σ2] Pr[v2 is unoccupied | σ2] Rσ3

3 =

Pr[v3 is occupied | σ3] Pr[v3 is unoccupied | σ3]

Rσ

v = λ d

Y

i=1

1 1 + Rσi

i

2

Self-Avoiding Walk Tree

1 2 3 4 5 6 6 5 5 6 4 3 3 5 6 5 6 4 1

v

T = T(G, v)

6 6 6 6 6

G

(Godsil 1981; Weitz 2006)

if cycle closing edge < cycle starting edge

σ

if cycle closing edge > cycle starting edge

SSM in trees of max-deg ≤Δ:

SSM and approx. inference in graphs with max-deg ≤Δ

µσ

v

µσ

root

=

in G in T

Rσ

T = λ d

Y

i=1

1 1 + Rσi

Ti

2

Self-Avoiding Walk Tree

1 2 3 4 5 6 6 5 5 6 4 3 3 5 6 5 6 4 1

v

T = T(G, v)

6 6 6 6 6

G

(Godsil 1981; Weitz 2006)

if cycle closing edge < cycle starting edge

σ

if cycle closing edge > cycle starting edge

hold for 2-spin systems µσ

v

µσ

root

=

in G in T

A = a00 a01 a10 a11

• b =

 b0 b1

• SSM in trees of max-deg ≤Δ:

SSM and approx. inference in graphs with max-deg ≤Δ

Rσ

T = b0

b1

d

Y

i=1

a00Rσi

Ti + a01

a10Rσi

Ti + a11

regular tree

v

σ: all leaves are occupied τ: all leaves are unoccupied

hardcore model:

Δ-

Correlation Decay

` uniqueness threshold:

λc(∆) = (∆ − 1)∆−1 (∆ − 2)∆

→ ∞ single-variate dynamical system:

f(x) = λ (1 + x)∆−1

hardcore recurrence:

Rσ

T = λ ∆−1

Y

i=1

1 1 + Rσi

Ti

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

unique fixed point: ˆ x = f(ˆ x)

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

|f 0(ˆ x)| > 1

( x+ = f(x−) x− = f(x+)

∃x− < ˆ x < x+

|f 0(ˆ x)| ≤ 1

|f 0(ˆ x)| ≤ 1

single-variate dynamical system:

f(x) = λ (1 + x)∆−1

λ ≤ λc(∆)

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

unique fixed point: ˆ x = f(ˆ x)

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

|f 0(ˆ x)| > 1

( x+ = f(x−) x− = f(x+)

∃x− < ˆ x < x+

|f 0(ˆ x)| ≤ 1

single-variate dynamical system:

f(x) = λ (1 + x)∆−1

λ ≤ (1 − δ)λc(∆)

|f 0(ˆ x)| ≤ 1 − Θ(δ)

Correlation Decay

hardcore recurrence:

∞

arbitrary initial values

` dynamical system:

Rσ

T = λ d

Y

i=1

1 1 + Rσi

Ti

variation at root

≤ (`)

∞ ∞ ∞

Rσi

Ti ∈ [0, ∞)

xi ∈ [0, ∞)

|f(~ x) f(~ x0)| = |hrf(~ ⇠), (~ x ~ x0)i|  krf(~ ⇠)k1k~ x ~ x0k1 = exp(−Ω(`))

if

(marginal ratio)

f(~ x) =

d

Y

i=1

1 1 + xi

d ≤ ∆ − 1

where

↵ , sup

~ x∈[0,∞)d d

X

i=1

• @f(~

x) @xi

• ≤ α · max

i

|xi − x0

i|

Mean-Value Thm:

λ <

1 ∆−1

= sup

~ x∈[0,∞)d λ d

Y

i=1

1 1 + xi

d

X

i=1

1 1 + xi

≤ λ · (∆ − 1)

|Rσ

T − Rτ T | =

Correlation Decay

hardcore recurrence:

f(~ x) =

d

Y

i=1

1 1 + xi

∞

arbitrary initial values

`

Rσ

T = λ d

Y

i=1

1 1 + Rσi

Ti

variation at root

≤ (`)

∞ ∞ ∞

Rσi

Ti ∈ [0, ∞)

xi ∈ [0, ∞)

d ≤ ∆ − 1

where

= exp(−Ω(`))

if :

|pσ

T − pτ T | ≤ |Rσ T − Rτ T |

SSM with exponential decay! R = 1−p

p

(marginal ratio)

dynamical system:

↵ , sup

~ x∈[0,∞)d d

X

i=1

• @f(~

x) @xi

• λ <

1 ∆−1

= sup

~ x∈[0,∞)d λ d

Y

i=1

1 1 + xi

d

X

i=1

1 1 + xi

≤ λ · (∆ − 1)

Correlation Decay

hardcore recurrence:

∞

arbitrary initial values

`

variation at root

≤ (`)

∞ ∞ ∞

xi ∈ [0, ∞)

d ≤ ∆ − 1

where

pσ

T =

1 1 + λ Qd

i=1 pσi Ti

pσi

Ti ∈ [0, 1]

f(~ x) = 1 1 + Qd

i=1 xi

unbounded!

when p 7! R = 1−p

p

: the decay rate α is reduced

(marginal probability)

dynamical system:

|f(~ x) f(~ x0)| = |hrf(~ ⇠), (~ x ~ x0)i|  krf(~ ⇠)k1k~ x ~ x0k1 ≤ α · max

i

|xi − x0

i|

Mean-Value Thm:

↵ , sup

~ x∈[0,1]d d

X

i=1

• @f(~

x) @xi

• =

sup

~ x∈[0,1]d f(~

x)(1 − f(~ x))

d

X

i=1

1 xi

• riginal:

potential (message): xi

yi yi = φ(xi)

f(~ x) =

d

Y

i=1

1 1 + xi

Mean Value Theorem:

(where , and denote )

ξi = φ(xi)

Φ(x) = φ0(x)

φ

The Potential Method

f φ(~ y) = (f(−1(y1), . . . , −1(yd)))

|f φ(~ y) − f φ(~ y0)| =

• hrf φ(~

⇠), (~ y ~ y0)i

•  krf φ(~

⇠)k1k~ y ~ y0k1

≤ f(~ x)Φ(f(~ x))

d

X

i=1

1 (1 + xi)Φ(xi)

Choose:

φ(x) = 2 sinh−1(√x) = 2 ln √x + √ x + 1

• Φ(x) = φ0(x) =

1

√

x(x+1)

so that

[Restrepo-Shin-Tetali-Vigoda-Yang 11] krf φ(~ ⇠)k1 =

d

X

i=1

• @f(~

x) @xi

• Φ(f(~

x)) Φ(xi)

f(~ x) =

d

Y

i=1

1 1 + xi

Mean Value Theorem:

(where , and denote )

ξi = φ(xi)

Φ(x) = φ0(x)

|f φ(~ y) − f φ(~ y0)| =

• hrf φ(~

⇠), (~ y ~ y0)i

• ≤ f(~

x)Φ(f(~ x))

d

X

i=1

1 (1 + xi)Φ(xi) φ(x) = 2 sinh−1(√x) = 2 ln √x + √ x + 1

• Φ(x) = φ0(x) =

1

√

x(x+1)

so that

f φ(~ y) = (f(−1(y1), . . . , −1(yd)))

• riginal:

potential:

≤ s d f(x) 1 + f(x) r dx 1 + x

f(x) = λ (1 + x)d

(where ) = s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi

(Jensen)

 ↵ · k~ y ~ y0k1

zi = − ln(x1 + 1)

let then

r xi xi + 1 = p ezi(e−zi − 1)

and is concave in zi

↵ =

d

X

i=1

• @f(~

x) @xi

• Φ(f(~

x)) Φ(xi)

f(~ x) = exp d X

i=1

zi !

↵ =

d

X

i=1

• @f(~

x) @xi

• Φ(f(~

x)) Φ(xi)

f(~ x) =

d

Y

i=1

1 1 + xi

Mean Value Theorem:

(where , and denote )

ξi = φ(xi)

Φ(x) = φ0(x)

|f φ(~ y) − f φ(~ y0)| =

• hrf φ(~

⇠), (~ y ~ y0)i

• ≤ f(~

x)Φ(f(~ x))

d

X

i=1

1 (1 + xi)Φ(xi) φ(x) = 2 sinh−1(√x) = 2 ln √x + √ x + 1

• Φ(x) = φ0(x) =

1

√

x(x+1)

so that

f φ(~ y) = (f(−1(y1), . . . , −1(yd)))

• riginal:

potential:

≤ s d f(x) 1 + f(x) r dx 1 + x

f(x) = λ (1 + x)d

(where ) = s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi

(Jensen) ≤ p |f 0(ˆ x)|

 ↵ · k~ y ~ y0k1

ˆ x = f(ˆ x) is fixpoint)

(where

= r dˆ x 1 + ˆ x

↵ =

d

X

i=1

• @f(~

x) @xi

• Φ(f(~

x)) Φ(xi)

f(~ x) =

d

Y

i=1

1 1 + xi

Mean Value Theorem:

(where , and denote )

ξi = φ(xi)

Φ(x) = φ0(x)

|f φ(~ y) − f φ(~ y0)| =

• hrf φ(~

⇠), (~ y ~ y0)i

• ≤ f(~

x)Φ(f(~ x))

d

X

i=1

1 (1 + xi)Φ(xi) φ(x) = 2 sinh−1(√x) = 2 ln √x + √ x + 1

• Φ(x) = φ0(x) =

1

√

x(x+1)

so that

f φ(~ y) = (f(−1(y1), . . . , −1(yd)))

• riginal:

potential:

≤ s d f(x) 1 + f(x) r dx 1 + x

f(x) = λ (1 + x)d

(where ) = s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi

(Jensen) ≤ p |f 0(ˆ x)|

 ↵ · k~ y ~ y0k1

ˆ x = f(ˆ x) is fixpoint)

(where

= r dˆ x 1 + ˆ x

↵ =

d

X

i=1

• @f(~

x) @xi

• Φ(f(~

x)) Φ(xi)

f(~ x) =

d

Y

i=1

1 1 + xi

Mean Value Theorem:

(where , and denote )

ξi = φ(xi)

Φ(x) = φ0(x)

|f φ(~ y) − f φ(~ y0)| =

• hrf φ(~

⇠), (~ y ~ y0)i

• ≤ f(~

x)Φ(f(~ x))

d

X

i=1

1 (1 + xi)Φ(xi) φ(x) = 2 sinh−1(√x) = 2 ln √x + √ x + 1

• Φ(x) = φ0(x) =

1

√

x(x+1)

so that

f φ(~ y) = (f(−1(y1), . . . , −1(yd)))

• riginal:

potential:

≤ s d f(x) 1 + f(x) r dx 1 + x

f(x) = λ (1 + x)d

(where ) = s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi

(Jensen) ≤ p |f 0(ˆ x)|

 ↵ · k~ y ~ y0k1

ˆ x = f(ˆ x) is fixpoint)

(where

< 1

(when )

λ < λc

f(~ x) =

d

Y

i=1

1 1 + xi

Mean Value Theorem:

|f φ(~ y) − f φ(~ y0)|

φ(x) = 2 sinh−1(√x) = 2 ln √x + √ x + 1

• Φ(x) = φ0(x) =

1

√

x(x+1)

so that

f φ(~ y) = (f(−1(y1), . . . , −1(yd)))

• riginal:

potential:

 ↵ · k~ y ~ y0k1

∞

arbitrary initial values

`

∞ ∞ ∞

xi ∈ [0, ∞)

≤ α`−1|φ(λ) − φ(0)|

|pσ

T − pτ T | ≤ |Rσ T − Rτ T |

≤ |φ (Rσ

T ) − φ (Rτ T )| · sup x∈[0,λ]

1 Φ(x)

when λ < λc :

≤ exp(−Ω(`)) · |() − (0)|

≤ exp(−Ω(`)) · |() − (0)| · sup

x∈[0,λ]

1 Φ(x)

= exp(−Ω(`)) SSM with exponential decay!

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

y

f(x) = λ (1 + x)d

x

φ

g(y) = φ(f(φ−1(y)))

φ

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

f(x) = λ (1 + x)d |f 0(x)| > 1

|f 0(ˆ x)| < 1

|g0(y)| < 1 everywhere!

Always contract! (if )

|f 0(ˆ x)| < 1

φ(x) = 2 sinh−1(√x)

= |f 0(ˆ x)|

ˆ x = f(ˆ x)

(at the fixpoint )

= 1

(when )

λ = λc =

dd (d−1)d+1

|g0(y)| = |f 0(x)|φ0(f(x)) φ0(x)

g(y) = 2arcsinh ✓√ λ · cosh ⇣y 2 ⌘−d◆

Good Potential Functions

, when :

λ = λc =

dd (d−1)d+1

monotone φ(x)

Φ(x) = φ0(x)

denote

f(x) = λ (1 + x)d

consider

|f 0(x)|φ0(f(x)) φ0(x) = d f(x)Φ(f(x)) (x + 1)Φ(x)

is maximized at fixpoint ˆ

x = f(ˆ x)

:

Good Potential Functions

, when :

λ = λc =

dd (d−1)d+1

monotone φ(x)

Φ(x) = φ0(x)

denote

f(x) = λ (1 + x)d

consider

|f 0(x)|φ0(f(x)) φ0(x) = d f(x)Φ(f(x)) (x + 1)Φ(x)

is maximized at fixpoint ˆ

x = f(ˆ x)

:

Good Potential Functions

, when :

λ = λc =

dd (d−1)d+1

monotone φ(x)

Φ(x) = φ0(x)

denote

f(x) = λ (1 + x)d

consider

|f 0(x)|φ0(f(x)) φ0(x) = d f(x)Φ(f(x)) (x + 1)Φ(x)

is maximized at fixpoint ˆ

x = f(ˆ x)

: : h(z) =

ez Φ(e−z − 1)

is concave

Good Potential Functions

, when :

λ = λc =

dd (d−1)d+1

monotone φ(x)

Φ(x) = φ0(x)

denote

f(x) = λ (1 + x)d

consider

|f 0(x)|φ0(f(x)) φ0(x) = d f(x)Φ(f(x)) (x + 1)Φ(x)

is maximized at fixpoint ˆ

x = f(ˆ x)

: : h(z) =

ez Φ(e−z − 1)

is concave

Good Potential Functions

, when :

λ = λc =

dd (d−1)d+1

monotone φ(x)

Φ(x) = φ0(x)

denote

f(x) = λ (1 + x)d

consider

|f 0(x)|φ0(f(x)) φ0(x) = d f(x)Φ(f(x)) (x + 1)Φ(x)

is maximized at fixpoint ˆ

x = f(ˆ x)

: : h(z) =

ez Φ(e−z − 1)

is concave : |φ(λ) − φ(0)| · sup

x∈[0,λ]

1 Φ(x) < ∞

Φ(x) = 1 p x(x + 1) [Li-Lu-Y. 13]:

Φ(x) = 1 ∆x + 1

Φ(x) = 1 (x + 1)(1 +

ln(x+1) 2ˆ xc−ln(ˆ xc+1))

[Restrepo-Shin-Tetali-Vigoda-Yang 11]: [Peters-Regts 17]:

[Li-Lu-Y. 12]: Φ(x) = x−

∆c(λ) 2(∆c(λ)−1)

An Abstract View

BP operator F : RN

≥0 → RN ≥0

Jacobian : J = J(~ x) converges to the unique fixpoint when

< 1

J(~ x)1 < 1 Jij =

• @Fi(~

x) @xj

• ∀i :

X

j

• @Fi(~

x) @xj

An Abstract View

BP operator F : RN

≥0 → RN ≥0

Jacobian : J = J(~ x) converges to the unique fixpoint when

φ : R≥0 → R≥0

monotone

Φ(x) = φ0(x)

denote Jφ = Jφ(~ x) :

Φ(Fi(~ x)) Φ(xj)

< 1

1 Φ(xj) < 1 Φ(Fi(~ x))

J φ(~ x)1 < 1 Jij =

• @Fi(~

x) @xj

• Jφ

ij =

• @Fi(~

x) @xj

• Φ(Fi(~

x)) Φ(xj)

∀i : X

j

• @Fi(~

x) @xj

• ∀i :

X

j

• @Fi(~

x) @xj

An Abstract View

BP operator F : RN

≥0 → RN ≥0

Jacobian : J = J(~ x) J(~ x)v(~ x) < v(F(~ x))

v : RN

≥0 → RN ≥0

for some where v(~ x) = (vi(xi))i

vi : R≥0 → R≥0

for converges to the unique fixpoint when

vi(x) = p x(x + 1)

= ( Fi(~

x) xj+1

j 2 Ni j 62 Ni

Jij =

• @Fi(~

x) @xj

• v = v1 ⊕ v2 ⊕ · · · ⊕ vN

2

Approximate Counting

1 2 3 4 5 6 6 5 5 6 4 3 3 5 6 5 6 4 1

v

T = T(G, v)

6 6 6 6 6

G

if cycle closing edge < cycle starting edge

σ

if cycle closing edge > cycle starting edge

µσ

v

µσ

root

=

in G in T

Rσ

T = λ d

Y

i=1

1 1 + Rσi

Ti

truncated

> ` levels

µσ

v is approximated within

additive error αΩ(`) ≤ O( ✏

n)

ε-approx. of ZG(λ)

`

2

Approximate Counting

1 2 3 4 5 6 6 5 5 6 4 3 3 5 6 5 6 4 1

v

T = T(G, v)

6 6 6 6 6

G

if cycle closing edge < cycle starting edge

σ

if cycle closing edge > cycle starting edge

µσ

v

µσ

root

=

in G in T

Rσ

T = λ d

Y

i=1

1 1 + Rσi

Ti

truncated

> ` levels

µσ

v is approximated within

additive error αΩ(`) ≤ O( ✏

n)

ε-approx. of ZG(λ)

`

2

Approximate Counting

1 2 3 4 5 6 6 5 5 6 4 3 3 5 6 5 6 4 1

v

T = T(G, v)

6 6 6 6 6

G

if cycle closing edge < cycle starting edge

σ

if cycle closing edge > cycle starting edge

µσ

v

µσ

root

=

in G in T

Rσ

T = λ d

Y

i=1

1 1 + Rσi

Ti

truncated

> ` levels

µσ

v is approximated within

additive error αΩ(`) ≤ O( ✏

n)

ε-approx. of ZG(λ)

`

when λ ≤ (1 − δ)λc(∆) for constant δ > 0 total time cost:

≤ n∆` = ⇣n ✏ ⌘O(log ∆)

Counting Independent Set

hardcore model: uniqueness regime:

undirected graph G(V,E) with max-degree ∆

deterministic approximate counting:

λ ≤ (1 − δ)λc(∆)

for constant δ > 0 ε-approx.:

(1 − ✏)ZG() ≤ ˆ Z ≤ (1 + ✏)ZG()

randomized approximate counting:

time cost time by Glauber dynamics

∆ ≥ ∆0(δ)

girth ≥7

)

[Efthymiou-Hayes-Štefankovič-Vigoda-Y. 16]

FPTAS when Δ = O(1) ˜ O

• n2

nO(log ∆)

Counting Matchings

monomer-dimer model: TSAW:

G(V,E) with max-degree ∆

deterministic approximate counting:

ε-approx.:

(1 − ✏)ZG() ≤ ˆ Z ≤ (1 + ✏)ZG()

randomized approximate counting:

time cost FPTAS when Δ = O(1)

SSM with rate: 1 − Θ

✓ 1 √ λ∆ ◆

f(~ x) = 1 1 + Pd

i=1 xi

ZG(λ) = X

M: matching in G

λ|M|

Poly(n,1/ε) time by Jerrum-Sinclair chain

[Bayati-Gamarnik-Katz-Nair-Tetali 2007] [Godsil 1981]

nO(

√ λ∆ log ∆)

Correlation Decay Method

• Deterministic approximate counting by recurrences

for marginal probabilities:

• SAW-tree; [Weitz ’06] [Bayati Gamarnik Katz Nair Tetali 2007]…
• computation tree. [Gamarnik Katz ’07] [Nair Tetali’07]…
• Deterministic approximate counting by non-

vanishing polynomials.

[Peters Regts ’17] [Bezakova Galanis Goldberg Štefankovič 18]…

• Rapid mixing of dynamics by coupling.

[Mossel Sly ’13] [Efthymiou Hayes Štefankovič Vigoda Y. ’16]…

Thank you!