Correlation Decay up to Uniqueness in Spin Systems Yitong Yin - - PowerPoint PPT Presentation

Correlation Decay up to Uniqueness in Spin Systems

Yitong Yin Nanjing University Joint work with Liang Li (Peking Univ) Pinyan Lu (Microsoft research Asia)

Two-State Spin System

A = A0,0 A0,1 A1,0 A1,1

• =

β 1 1 γ

• 2 states {0,1}

configuration σ : V → {0, 1} graph G=(V,E) edge activity:

β γ 1

external field:

1 λ

b = (b0, b1) = (λ, 1)

w(σ) =

• (u,v)∈E

Aσ(u),σ(v)

• v∈V

bσ(v)

Two-State Spin System

A = A0,0 A0,1 A1,0 A1,1

• =

β 1 1 γ

• 2 states {0,1}

configuration σ : V → {0, 1} graph G=(V,E)

w(σ) =

• (u,v)∈E

Aσ(u),σ(v)

• v∈V

bσ(v)

Gibbs measure:

=

• σ∈{0,1}V

w(σ)

partition function:

Z(G)

(σ) = w(σ) Z(G)

b = (b0, b1) = (λ, 1)

w(σ) =

• (u,v)∈E

Aσ(u),σ(v)

• v∈V

bσ(v) =

• σ∈{0,1}V

Z(G) w(σ)

A = A0,0 A0,1 A1,0 A1,1

• =

β 1 1 γ

• partition function:

marginal probability:

b = (b0, b1) = (λ, 1)

• σ [σ(v) = 0 | σ(v1), . . . , σ(vk)]

(τ) =

n

• k=1
• σ [σ(vk) = τ(vk) | σ(vi) = τ(vi), 1 ≤ i < k]

= w(τ) Z

1/poly(n) additive error for marginal in poly-time

FPTAS for Z(G)

ferromagnetic:

[Jerrum-Sinclair’93]

βγ > 1

FPRAS:

[Goldberg-Jerrum-Paterson’03]

anti-ferromagnetic:

βγ < 1

hardcore model: Ising model:

β = 0, γ = 1 β = γ

∃ FPTAS for graphs

• f max-degree Δ

(β, γ, λ) lies in the interior of uniqueness region of Δ-regular tree [Sinclair-Srivastava-Thurley’12] [Weitz’06]

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3

• 0< , <1

= 1

[Goldberg-Jerrum-Paterson’03]

β γ

FPRAS for arbitrary graphs

[Li-Lu-Y. ’12]: no external field

FPTAS for arbitrary graphs

anti-ferromagnetic:

βγ < 1

∃ FPTAS for graphs of max-degree Δ (β, γ, λ) lies in the interiors of uniqueness regions of d-regular trees for all d ≤ Δ. ∄ FPRAS for graphs of max-degree Δ

(β, γ, λ) lies in the interiors of non-uniqueness regions of d-regular trees for some d ≤ Δ.

assuming NP ≠RP

[Sly-Sun’12] [Galanis-Stefankovic-Vigoda’12]: bounded Δ or Δ=∞

Uniqueness Condition

(d+1)-regular tree

reg. tree

t

arbitrary boundary config

marginal at root ± exp(-t) fd(x) = λ βx + 1 x + γ d

ˆ xd = fd(ˆ xd) |f

d(ˆ

xd)| < 1

anti-ferromagnetic:

βγ < 1

∃ FPTAS for graphs of max-degree Δ ∄ FPRAS for graphs of max-degree Δ

assuming NP ≠RP

[Sly-Sun’12] [Galanis-Stefankovic-Vigoda’12]: bounded Δ or Δ=∞ ∀d < ∆, |f

d(ˆ

xd)| < 1 ∃d < ∆, |f

d(ˆ

xd)| > 1

fd(x) = λ βx + 1 x + γ d

Correlation Decay

strong spatial mixing (SSM): B

∂B

G v

∀σ∂B, τ∂B ∈ {0, 1}∂B Λ

• σ [σ(v) = 0 | σ∂B] ≈

σ [σ(v) = 0 | τ∂B]

• σ [σ(v) = 0 | σ∂B, σΛ] ≈

σ [σ(v) = 0 | τ∂B, σΛ]

t error < exp (-t) exponential correlation decay weak spatial mixing (WSM): uniqueness: WSM in reg. tree

1

Self-Avoiding Walk Tree

due to Weitz (2006)

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

G=(V,E) v

T = T(G, v) 6 6 6 6 6 σΛ preserve the marginal dist. at v

SSM FPTAS

• n bounded degree graphs:

SSM in Δ-reg. tree SSM in graphs

• f degree ≤Δ

in reg. trees: WSM SSM

hardcore model, anti-ferro Ising model:

β, γ < 1

(for )

WSM in Δ-reg. tree SSM in graphs

• f degree ≤Δ

SSM in trees

• f degree ≤Δ

SSM in graphs

• f degree ≤Δ

SSM in trees

• f degree ≤Δ

SSM in Δ-reg. tree

SAW-tree fixing hardcore model, anti-ferro Ising model:

β, γ < 1

(for )

in reg. trees: WSM SSM

SSM in trees

• f degree ≤Δ

SSM in graphs

• f degree ≤Δ

SSM in trees

• f degree ≤Δ

SSM in Δ-reg. tree

SAW-tree for general anti-ferro 2-state spin systems:

in reg. trees: WSM SSM

WSM in d-reg. trees for d≤Δ SSM in trees

• f degree ≤Δ

SSM in trees

• f degree ≤Δ

SSM in graphs

• f degree ≤Δ

in reg. trees: WSM SSM

WSM in d-reg. trees for d≤Δ SSM in trees

• f degree ≤Δ

SSM in trees

• f degree ≤Δ

SSM in graphs

• f degree ≤Δ

for general anti-ferro 2-state spin systems:

SSM in graphs

• f degree ≤Δ

WSM in d-reg. trees for d≤Δ

v T v1 v2 vd T1 Td x = f(x1, . . . , xd) = λ

d

• i=1

βxi + 1 xi + γ

• x = [σ(v) = 0 | σΛ]

[σ(v) = 1 | σΛ] x n ∈ [0, ∞) x ∈ [R, R + δ] δ = (−Ω(n))

Potential Analysis

x f(x) f Fn(x + δ) − Fn(x) = F

n(x0) · δ

Fn(x) = f f · · · f

• n

(x) = δ ·

n1

• t=0

f (xt) xt = f(xt−1) = δ · Φ(x0) Φ(xn) ·

n1

• t=0

Φ(f(xt)) Φ(xt) f (xt)

Potential Analysis

x f(x) y g(y) g f φ φ−1 φ(x) = Φ(x) G

n(x0) = n1

• t=0

g(xt) =

n1

• t=0

[φ(f(φ1(yt))] =

n1

• t=0

Φ(f(xt)) Φ(xt) f (xt) Gn(x) = g g · · · g

• n

(x) Gn(x + δ) − Gn(x) = G

n(x0) · δ

v v1 v2 vd x1 xd x v v1 v2 vd y y1 yd φ

f(x1, . . . , xd) = λ

d

• i=1

βxi + 1 xi + γ

• g(y1, . . . , yd) g(y1 + δ1, . . . , yd + δd)

= φ(f(φ−1(y1), . . . , φ−1(yd))) · (δ1, . . . , δd)

φ(x) = Φ(x) = 1

• x(βx + 1)(x + γ)

≤ α(d; x1, . . . , xd) ·

1≤i≤d{δi}

+δ1 +δd

amortized decay rate

g(y1, . . . , yd) = φ(f(φ−1(y1), . . . , φ−1(yd)))

α(d; x1, , xd) αd(x) α(d; x, . . . , x

d

)

Cauchy-Schwarz arithmetic and geometric means

amortized decay rate

= Φ(f(x)) Φ(x) |f (x)|

= (1 − βγ)

• λ d

i=1 βxi+1 xi+γ

1

2

• βλ d

i=1 βxi+1 xi+γ + 1

1

2

λ d

i=1 βxi+1 xi+γ + γ

1

2 ·

d

• i=1

x

1 2

i

(βxi + 1)

1 2 (xi + γ) 1 2

=

• d(1 − βγ)x

(βx + 1)(x + γ)

• d(1 − βγ)λ
• βx+1

x+γ

d

• βλ
• βx+1

x+γ

d + 1 λ

• βx+1

x+γ

d + γ

αd(x)

=

• d(1 − βγ)x

(βx + 1)(x + γ)

• d(1 − βγ)λ
• βx+1

x+γ

d

• βλ
• βx+1

x+γ

d + 1 λ

• βx+1

x+γ

d + γ

• fd(x) = λ

βx + 1 x + γ d

ˆ xd = fd(ˆ xd) |f

d(ˆ

xd)| < 1

≤

• d(1 − βγ)ˆ

x (βˆ x + 1)(ˆ x + γ) =

• |f

d(ˆ

xd)|

=

• d(1 − βγ)x

(βx + 1)(x + γ)

• d(1 − βγ)fd(x)

(βfd(x) + 1) (fd(x) + γ)

v v1 v2 vd y y1 yd +δ1 +δd +δ

δ ≤ α ·

1≤i≤d{δi}

α < 1

anti-ferromagnetic:

βγ < 1 ∃ FPTAS for graphs of max-degree Δ ∀d < ∆, |f

d(ˆ

xd)| < 1

fd(x) = λ βx + 1 x + γ d

SSM in graphs of max-degree Δ SSM in trees of max-degree Δ bounded Δ |f

∆(ˆ

x∆)| < 1 SSM in Δ-reg. tree SSM in reg. trees: WSM

[Weitz’06] + [Sinclair-Srivastava-Thurley’12] + translation

uniqueness:

|f (ˆ x)| < 1

amortized decay: requirement of potential function:

|f (x)| · Φ (f(x)) Φ(x) < 1

f(x) = λ βx + 1 x + γ d

ˆ x = f(ˆ x)

f(x) = λ βx + 1 x + γ d

ˆ x = f(ˆ x)

amortized decay: requirement of potential function: phase-trans: |f (ˆ

x)| = 1

• f (x) · Φ (f(x))

Φ(x)

• x=ˆ

x

= 0 (ln(Φ(ˆ x))) = −f (ˆ x) 2 = 1 2 1 ˆ x + 1 ˆ x + γ + β βˆ x + 1

• |f (x)| · Φ (f(x))

Φ(x) |f (ˆ x)| · Φ (f(ˆ x)) Φ(ˆ x) = 1

requirement of potential function:

(ln(Φ(ˆ x))) = 1 2 1 ˆ x + 1 ˆ x + γ + β βˆ x + 1

• (ln(Φ(x))) = 1

2 1 x + 1 x + γ + β βx + 1

• strengthen the requirement:

Φ(x) = C

• x(βx + 1)(x + γ)

Computationally Efficient Correlation Decay

for some

αd(x) =

• d(1 − βγ)x

(βx + 1)(x + γ)

• d(1 − βγ)fd(x)

(βfd(x) + 1) (fd(x) + γ)

v v1 v2 vd y y1 yd +δ1 +δd +δ δ ≤ αd(x) ·

1≤i≤d{δi}

Computationally Efficient Correlation Decay

for some

for small

• ne-step recursion decays

for large

• ne-step recursion decays

behaves like steps!

αd(x) v v1 v2 vd y y1 yd +δ1 +δd +δ δ ≤ αd(x) ·

1≤i≤d{δi}

Computationally Efficient Correlation Decay

v v1 v2 vd “span” d leaves M-ary

• ld metric

v v1 v2 vd

new metric correlation decay in size grows exponentially:

distance = O(log n) 1/poly-precision in poly-time

anti-ferromagnetic:

βγ < 1 ∃ FPTAS for graphs of max-degree Δ bounded Δ or Δ=∞ WSM in d-reg. trees for d ≤ Δ

hardcore model Ising model

[Sinclair-Srivastava-Thurley’12] [Weitz’06]

uniqueness condition:

WSM in Δ-reg. tree

fd(x) = λ βx + 1 x + γ d

ˆ xd = fd(ˆ xd) |f

d(ˆ

xd)| < 1

500 1000 1500 2000 0.5 1.0 1.5

d |f

d(ˆ

xd)| due to [Guo’12]

β, γ ≤ 1

γ > 1

βγ < 1

monotone single-peak

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3

• 0< , <1

= 1

uniqueness threshold

threshold achieved by heatbath random walk

β

γ

β, γ ≤ 1

WSM in Δ-reg. tree ⇒ WSM in d-reg. tree for d ≤ Δ WSM in D-reg. tree ⇒ WSM in all d-reg. trees

γ > 1

βγ < 1

D

500 1000 1500 2000 0.5 1.0 1.5

d |f

d(ˆ

xd)|

β, γ ≤ 1

γ > 1

βγ < 1

monotone single-peak

(β, γ, λ) that WSM in Δ-reg. tree but not in (Δ-1)-reg. tree

1

SSM in reg. trees: WSM

[Weitz’06] + [Sinclair-Srivastava-Thurley’12] + translation

SSM in graphs

• f degree ≤Δ

SSM in Δ-reg. tree

Open Problems

• Characterization of SSM in ferromagnetic 2-state spin

systems.

• SSM in multi-state spin systems:
• difficulty: no SAW-tree;
• implications: WSM vs. SSM in reg. trees, monotonicity of

WSM/SSM w.r.t degree.

• Apply potential analysis and computationally efficient

correlation decay to other problems.

Thank you!