Rapid mixing and Markov bases Tobias Windisch OvGU Magdeburg AMS - - PowerPoint PPT Presentation

Rapid mixing and Markov bases

Tobias Windisch

OvGU Magdeburg

AMS Sectional Meeting, Chicago, USA October 4, 2015

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Integer points of polytopes

Fibers

Let A ∈ Zm×d and b ∈ Zm, the b-fiber of A is

FA,b := {u ∈ Nd : A · u = b}.

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Integer points of polytopes

Fibers

Let A ∈ Zm×d and b ∈ Zm, the b-fiber of A is

FA,b := {u ∈ Nd : A · u = b}. Fiber graphs

Let M ⊆ ker(A) ∩ Zd. The graph FA,b(M) has vertices FA,b and u, v ∈ FA,b are adjacent if u − v ∈ ±M.

Markov bases M is a Markov basis if FA,b(M) is connected for all b ∈ Zd.

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Integer points of polytopes

Fibers

Let A ∈ Zm×d and b ∈ Zm, the b-fiber of A is

FA,b := {u ∈ Nd : A · u = b}. Fiber graphs

Let M ⊆ ker(A) ∩ Zd. The graph FA,b(M) has vertices FA,b and u, v ∈ FA,b are adjacent if u − v ∈ ±M.

Markov bases M is a Markov basis if FA,b(M) is connected for all b ∈ Zd.

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Integer points of polytopes

Fibers

Let A ∈ Zm×d and b ∈ Zm, the b-fiber of A is

FA,b := {u ∈ Nd : A · u = b}. Fiber graphs

Let M ⊆ ker(A) ∩ Zd. The graph FA,b(M) has vertices FA,b and u, v ∈ FA,b are adjacent if u − v ∈ ±M.

Markov bases M is a Markov basis if FA,b(M) is connected for all b ∈ Zd. Theorem (Diaconis, Sturmfels, 1996) M is a Markov basis of A if and only if IM = IA.

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Walking randomly on fibers

Algorithm (Simple Walk) Input: M Markov basis, u0 ∈ FA,b, r ∈ N Output: Uniform sample from FA,b

◮ FOR i = 1..r

◮ Sample m ∈ ±M uniformly ◮ IF ui + m ∈ Nd ◮ THEN ui = ui−1 + m ◮ ELSE ui = ui−1

◮ RETURN ur

→ This gives an irreducible and aperiodic Markov chain. Convergence Theorem of Markov chains

◮ After r = O

• −1

log(|λ|)

• steps, ur can be regarded as an uniform sample

from FA,b

◮ −1 < λ < 1: SLEM of transition matrix of the random walk.

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Analysing the speed

What is fast? (Gn)n∈N sequence of graphs with simple walk

◮ Expanders: number of steps needed is independent of n ◮ Rapid mixing: number of steps needed is polynomial in log |V(Gn)|

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Analysing the speed

What is fast? (Gn)n∈N sequence of graphs with simple walk

◮ Expanders: number of steps needed is independent of n ◮ Rapid mixing: number of steps needed is polynomial in log |V(Gn)|

→ Consider (n · b)n∈N ⊆ NA

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Analysing the speed

What is fast? (Gn)n∈N sequence of graphs with simple walk

◮ Expanders: number of steps needed is independent of n ◮ Rapid mixing: number of steps needed is polynomial in log |V(Gn)|

→ Consider (n · b)n∈N ⊆ NA Expander and rapid mixing

For any n ∈ N, let λn be the SLEM of the simple walk on FA,n·b(Mn)

◮ (FA,n·b(Mn))n∈N is an expander if λn ≤ 1 − ǫ for all n ∈ N ◮ (FA,n·b(Mn))n∈N is rapidly mixing if λn ≤ 1 − 1 p(log n) for all n ∈ N

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Analysing the speed

What is fast? (Gn)n∈N sequence of graphs with simple walk

◮ Expanders: number of steps needed is independent of n ◮ Rapid mixing: number of steps needed is polynomial in log |V(Gn)|

→ Consider (n · b)n∈N ⊆ NA Expander and rapid mixing

For any n ∈ N, let λn be the SLEM of the simple walk on FA,n·b(Mn)

◮ (FA,n·b(Mn))n∈N is an expander if λn ≤ 1 − ǫ for all n ∈ N ◮ (FA,n·b(Mn))n∈N is rapidly mixing if λn ≤ 1 − 1 p(log n) for all n ∈ N

Algebraic statistics

Use the same moves for every right-hand side: Mn := M

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Fiber walks can be slow!

No expander

◮ H :=

 

I3 I3

−13

I3 I3

−13

1 1

 

◮ Ray {n · e7 : n ∈ N}

2 4 6 8 10 0.4 0.6 0.8 1 n SLEM Graver Gr¨

• bner

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Fiber walks can be slow!

No expander

◮ H :=

 

I3 I3

−13

I3 I3

−13

1 1

 

◮ Hemmecke-Ray {n · e7 : n ∈ N}

2 4 6 8 10 0.4 0.6 0.8 1 n SLEM Graver Gr¨

• bner

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Fiber walks can be slow!

No expander

◮ H :=

 

I3 I3

−13

I3 I3

−13

1 1

 

◮ Hemmecke-Ray {n · e7 : n ∈ N} ◮ limn→∞ λ(Gr¨

• bner) = 1

◮ limn→∞ λ(Graver) = 1

2 4 6 8 10 0.4 0.6 0.8 1 n SLEM Graver Gr¨

• bner

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Fiber walks can be slow!

No expander

◮ H :=

 

I3 I3

−13

I3 I3

−13

1 1

 

◮ Hemmecke-Ray {n · e7 : n ∈ N} ◮ limn→∞ λ(Gr¨

• bner) = 1

◮ limn→∞ λ(Graver) = 1

2 4 6 8 10 0.4 0.6 0.8 1 n SLEM Graver Gr¨

• bner

Not rapidly mixing

◮ A := (1, 1) ∈ Z1×2, M := {(1, −1)t}, b = 1 ◮ FA,n(M) ∼

=

. . .

⇒ SLEM of FA,n(M) ≥ 1 −

1 n+1

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Fiber walks can be slow!

No expander

◮ H :=

 

I3 I3

−13

I3 I3

−13

1 1

 

◮ Hemmecke-Ray {n · e7 : n ∈ N} ◮ limn→∞ λ(Gr¨

• bner) = 1

◮ limn→∞ λ(Graver) = 1

2 4 6 8 10 0.4 0.6 0.8 1 n SLEM Graver Gr¨

• bner

Not rapidly mixing

◮ A := (1, 1) ∈ Z1×2, M := {(1, −1)t}, b = 1 ◮ FA,n(M) ∼

=

. . .

⇒ SLEM of FA,n(M) ≥ 1 −

1 n+1 ◮ Is this (asymptotic) behaviour typical for fiber walks?

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Edge-expansion

Edge-expansion, Cheeger constant

h(G) := min|U|≤ 1

2|V|

#(edges leaving U) |U|

Expander Mixing Lemma

Simple walk on d-regular graph G: 1 − h(G)

d

≤ |λ| ≤ 1 − h(G)2

d2

Why is this convenient for fibers? FA,n1·b(M) FA,n2·b(M)

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How fast are fiber walks?

Theorem (W.; 2015)

Let M be a Markov basis of A and let b ∈ NA with dim(FA,b) > 0. Then:

◮ limn→∞ h(FA,n·b(M)) = 0

(no expander)

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How fast are fiber walks?

Theorem (W.; 2015)

Let M be a Markov basis of A and let b ∈ NA with dim(FA,b) > 0. Then:

◮ limn→∞ h(FA,n·b(M)) = 0

(no expander)

◮ h(FA,n·b(M)) ∈ O( 1 n)

(not rapidly mixing)

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How fast are fiber walks?

Theorem (W.; 2015)

Let M be a Markov basis of A and let b ∈ NA with dim(FA,b) > 0. Then:

◮ limn→∞ h(FA,n·b(M)) = 0

(no expander)

◮ h(FA,n·b(M)) ∈ O( 1 n)

(not rapidly mixing)

Graver vs. Gr¨

• bner vs. Markov

Using more structural moves does not improve mixing asymptotically.

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How fast are fiber walks?

Theorem (W.; 2015)

Let M be a Markov basis of A and let b ∈ NA with dim(FA,b) > 0. Then:

◮ limn→∞ h(FA,n·b(M)) = 0

(no expander)

◮ h(FA,n·b(M)) ∈ O( 1 n)

(not rapidly mixing)

Graver vs. Gr¨

• bner vs. Markov

Using more structural moves does not improve mixing asymptotically.

A way out?

Use an “adapted” Markov basis Mb

n for

every FA,n·b and control h(FA,n·b(Mb

n))

|Mb

n|

.

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How fast are fiber walks?

Theorem (W.; 2015)

Let M be a Markov basis of A and let b ∈ NA with dim(FA,b) > 0. Then:

◮ limn→∞ h(FA,n·b(M)) = 0

(no expander)

◮ h(FA,n·b(M)) ∈ O( 1 n)

(not rapidly mixing)

Graver vs. Gr¨

• bner vs. Markov

Using more structural moves does not improve mixing asymptotically.

A way out?

Use an “adapted” Markov basis Mb

n for

every FA,n·b and control h(FA,n·b(Mb

n))

|Mb

n|

.

◮ In general, |Mb n| = O(log n) does not suffice to obtain rapid mixing.

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Can we construct expanders on fibers?

Adaption

Let M = {m1, . . . , mr} ⊆ Zd be a Markov basis , n ∈ N, and b ∈ NA.

Mb

n :=

  

r

• j=1

λjmj : λ1, . . . , λr ∈ Z,

r

• j=1

|λj| ≤ Diam(FA,n·b(M))    .

SLEM: 1 − |FA,n·b|

|Mb

n| 8 / 11

Can we construct expanders on fibers?

Adaption

Let M = {m1, . . . , mr} ⊆ Zd be a Markov basis , n ∈ N, and b ∈ NA.

Mb

n :=

  

r

• j=1

λjmj : λ1, . . . , λr ∈ Z,

r

• j=1

|λj| ≤ Diam(FA,n·b(M))    .

SLEM: 1 − |FA,n·b|

|Mb

n|

Theorem (W.; 2015)

Diam(FA,nb(M)) ∈ O(n) ⇒ (FA,njb(Mb

nj))j∈N is an expander.

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How to interpret this?

0.8 0.9 1 n SLEM conventional adapted

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Computational results

Sampling from Mb

i ◮ Sample coefficients λi ∈ [li, ui] and use the move r i=1 λimi.

Hemmecke-Ray FH3,n·e7(Gr¨

• bnere7

n )

n = 103: 3.551.720 n = 106: 4.058.733 n = 1050: 4.059.281

3 × 3 independence model FA33,n·16(M16

i )

n = 1010: 21.062.343 n = 10100: 37.255.074 n = 101000: 37.255.074

Adapted vs. conventional

◮ A = (1, 1, 1) ◮ M = {e1 − e2, e1 − e3} ◮ bn = n · 1

5 10 15 20 0.8 0.9 1 n SLEM M M1

2n 10 / 11

Wrap up

This talk was about

◮ Fiber walks are (asymptotically) slow. ◮ Adaption of Markov basis can lead to expanders.

Open problems

◮ How can mixing behaviour be improved for single fibers? ◮ What happens with the SLEM when using the Metropolis-Hastings

Algorithm? arXiv:1505.03018

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Wrap up

This talk was about

◮ Fiber walks are (asymptotically) slow. ◮ Adaption of Markov basis can lead to expanders.

Open problems

◮ How can mixing behaviour be improved for single fibers? ◮ What happens with the SLEM when using the Metropolis-Hastings

Algorithm? arXiv:1505.03018 Thanks!

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