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 1 / 11

Integer points of polytopes Fibers Let A ∈ Z m × d and b ∈ Z m , the b-fiber of A is F A , b := { u ∈ N d : A · u = b } . 2 / 11

Integer points of polytopes Fibers Let A ∈ Z m × d and b ∈ Z m , the b-fiber of A is F A , b := { u ∈ N d : A · u = b } . Fiber graphs Let M ⊆ ker ( A ) ∩ Z d . The graph F A , b ( M ) has vertices F A , b and u , v ∈ F A , b are adjacent if u − v ∈ ±M . Markov bases M is a Markov basis if F A , b ( M ) is connected for all b ∈ Z d . 2 / 11

Integer points of polytopes Fibers Let A ∈ Z m × d and b ∈ Z m , the b-fiber of A is F A , b := { u ∈ N d : A · u = b } . Fiber graphs Let M ⊆ ker ( A ) ∩ Z d . The graph F A , b ( M ) has vertices F A , b and u , v ∈ F A , b are adjacent if u − v ∈ ±M . Markov bases M is a Markov basis if F A , b ( M ) is connected for all b ∈ Z d . 2 / 11

Integer points of polytopes Fibers Let A ∈ Z m × d and b ∈ Z m , the b-fiber of A is F A , b := { u ∈ N d : A · u = b } . Fiber graphs Let M ⊆ ker ( A ) ∩ Z d . The graph F A , b ( M ) has vertices F A , b and u , v ∈ F A , b are adjacent if u − v ∈ ±M . Markov bases M is a Markov basis if F A , b ( M ) is connected for all b ∈ Z d . Theorem (Diaconis, Sturmfels, 1996) M is a Markov basis of A if and only if I M = I A . 2 / 11

Walking randomly on fibers Algorithm (Simple Walk) Input : M Markov basis, u 0 ∈ F A , b , r ∈ N Output : Uniform sample from F A , b ◮ FOR i = 1 .. r ◮ Sample m ∈ ±M uniformly ◮ IF u i + m ∈ N d ◮ THEN u i = u i − 1 + m ◮ ELSE u i = u i − 1 ◮ RETURN u r → This gives an irreducible and aperiodic Markov chain. Convergence Theorem of Markov chains � � − 1 ◮ After r = O steps, u r can be regarded as an uniform sample log ( | λ | ) from F A , b ◮ − 1 < λ < 1: SLEM of transition matrix of the random walk. 3 / 11

Analysing the speed What is fast? ( G n ) 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 ( G n ) | 4 / 11

Analysing the speed What is fast? ( G n ) 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 ( G n ) | → Consider ( n · b ) n ∈ N ⊆ N A 4 / 11

Analysing the speed What is fast? ( G n ) 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 ( G n ) | → Consider ( n · b ) n ∈ N ⊆ N A Expander and rapid mixing For any n ∈ N , let λ n be the SLEM of the simple walk on F A , n · b ( M n ) ◮ ( F A , n · b ( M n )) n ∈ N is an expander if λ n ≤ 1 − ǫ for all n ∈ N 1 ◮ ( F A , n · b ( M n )) n ∈ N is rapidly mixing if λ n ≤ 1 − p ( log n ) for all n ∈ N 4 / 11

Analysing the speed What is fast? ( G n ) 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 ( G n ) | → Consider ( n · b ) n ∈ N ⊆ N A Expander and rapid mixing For any n ∈ N , let λ n be the SLEM of the simple walk on F A , n · b ( M n ) ◮ ( F A , n · b ( M n )) n ∈ N is an expander if λ n ≤ 1 − ǫ for all n ∈ N 1 ◮ ( F A , n · b ( M n )) n ∈ N is rapidly mixing if λ n ≤ 1 − p ( log n ) for all n ∈ N Algebraic statistics Use the same moves for every right-hand side: M n := M 4 / 11

Fiber walks can be slow! No expander  − 1 3  I 3 I 3 0 0 0 1 ◮ H := 0 0 I 3 I 3 0 − 1 3   0 0 0 0 1 1 0 . 8 SLEM ◮ Ray { n · e 7 : n ∈ N } 0 . 6 Graver Gr¨ obner 0 . 4 2 4 6 8 10 n 5 / 11

Fiber walks can be slow! No expander  − 1 3  I 3 I 3 0 0 0 1 ◮ H := 0 0 I 3 I 3 0 − 1 3   0 0 0 0 1 1 0 . 8 SLEM ◮ Hemmecke-Ray { n · e 7 : n ∈ N } 0 . 6 Graver Gr¨ obner 0 . 4 2 4 6 8 10 n 5 / 11

Fiber walks can be slow! No expander  − 1 3  I 3 I 3 0 0 0 1 ◮ H := 0 0 I 3 I 3 0 − 1 3   0 0 0 0 1 1 0 . 8 SLEM ◮ Hemmecke-Ray { n · e 7 : n ∈ N } 0 . 6 Graver ◮ lim n →∞ λ ( Gr¨ obner ) = 1 Gr¨ obner 0 . 4 ◮ lim n →∞ λ ( Graver ) = 1 2 4 6 8 10 n 5 / 11

Fiber walks can be slow! No expander  − 1 3  I 3 I 3 0 0 0 1 ◮ H := 0 0 I 3 I 3 0 − 1 3   0 0 0 0 1 1 0 . 8 SLEM ◮ Hemmecke-Ray { n · e 7 : n ∈ N } 0 . 6 Graver ◮ lim n →∞ λ ( Gr¨ obner ) = 1 Gr¨ obner 0 . 4 ◮ lim n →∞ λ ( Graver ) = 1 2 4 6 8 10 n Not rapidly mixing ◮ A := ( 1 , 1 ) ∈ Z 1 × 2 , M := { ( 1 , − 1 ) t } , b = 1 . . . ◮ F A , n ( M ) ∼ = 1 ⇒ SLEM of F A , n ( M ) ≥ 1 − n + 1 5 / 11

Fiber walks can be slow! No expander  − 1 3  I 3 I 3 0 0 0 1 ◮ H := 0 0 I 3 I 3 0 − 1 3   0 0 0 0 1 1 0 . 8 SLEM ◮ Hemmecke-Ray { n · e 7 : n ∈ N } 0 . 6 Graver ◮ lim n →∞ λ ( Gr¨ obner ) = 1 Gr¨ obner 0 . 4 ◮ lim n →∞ λ ( Graver ) = 1 2 4 6 8 10 n Not rapidly mixing ◮ A := ( 1 , 1 ) ∈ Z 1 × 2 , M := { ( 1 , − 1 ) t } , b = 1 . . . ◮ F A , n ( M ) ∼ = 1 ⇒ SLEM of F A , n ( M ) ≥ 1 − n + 1 ◮ Is this (asymptotic) behaviour typical for fiber walks? 5 / 11

Edge-expansion Edge-expansion, Cheeger constant #( edges leaving U ) h ( G ) := min | U |≤ 1 2 | V | | U | Expander Mixing Lemma ≤ | λ | ≤ 1 − h ( G ) 2 Simple walk on d -regular graph G : 1 − h ( G ) d d 2 Why is this convenient for fibers? F A , n 1 · b ( M ) F A , n 2 · b ( M ) 6 / 11

How fast are fiber walks? Theorem (W.; 2015) Let M be a Markov basis of A and let b ∈ N A with dim ( F A , b ) > 0 . Then: ◮ lim n →∞ h ( F A , n · b ( M )) = 0 (no expander) 7 / 11

How fast are fiber walks? Theorem (W.; 2015) Let M be a Markov basis of A and let b ∈ N A with dim ( F A , b ) > 0 . Then: ◮ lim n →∞ h ( F A , n · b ( M )) = 0 (no expander) ◮ h ( F A , n · b ( M )) ∈ O ( 1 n ) (not rapidly mixing) 7 / 11

How fast are fiber walks? Theorem (W.; 2015) Let M be a Markov basis of A and let b ∈ N A with dim ( F A , b ) > 0 . Then: ◮ lim n →∞ h ( F A , n · b ( M )) = 0 (no expander) ◮ h ( F A , n · b ( M )) ∈ O ( 1 n ) (not rapidly mixing) Graver vs. Gr¨ obner vs. Markov Using more structural moves does not improve mixing asymptotically. 7 / 11

How fast are fiber walks? Theorem (W.; 2015) Let M be a Markov basis of A and let b ∈ N A with dim ( F A , b ) > 0 . Then: ◮ lim n →∞ h ( F A , n · b ( M )) = 0 (no expander) ◮ h ( F A , n · b ( M )) ∈ O ( 1 n ) (not rapidly mixing) Graver vs. Gr¨ obner vs. Markov Using more structural moves does not improve mixing asymptotically. A way out? Use an “adapted” Markov basis M b n for every F A , n · b and control h ( F A , n · b ( M b n )) . |M b n | 7 / 11

How fast are fiber walks? Theorem (W.; 2015) Let M be a Markov basis of A and let b ∈ N A with dim ( F A , b ) > 0 . Then: ◮ lim n →∞ h ( F A , n · b ( M )) = 0 (no expander) ◮ h ( F A , n · b ( M )) ∈ O ( 1 n ) (not rapidly mixing) Graver vs. Gr¨ obner vs. Markov Using more structural moves does not improve mixing asymptotically. A way out? Use an “adapted” Markov basis M b n for every F A , n · b and control h ( F A , n · b ( M b n )) . |M b n | ◮ In general, |M b n | = O ( log n ) does not suffice to obtain rapid mixing. 7 / 11

Can we construct expanders on fibers? Adaption Let M = { m 1 , . . . , m r } ⊆ Z d be a Markov basis , n ∈ N , and b ∈ N A .   r r   M b � � n := λ j m j : λ 1 , . . . , λ r ∈ Z , | λ j | ≤ Diam ( F A , n · b ( M ))  .  j = 1 j = 1 SLEM: 1 − |F A , n · b | |M b n | 8 / 11

Can we construct expanders on fibers? Adaption Let M = { m 1 , . . . , m r } ⊆ Z d be a Markov basis , n ∈ N , and b ∈ N A .   r r   M b � � n := λ j m j : λ 1 , . . . , λ r ∈ Z , | λ j | ≤ Diam ( F A , n · b ( M ))  .  j = 1 j = 1 SLEM: 1 − |F A , n · b | |M b n | Theorem (W.; 2015) Diam ( F A , nb ( M )) ∈ O ( n ) ⇒ ( F A , n j b ( M b n j )) j ∈ N is an expander. 8 / 11

How to interpret this? 1 0 . 9 SLEM conventional 0 . 8 adapted n 9 / 11

Computational results Sampling from M b i ◮ Sample coefficients λ i ∈ [ l i , u i ] and use the move � r i = 1 λ i m i . 3 × 3 independence model Hemmecke-Ray obner e 7 F A 33 , n · 1 6 ( M 1 6 F H 3 , n · e 7 ( Gr¨ n ) i ) n = 10 3 : n = 10 10 : 3 . 551 . 720 21 . 062 . 343 n = 10 6 : n = 10 100 : 4 . 058 . 733 37 . 255 . 074 n = 10 50 : n = 10 1000 : 4 . 059 . 281 37 . 255 . 074 1 Adapted vs. conventional 0 . 9 SLEM ◮ A = ( 1 , 1 , 1 ) ◮ M = { e 1 − e 2 , e 1 − e 3 } M 0 . 8 M 1 ◮ b n = n · 1 2 n 0 5 10 15 20 n 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 11 / 11