Echantillonnage exact de distributions de Gibbs d énergies sous-modulaires. Exact sampling of Gibbs distributions with submodular energies

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1 Echantillonnage exact de distributions de Gibbs d énergies sous-modulaires Exact sampling of Gibbs distributions with submodular energies Marc Sigelle 1 and Jérôme Darbon 2 1 Institut TELECOM TELECOM ParisTech CNRS UMR 5141, rue Dareau 754 Paris Cedex 13 France 2 CMLA, ENS Cachan, CNRS, UniverSud, 61 Av. Président Wilson, F Cachan, France 2 Department of Mathematics, University of California, Los Angeles, CA, USA. CNRS PEPS INS2II Approches combinatoires et stochastiques pour le traitement des images sur architectures parallèles embarquées Financement de l Office of Naval Research ONR-N

Av. Président Wilson, F-94230 Cachan, France 2 Department of Mathematics, University of California, Los Angeles, CA, USA.

2 Plan motivation recalls Markov Random Field (MRF) Markov Chain Monte Carlo used for sampling MRF exact sampling: Coupling From the Past principle monotony our contribution this works as well for submodular energies! conclusion and perspectives MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #1

monotony our contribution this works as well for submodular energies!

3 Motivation sampling a MRF prior distribution testing a MRF energy model sampling a MRF posterior distribution: denoising, deblurring... statistical estimate based on samples of the distribution Louchet and Moisan (2008, 22) hyperparameter estimation for prior/posterior models (iterative) hyperparameter update based on current samples MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #2

.. statistical estimate based on samples of the distribution Louchet and Moisan (2008, 22)

4 Recalls: Markov Random Field (MRF) definition x Ω finite (ex: 2 ( ) ) π(x = x) = exp U(x) Z Gibbs distribution U(x) = c C U c (x) total energy U (s,t) (x) = V (x s, x t ) clique energy Z = x Ω exp U(x) partition function Z unavailable impossible to draw (sample) directly in Ω MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #3

) clique energy Z = x Ω exp U(x) partition function Z unavailable impossible to draw

5 Recalls: Markov Chain Monte Carlo (MCMC) Gibbs / Metropolis sampler X (0) X (1) X (n-1) X (n)..... ergodicity sampler convergence burn-up period for how long? when to assess this convergence? MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #4

6 Coupling From the Past (CFTP) J. D. Propp and D.B. Wilson (1996) James D. Propp and D. B. Wilson. Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms, 9(1,2): , Annotated Bibliography of Perfectly Random Sampling with Markov Chains MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #5

Exact sampling with coupled Markov chains and applications to statistical mechanics.

7 Coupling From the Past (CFTP) (followed) looking backward what can happen at fixed time 0 ξ Ω may be obtained from many paths many simultaneous Markov chains ( Ω ) launched at some time (t < 0) random map f t : Ω Ω Markov(t) from time t t + 1 random map F t = f 1 o f 2... o f t+1 o f t from time t 0 recursive relation F t = F t+1 o f t F t = constant coalescence MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #6

: Ω Ω Markov(t) from time t t + 1 random map F t = f 1 o f 2.

8 Coupling From the Past (CFTP) (followed) f 1 MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #7

9 Coupling From the Past (CFTP) (followed) f 2 f 1 MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #8

10 Coupling From the Past (CFTP) (followed) f 3 f 2 f 1 MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #9

11 Coupling From the Past (CFTP) (followed) t = f 3 f 2 f 1 MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #10

12 CFTP : main results (Propp and Wilson (1996)) with probability 1: finite time t = M < 0 s.t. coalescence occurs when starting at time M the state ξ so obtained is a perfect sample of distribution π MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #11

13 CFTP : a result from Murdoch and Green (1998) MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #12

14 CFTP : monotony (Propp and Wilson (1996)) launch Ω chains? partial ordering on Ω x y x s y s s S 2 extremal elements: ˆ0 x ˆL x Ω if f t = Markov(t) preserves monotony t: x, y Ω x y f t (x) f t (y) coupled Gibbs sampler then 2 chains starting from ˆ0 and ˆL are enough until they coalesce! MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #13

preserves monotony t: x, y Ω x y f t (x) f t (y) coupled Gibbs sampler then 2 chains

15 Monotony for the ferromagnetic Ising model experiment at Ising critical temperature: M = 30 sweeps claimed! ^ 1 x ^ u t ˆ0 x ˆ1 x Ω equivalent monotony condition for binary models N s N s π(x s = 0 N s ) π(x s = 0 N s) MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #14

11111 11111 11111 11111 11111 11111 11111 11111 11111 u t ˆ0 x ˆ1 x Ω equivalent monotony condition for binary

16 Our contribution: CFTP extends to submodular energies! U(x s + 1, x t + 1) + U(x s, x t ) U(x s + 1, x t ) + U(x s, x t + 1) condition on interaction energy only independent of attachment to data term!! example : L 2 + (anisotropic) TV U = (x s y s ) 2 + w st x s x t w st 0 s S (s,t) coalescence obtained from ˆ0 and ˆL M = 100 sweeps claimed for a 256 greylevel image (Moisan Darbon (22))!! MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #15

17 Conclusion and perspectives other models other estimates in progress (Tupin, Darbon, Sigelle (23)) other samplers Metropolis-Hastings done efficient hardware/sofware implementation in progress... MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #16

Metropolis-Hastings done efficient hardware/sofware implementation

18 Recalls: Markov Chain Monte Carlo (MCMC) Doeblin coupling a b a b u t Q((a, a ), (b, b )) = Q(a, b). Q(a, b ) if a b = Q(a, b) if a = a, b = b = 0 else marginally each Markov chain follows kernel Q Louchet and Moisan (2008, 2009, 22) MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #17

19 Recalls: Gibbs Sampler classical Gibbs sampler x (n) s N (n) s current configuration current site current neighborhood x (n+1) s π(x s =. N (n) s ) coupled Gibbs sampler (M arkov(t)) u t x (n+1) s x (n+1) s MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #18

s =. N (n) s ) coupled Gibbs sampler (M arkov(t)) u t x (n+1) s x

20 Recalls: Gibbs Sampler (followed) a remark N s x s u t s x s u t 0 1 N s MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #19

21 References Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1), Lalley, S. P. (2009). Convergence Rates of Markov Chains. Louchet, C. and Moisan, L. (2008). Total Variation denoising using posterior expectation. In Proceedings of EUropean Signal and Image Processing Conference EUSIPCO. Louchet, C. and Moisan, L. (22). Posterior Expectation of the Total Variation model: Properties and Experiments. Murdoch, D. J. and Green, P. J. (1998). Exact sampling from a continuous state space. Scandinavian Journal of Statistics, 25(3), Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and statistical mechanics. Random Structures and Algorithms, 9(1), MMTI Journée thématique : Autour de l échantillonnage 27th June 23 #20

Introduction to Markov Chain Monte Carlo

Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution to estimate the distribution to compute max, mean Markov Chain Monte Carlo: sampling using local information Generic problem