Random cluster dynamics for the Ising model is rapidly mixing

Guo, Heng; Jerrum, Mark

doi:10.1137/1.9781611974782.118

Cited by 55 publications

(80 citation statements)

References 27 publications

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“…Previously, the famous Jerrum-Sinclair chain on even subgraphs [9] gives a poly-time approximate sampler for the ferro-Ising model. The same can also be obtained by a recent result of Guo and Jerrum [18] for the rapid mixing of the random cluster model, which combined with the coupling from the past (CFTP) of Propp and Wilson [41] also gives a poly-time perfect sampler for the ferro-Ising model. These Ising samplers work in the entire ferromagnetic regime, but require global translations of the probability space and has large polynomial running times.…”

Section: Dynamic Sampling From the Spin Systemssupporting

confidence: 71%

Dynamic sampling from graphical models

Feng

Vishnoi

Yin

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update, on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or anti-ferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, change of functions), both efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new "equilibrium" property that is shown to be satisfied by our algorithm at each step, and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest.

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Section: Dynamic Sampling From the Spin Systemssupporting

confidence: 71%

Dynamic sampling from graphical models

Feng

Vishnoi

Yin

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…For B = B u , by (17) and Item 2 of Lemma 7 we have that Ψ 1 has exactly two critical points, at u = 1/q and a > 1/q. By (16), we have that Ψ 1 has a local maximum at u = 1/q. Ψ 1 cannot have a local maximum at a, otherwise Ψ 1 must have at least one critical point in the interval (u, a) which contradicts the fact that Ψ 1 has exactly one critical point in (1/q, 1] (Item 2 of Lemma 7).…”

Section: Lemmamentioning

confidence: 96%

“…For B ≥ B rc , by (17) and Item 4 of Lemma 7, we have that Ψ 1 has exactly two critical points in the interval [1/q, 1], at u = 1/q and a > 1/q. By (16), we have that Ψ 1 does not have a local maximum at u = 1/q. Since Ψ 1 (α) ↓ −∞ for α ↑ 1, we obtain that Ψ 1 cannot have a local minimum at a (otherwise there would be a critical point in the interval (a, 1)).…”

Section: Lemmamentioning

confidence: 96%

“…Case I. For B < B u , by (16), (17) and Item 1 of Lemma 7, we have that u = 1/q is the unique critical point of Ψ 1 and it is a local maximum of Ψ 1 .…”

Section: Lemmamentioning

confidence: 97%

“…For square boxes of

ℤ^{2}

, Ullrich proved that the mixing time of Swendsen‐Wang is polynomial for all temperatures (building upon results for the Glauber dynamics by Martinelli and Olivieri and Lubetzky and Sly ). Very recently, Guo and Jerrum showed that the mixing time of Swendsen‐Wang is polynomial for any graph G for all temperatures.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Swendsen‐Wang algorithm on the mean‐field Potts model

Galanis

Štefankovič

Vigoda

2018

Random Struct Algorithms

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We study the q‐state ferromagnetic Potts model on the n‐vertex complete graph known as the mean‐field (Curie‐Weiss) model. We analyze the Swendsen‐Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single‐site Glauber dynamics. Long et al. studied the case q = 2, the Swendsen‐Wang algorithm for the mean‐field ferromagnetic Ising model, and showed that the mixing time satisfies: (i) Θ(1) for βnormalβc, where βc is the critical temperature for the ordered/disordered phase transition. In contrast, for q≥3 there are two critical temperatures 0

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Spatial mixing and nonlocal Markov chains

Blanca

Caputo

Sinclair

et al. 2019

Random Struct Algorithms

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We consider spin systems with nearest-neighbor interactions on an n-vertex -dimensional cube of the integer lattice graph Z . We study the effects that the strong spatial mixing condition (SSM) has on the rate of convergence to equilibrium of nonlocal Markov chains. We prove that when SSM holds, the relaxation time (i.e., the inverse spectral gap) of general block dynamics is O(r), where r is the number of blocks. As a second application of our technology, it is established that SSM implies an O(1) bound for the relaxation time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is O(log n(log log n) 2 ). Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.

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Random cluster dynamics for the Ising model is rapidly mixing

Cited by 55 publications

References 27 publications

Dynamic sampling from graphical models

Dynamic sampling from graphical models

Swendsen‐Wang algorithm on the mean‐field Potts model

Spatial mixing and nonlocal Markov chains

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