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Mixing properties of the Swendsen–Wang process on classes of graphs

Colin Cooper, Alan M. Frieze
1999 Random structures & algorithms (Print)  
We consider the mixing properties of the Swendsen-Wang process for the 2state Potts model or Ising model, on the complete n vertex graph K n and for the Q-state model on an a × n grid where a is bounded  ...  Gore and Jerrum [11] obtain negative results for the mixing properties of the Swendsen-Wang process on the complete graph K n on n vertices (the Curie-Weiss or Mean-Field model) for Q ≥ 3 and certain  ...  Introduction We consider the mixing properties of the Swendsen-Wang process for the Markov chain Monte Carlo estimation of the partition function of the ferromagnetic Q-state Potts model, for two classes  ... 
doi:10.1002/(sici)1098-2418(199910/12)15:3/4<242::aid-rsa4>3.3.co;2-3 fatcat:4vz2l5rpyfe7hazsqzrwtbnmdq

Mixing properties of the Swendsen-Wang process on classes of graphs

Colin Cooper, Alan M. Frieze
1999 Random structures & algorithms (Print)  
We consider the mixing properties of the Swendsen-Wang process for the 2state Potts model or Ising model, on the complete n vertex graph K n and for the Q-state model on an a × n grid where a is bounded  ...  Gore and Jerrum [11] obtain negative results for the mixing properties of the Swendsen-Wang process on the complete graph K n on n vertices (the Curie-Weiss or Mean-Field model) for Q ≥ 3 and certain  ...  Introduction We consider the mixing properties of the Swendsen-Wang process for the Markov chain Monte Carlo estimation of the partition function of the ferromagnetic Q-state Potts model, for two classes  ... 
doi:10.1002/(sici)1098-2418(199910/12)15:3/4<242::aid-rsa4>3.0.co;2-c fatcat:tkuizlopsvegxjj44h554u77iu

The Swendsen-Wang process does not always mix rapidly

Vivek K. Gore, Mark R. Jerrum
1997 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC '97  
Empirical observations suggest that the Swendsen-Wang process mixes rapidly in many instances of practical interest.  ...  The Swendsen-Wang process provides one possible dynamics for the Qstate Potts model in statistical physics.  ...  for his careful reading and constructive criticism of an early version of this paper.  ... 
doi:10.1145/258533.258662 dblp:conf/stoc/GoreJ97 fatcat:uwxk3c5gnrcnnaghavra5jav3q

Page 1216 of Mathematical Reviews Vol. , Issue 2001B [page]

2001 Mathematical Reviews  
The following results are proved concerning the mixing-time of the Swendsen-Wang process on various classes of graphs: (i) for graphs of bounded degree, the mixing time is polynomial in n, the number of  ...  Guiol (BR-ECPM; Campinas) 2001b:60117 60K35 05C80 60J10 68Q25 82B20 Cooper, Colin (4-NLON; London); Frieze, Alan M. (1-CMU; Pittsburgh, PA) Mixing properties of the Swendsen-Wang process on classes of  ... 

Rapid Mixing Swendsen-Wang Sampler for Stochastic Partitioned Attractive Models [article]

Sejun Park, Yunhun Jang, Andreas Galanis, Jinwoo Shin, Daniel Stefankovic, Eric Vigoda
2017 arXiv   pre-print
Our experiments also confirm that the Swendsen-Wang sampler significantly outperforms the Gibbs sampler when they are used for learning parameters of attractive GMs.  ...  We prove O(\log n) mixing time for attractive binary pairwise GMs (i.e., ferromagnetic Ising models) on stochastic partitioned graphs having n vertices, under some mild conditions, including low temperature  ...  The authors of [10] establish the rapid mixing property of the Swendsen-Wang chain for the complete graph by analyzing a one-dimensional function, the so-called simplified Swendsen-Wang (see Appendix  ... 
arXiv:1704.02232v1 fatcat:holxuekydra4bjdgnfhi73dgxi

Comparison of Swendsen-Wang and heat-bath dynamics

Mario Ullrich
2012 Random structures & algorithms (Print)  
We prove that the spectral gap of the Swendsen-Wang process for the Potts model on graphs with bounded degree is bounded from below by some constant times the spectral gap of any single-spin dynamics.  ...  This implies rapid mixing of the Swendsen-Wang process for the two-dimensional Potts model at all temperatures above the critical one, as well as rapid mixing at the critical temperature for the Ising  ...  Acknowledgements The author thanks Alan Sokal for carefully reading the paper and for his valuable comments.  ... 
doi:10.1002/rsa.20431 fatcat:yaqz4dd4uzcczcc7gz6kml4crq

Rapid mixing of Swendsen–Wang dynamics in two dimensions

Mario Ullrich
2014 Dissertationes Mathematicae  
As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional  ...  We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs.  ...  Finally, we restrict ourself in Chapter 5 to a special class of graphs, namely planar graphs, and, using the notion of dual graphs, it will be possible to relate the mixing properties of the Swendsen-Wang  ... 
doi:10.4064/dm502-0-1 fatcat:ahlkv7qe4ba23fom7svvgwgalu

A bounding chain for Swendsen-Wang

Mark Huber
2002 Random structures & algorithms (Print)  
The Swendsen-Wang bounding chain allow us to efficiently obtain exact samples from the ferromagnetic Q-state Potts model for certain classes of graphs.  ...  Here we present a bounding chain for the Swendsen-Wang process.  ...  Cooper and Frieze [3] did in fact show Swendsen-Wang to be rapidly mixing on these graphs.  ... 
doi:10.1002/rsa.10071 fatcat:vytu7otxezd4zbq6ikofzv2ika

Swendsen‐Wang dynamics for general graphsin the tree uniqueness region

Antonio Blanca, Zongchen Chen, Eric Vigoda
2019 Random structures & algorithms (Print)  
We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log |V |) and relaxation time Θ(1) on any graph of maximum degree d for all β < β c (d).  ...  We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model.  ...  If β < β c (d), then the relaxation time of the Swendsen-Wang dynamics is Θ(1).  ... 
doi:10.1002/rsa.20858 fatcat:hzralg762rb63dj3mrxdvgk4xa

Cutoff for the Swendsen-Wang dynamics on the lattice [article]

Danny Nam, Allan Sly
2019 arXiv   pre-print
In this work we derive strong enough bounds on the mixing time, proving that the Swendsen-Wang dynamics on the lattice at sufficiently high temperatures exhibits a sharp transition from "unmixed" to "well-mixed  ...  We study the Swendsen-Wang dynamics for the q-state Potts model on the lattice.  ...  In §3, we introduce a coupling of the Swendsen-Wang dynamics and deduce estimates on the spectral gap and the mixing time.  ... 
arXiv:1805.04227v2 fatcat:stkspp5rqvbe3fow6eieykz6c4

Rate of convergence of the Swendsen-Wang dynamics in image segmentation problems: a theoretical and experimental study

Isabelle Gaudron
1997 E S A I M: Probability & Statistics  
We study in this paper the convergence rate of the Swendsen-Wang dynamics towards its equilibrium law, when the energy belongs to a large family of energies used in image segmentation problems.  ...  We compute the exponential equivalents of the transitions which c o n trol the process at low temperature, as well as the critical constant w h i c h gives its convergence rate.  ...  process at low temperature, and deduce that there exist, at least at low temperature, a class of energies for which the Metropolis dynamics runs faster, and an other one where the Swendsen-Wang dynamics  ... 
doi:10.1051/ps:1997110 fatcat:3qvwdpmxdvb5zgdfw2576wtmty

Iterative simulation methods

B.D. Ripley, M.D. Kirkland
1990 Journal of Computational and Applied Mathematics  
The standard methods of generating sample from univariate distributions often become hopelessly inefficient when applied to realizations of stochastic processes .  ...  The methods are surveyed and compared, with particular reference to their convergence properties .  ...  For example, we can compare the clock method and the Swendsen-Wang method on the number of "sweeps" across the graph .  ... 
doi:10.1016/0377-0427(90)90347-3 fatcat:wd6foqhdwnepfpuuv24rkkzjiu

Self-Avoiding Random Dynamics on Integer Complex Systems [article]

Firas Hamze, Ziyu Wang, Nando de Freitas
2011 arXiv   pre-print
the design of quantum computers.  ...  We name the algorithm SARDONICS, an acronym for Self-Avoiding Random Dynamics on Integer Complex Systems.  ...  Acknowledgments The authors would like to thank Misha Denil and Helmut Katzgraber. The Swendsen-Wang code was modified from a slower version kindly made available by Iain Murray.  ... 
arXiv:1111.5379v2 fatcat:t4rkag4mbnex7e6ic3uai5qy6y

Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region [article]

Antonio Blanca, Zongchen Chen, Eric Vigoda
2018 arXiv   pre-print
We establish that this variant of the Swendsen-Wang dynamics has mixing time O(|V|) and relaxation time Θ(1) on any graph of maximum degree d for all β < β_c(d).  ...  We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model.  ...  Let G be an arbitrary n-vertex graph of maximum degree d. If β < β c (d), then the relaxation time of the Swendsen-Wang dynamics is Θ(1).  ... 
arXiv:1806.04602v1 fatcat:vjdnk6mrrfdejgikep2b6cgo6m

Generalizing Swendsen-Wang to sampling arbitrary posterior probabilities

A. Barbu, Song-Chun Zhu
2005 IEEE Transactions on Pattern Analysis and Machine Intelligence  
In this paper, we present a new inference algorithm that generalizes the Swendsen-Wang method [25] to arbitrary probabilities defined on graph partitions.  ...  Thus, it realizes the split, merge, and regrouping of a "chunk" of the graph, in contrast to Gibbs sampler that flips a single vertex.  ...  An earlier version of this paper appeared in Proceedings of the Ninth International Conference on Computer Vision, Nice, France, 2003.  ... 
doi:10.1109/tpami.2005.161 pmid:16119263 fatcat:bofq6faz7jcfpkjh2rxm4jv4fe
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