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Generalizing swendsenwang to sampling arbitrary posterior probabilities
, 2007
"... Abstract—Many vision tasks can be formulated as graph partition problems that minimize energy functions. For such problems, the Gibbs sampler [9] provides a general solution but is very slow, while other methods, such as Ncut [24] and graph cuts [4], [22], are computationally effective but only work ..."
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Cited by 77 (17 self)
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Abstract—Many vision tasks can be formulated as graph partition problems that minimize energy functions. For such problems, the Gibbs sampler [9] provides a general solution but is very slow, while other methods, such as Ncut [24] and graph cuts [4], [22], are computationally effective but only work for specific energy forms [17] and are not generally applicable. In this paper, we present a new inference algorithm that generalizes the SwendsenWang method [25] to arbitrary probabilities defined on graph partitions. We begin by computing graph edge weights, based on local image features. Then, the algorithm iterates two steps. 1) Graph clustering: It forms connected components by cutting the edges probabilistically based on their weights. 2) Graph relabeling: It selects one connected component and flips probabilistically, the coloring of all vertices in the component simultaneously. Thus, it realizes the split, merge, and regrouping of a “chunk ” of the graph, in contrast to Gibbs sampler that flips a single vertex.We prove that this algorithm simulates ergodic and reversibleMarkov chain jumps in the space of graph partitions and is applicable to arbitrary posterior probabilities or energy functions defined on graphs. We demonstrate the algorithm on two typical problems in computer vision—image segmentation and stereo vision. Experimentally, we show that it is 100400 times faster in CPU time than the classical Gibbs sampler and 2040 times faster then the DDMCMC segmentation algorithm [27]. For stereo, we compare performance with graph cuts and belief propagation. We also show that our algorithm can automatically infer generativemodels and obtain satisfactory results (better than the graphic cuts or belief propagation) in the same amount of time.
Mathematical foundations of the Markov chain Monte Carlo method
 in Probabilistic Methods for Algorithmic Discrete Mathematics
, 1998
"... 7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that a ..."
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Cited by 34 (1 self)
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7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that are hard to compute exactly. The quantity z of interest is expressed as the expectation z = ExpZ of a random variable (r.v.) Z for which some efficient sampling procedure is available. By taking the mean of some sufficiently large set of independent samples of Z, one may obtain an approximation to z. For example, suppose S = \Phi (x; y) 2 [0; 1] 2 : p i (x; y) 0; for all i \Psi<F12
Mixing Properties of the SwendsenWang Process on the Complete Graph and Narrow Grids
 IN PROCEEDINGS OF DIMACS WORKSHOP ON STATISTICAL PHYSICS METHODS IN DISCRETE PROBABILITY, COMBINATORICS AND THEORETICAL COMPUTER SCIENCE
, 2000
"... We consider the mixing properties o the SwendsenWang process or the 2state Potts model or Ising model, on the complete n vertex graph Kn and for the Qstate model on an a x n grid where a is bounded as n  . ..."
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Cited by 34 (0 self)
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We consider the mixing properties o the SwendsenWang process or the 2state Potts model or Ising model, on the complete n vertex graph Kn and for the Qstate model on an a x n grid where a is bounded as n  .
Torpid Mixing of Simulated Tempering on the Potts Model
 Proceedings of the 15th ACMSIAM Symposium on Discrete Algorithms
, 2004
"... Simulated tempering and swapping are two families of sampling algorithms in which a parameter representing temperature varies during the simulation. The hope is that this will overcome bottlenecks that cause sampling algorithms to be slow at low temperatures. Madras and Zheng demonstrate that the sw ..."
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Cited by 26 (4 self)
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Simulated tempering and swapping are two families of sampling algorithms in which a parameter representing temperature varies during the simulation. The hope is that this will overcome bottlenecks that cause sampling algorithms to be slow at low temperatures. Madras and Zheng demonstrate that the swapping and tempering algorithms allow efficient sampling from the lowtemperature meanfield Ising model, a model of magnetism, and a class of symmetric bimodal distributions [10]. Local Markov chains fail on these distributions due to the existence of bad cuts in the state space.
Torpid Mixing of Some Monte Carlo Markov Chain Algorithms
 in Statistical Physics, Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer
, 1999
"... We study two widely used algorithms, Glauber dynamics and the SwendsenWang algorithm, on rectangular subsets of the hypercubic lattice Z d. We prove that under certain circumstances, the mixing time in a box of side length L with periodic boundary conditions can be exponential in L d−1. In other wo ..."
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Cited by 25 (3 self)
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We study two widely used algorithms, Glauber dynamics and the SwendsenWang algorithm, on rectangular subsets of the hypercubic lattice Z d. We prove that under certain circumstances, the mixing time in a box of side length L with periodic boundary conditions can be exponential in L d−1. In other words, under these circumstances, the mixing in these widely used algorithms is not rapid; instead, it is torpid. The models we study are the independent set model and the qstate Potts model. For both models, we prove that Glauber dynamics is torpid in the region with phase coexistence. For the Potts model, we prove that SwendsenWang is torpid at the phase transition point. 1
Random walks on combinatorial objects
 Surveys in Combinatorics 1999
, 1999
"... Summary Approximate sampling from combinatoriallydefined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we reexamine the unde ..."
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Cited by 22 (8 self)
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Summary Approximate sampling from combinatoriallydefined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we reexamine the underlying formal foundational framework in the light of this. We give a detailed treatment of the coupling technique, a classical method for analysing the convergence rates of Markov chains. The related topic of perfect sampling is examined. In perfect sampling, the goal is to sample exactly from the target set. We conclude with a discussion of negative results in this area. These are results which imply that there are no polynomial time algorithms of a particular type for a particular problem. 1
The complexity of ferromagnetic Ising with local fields
, 2005
"... We consider the complexity of approximating the partition function of the ferromagnetic Ising model with varying interaction energies and local external magnetic fields. Jerrum and Sinclair provided a fully polynomial randomised approximation scheme for the case in which the system is consistent in ..."
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Cited by 18 (6 self)
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We consider the complexity of approximating the partition function of the ferromagnetic Ising model with varying interaction energies and local external magnetic fields. Jerrum and Sinclair provided a fully polynomial randomised approximation scheme for the case in which the system is consistent in the sense that the local external fields all favour the same spin. We characterise the complexity of the general problem by showing that it is equivalent in complexity to the problem of approximately counting independent sets in bipartite graphs, thus it is complete in a logicallydefined subclass of #P previously studied by Dyer, Goldberg, Greenhill and Jerrum. By contrast, we show that the corresponding computational task for the qstate Potts model with local external magnetic fields and q> 2 is complete for all of #P with respect to approximationpreserving reductions.
On the swapping algorithm
 Random Structures and Algorithms
, 2003
"... ABSTRACT: The Metropoliscoupled Markov chain method (or “Swapping Algorithm”) is an empirically successful hybrid Monte Carlo algorithm. It alternates between standard transitions on parallel versions of the system at different parameter values, and swapping two versions. We prove rapid mixing for ..."
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ABSTRACT: The Metropoliscoupled Markov chain method (or “Swapping Algorithm”) is an empirically successful hybrid Monte Carlo algorithm. It alternates between standard transitions on parallel versions of the system at different parameter values, and swapping two versions. We prove rapid mixing for two bimodal examples, including the meanfield Ising model. © 2002 Wiley
Tight Bounds for Mixing of the SwendsenWang Algorithm at the Potts Transition Point
, 2008
"... We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice Z d – heat bath dynamics and the SwendsenWang algorithm – and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath ..."
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Cited by 12 (0 self)
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We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice Z d – heat bath dynamics and the SwendsenWang algorithm – and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the SwendsenWang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L d−1. This work provides the first upper bound of this form for the SwendsenWang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L) 2. 1