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13
What do we know about the Metropolis algorithm
 J. Comput. System. Sci
, 1998
"... The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability and statistics. Some new results are given ae an il ..."
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Cited by 86 (13 self)
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The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability and statistics. Some new results are given ae an illustration of the geometric theory of Markov chains. 1. Introduction. Let % be a finite set and m(~)> 0 a probability distribution on %. The Metropolis algorithm is a procedure for drawing samples from X. It was introduced by Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller [1953]. The algorithm requires the user to specify a connected, aperiodic Markov chain 1<(z, y) on %. This chain need not be symmetric but if K(z, y)>0, one needs 1<(Y, z)>0. The chain K is modified
Analysis of systematic scan Metropolis algorithms using Iwahori–Hecke algebra techniques
, 2004
"... We give the first analysis of a systematic scan version of the Metropolis algorithm. Our examples include generating random elements of a Coxeter group with probability determined by the length function. The analysis is based on interpreting Metropolis walks in terms of the multiplication in the Iwa ..."
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Cited by 34 (8 self)
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We give the first analysis of a systematic scan version of the Metropolis algorithm. Our examples include generating random elements of a Coxeter group with probability determined by the length function. The analysis is based on interpreting Metropolis walks in terms of the multiplication in the IwahoriHecke algebra.
Separation cutoffs for birth death chains
, 2006
"... This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cutoff. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there ..."
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Cited by 21 (8 self)
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This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cutoff. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there is a cutoff if and only if the product of spectral gap and mixing time tends to infinity. We establish this for arbitrary birth and death chains in continuous time when the convergence is measured in separation and the chains all start at 0.
Applications of geometric bounds to the convergence rate of Markov chains on R^n
, 2001
"... Quantitative geometric rates of convergence for reversible Markov chains are closely related to the spectral gap of the corresponding operator, which is hard to calculate for general state spaces. This thesis describes a geometric argument to give different types of bounds for spectral gaps of Marko ..."
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Cited by 10 (1 self)
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Quantitative geometric rates of convergence for reversible Markov chains are closely related to the spectral gap of the corresponding operator, which is hard to calculate for general state spaces. This thesis describes a geometric argument to give different types of bounds for spectral gaps of Markov chains on bounded subsets of Rn and to compare the rates of convergence of different Markov chains. We also extend the discretetime results to homogeneous continuoustime reversible Markov processes. The limit path bounds and the limit Cheeger's bounds are introduced. Two quantitative examples of 1dimensional diffusions are studied for the limit Cheeger's bounds and a ndimensional diffusion is studied for the limit path bounds.
Stein’s method and the rank distribution of random matrices over finite fields
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Random Walks and Graph Homomorphisms
"... In this report (whose basic approach is based on [12]) we introduce a general idea which gives rise to some necessary conditions for the existence of graph homomorphisms (directed and undirected), which is mainly based on available comparison techniques for Markov chains. We focus on the nite st ..."
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Cited by 1 (1 self)
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In this report (whose basic approach is based on [12]) we introduce a general idea which gives rise to some necessary conditions for the existence of graph homomorphisms (directed and undirected), which is mainly based on available comparison techniques for Markov chains. We focus on the nite strongly{ connected case to propose the main ideas, however, there are also a variety of conceivable extensions to weaker conditions or the in nite case.
Applications Of Cheeger's Constant To The Convergence Rate Of Markov Chains On R^n
, 1997
"... Quantitative geometric rates of convergence for reversible Markov chains are closely related to the Cheeger's constant, which is hard to calculate for general state spaces. This article describes a geometric argument to bound the Cheeger's constant for chains on bounded subsets of R^n. ..."
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Quantitative geometric rates of convergence for reversible Markov chains are closely related to the Cheeger's constant, which is hard to calculate for general state spaces. This article describes a geometric argument to bound the Cheeger's constant for chains on bounded subsets of R^n.