Group representations in probability and statistics
Citations
2,573 citations
Cites background from "Group representations in probabilit..."
...The survey by Diaconis and SaloffCoste (1998) contains more on the Metropolis algorithm....
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...The effectiveness of the `(2) bound was first demonstrated by Diaconis and Shahshahani (1981). Diaconis (1988) uses representation theory to calculate eigenvalues and eigenfunctions for random walks on groups. Spielman and Teng (1996) show that for any planar graph with n vertices and maximum degree ∆, the relaxation time is at least c(∆)n, where c(∆) is a constant depending on the ∆....
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...The survey by Diaconis and SaloffCoste (1998) contains more on the Metropolis algorithm. The textbook Brémaud (1999) also discusses the use of the Metropolis algorithm for Monte Carlo sampling....
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...In fact, as Diaconis and Shahshahani (1981) proved, the random transpositions walk has a sharp cutoff of width O(n) at (1/2)n log n....
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...Diaconis (1988) is a starting place. Pólya’s urn was introduced in Eggenberger and Pólya (1923) and Pólya (1931). Urns are fundamental models for reinforced process. See Pemantle (2007) for a wealth of information and many references on urn processes and more generally processes with reinforcement....
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1,564 citations
Cites background from "Group representations in probabilit..."
...We also sketch recent algorithmic applications of random walks, in particular to the problem of sampling....
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1,271 citations
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Cites background or methods from "Group representations in probabilit..."
...See [123, 127] and [112] Chapter 3F for details on convergence to equilibrium and [110] for hitting and cover times....
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...Brief accounts can be found in Letac [224, 225] and Diaconis [112] Chapter 3F, which contains an extensive annotated bibliography....
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...I lack the space (and, more importantly, the knowledge) to give a worthwhile treatment here, and in any case an account which is both introductory and gets to interesting results is available in Diaconis [112]....
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...jjj Cite Diaconis book [112]? Continue same paragraph: So the relaxation time is...
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...3 with random transpositions benchmark, it is known from group representation methods [112, 122] which make essential use of all the eigenvalues λ̃r, not just λ̃2, that ̃̂ τ ∼ 1 2m logm as m→∞....
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References
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