DMTCS Proceedings, 2005 International Conference on Analysis of Algorithms

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DMTCS Conference vol AD (2005), pp. 371-382

DMTCS

2005 International Conference on Analysis of Algorithms

Conrado Martínez (ed.)

DMTCS Conference Volume AD (2005), pp. 371-382


author: Shuji Kijima and Tomomi Matsui
title: Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex
keywords: Markov chain, Mixing time, Path coupling, Coupling from the past, Log-concave function.
abstract: In this paper, we are concerned with random sampling of an
n
dimensional integral point on an
(n-1)
dimensional simplex according to a multivariate discrete distribution. We employ sampling via Markov chain and propose two ``hit-and-run'' chains, one is for approximate sampling and the other is for perfect sampling. We introduce an idea of alternating inequalities and show that a logarithmic separable concave function satisfies the alternating inequalities. If a probability function satisfies alternating inequalities, then our chain for approximate sampling mixes in
O(n
2
ln(Kɛ
-1
))
, namely
(1/2)n(n-1) ln(K ɛ
-1
)
, where
K
is the side length of the simplex and
ɛ
(
0<ɛ<1
) is an error rate. On the same condition, we design another chain and a perfect sampler based on monotone CFTP (Coupling from the Past). We discuss a condition that the expected number of total transitions of the chain in the perfect sampler is bounded by
O(n
3
ln(Kn))
.
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reference: Shuji Kijima and Tomomi Matsui (2005), Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex , in 2005 International Conference on Analysis of Algorithms, Conrado Martínez (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AD, pp. 371-382
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