Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex

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 <i>alternating inequalities </i> and show that a <i>logarithmic separable concave</i> function satisfies the alternating inequalities. If a probability function satisfies alternating inequalities, then our chain for approximate sampling mixes in $\textit{O}(n^2 \textit{ln}(Kɛ^{-1}))$, namely $(1/2)n(n-1) \textit{ln}(K ɛ^{-1})$, 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 $\textit{O}(n^3 \textit{ln}(Kn))$.