具有固定边缘的二进抽样矩阵的Switch和Curveball Markov链的比较
Comparing the Switch and Curveball Markov Chains for Sampling Binary Matrices with Fixed Marginals
摘要：
曲线球算法是一种著名的基于开关的马尔可夫链方法的变体，用于对具有固定行和列和的二进制矩阵进行均匀采样。曲线球算法在算法的每次迭代中执行所谓的二项式交易，而不是开关。直觉上，这可能导致在某些情况下达到平稳（均匀）分布的更好收敛速度。在文献中已经给出了一些实验证据。在本文中，我们给出了两个基于开关的链和曲线球链之间的谱隙比较。特别是，这种比较使我们能够得出结论，曲球马尔可夫链是快速混合时，两个开关链之一是快速混合。我们的分析直接延伸到的情况下，采样的二进制矩阵的禁止条目（假设不可约）。这特别是捕获的情况下，采样简单的有向图与给定的度。作为我们的分析的副产品，我们表明，开关马尔可夫链的Kannan-Tetali-Vempala猜想只有非负的特征值，如果采样的二元矩阵至少有三列。这表明，马尔可夫链不一定是懒惰的，这是独立的利益。对于均匀抽样简单有向正则图，我们还得到了Greenhill研究的开关马尔可夫链的最小特征值的一个改进的界。
英文摘要：
The Curveball algorithm is a variation on well-known switch-based Markov chain approaches for uniformly sampling binary matrices with fixed row and column sums. Instead of a switch, the Curveball algorithm performs a so-called binomial trade in every iteration of the algorithm. Intuitively, this could lead to a better convergence rate for reaching the stationary (uniform) distribution in certain cases. Some experimental evidence for this has been given in the literature. In this note we give a spectral gap comparison between two switch-based chains and the Curveball chain. In particular, this comparison allows us to conclude that the Curveball Markov chain is rapidly mixing whenever one of the two switch chains is rapidly mixing. Our analysis directly extends to the case of sampling binary matrices with forbidden entries (under the assumption of irreducibility). This in particular captures the case of sampling simple directed graphs with given degrees. As a by-product of our analysis, we show that the switch Markov chain of the Kannan-Tetali-Vempala conjecture only has non-negative eigenvalues if the sampled binary matrices have at least three columns. This shows that the Markov chain does not have to be made lazy, which is of independent interest. We also obtain an improved bound on the smallest eigenvalue for the switch Markov chain studied by Greenhill for uniformly sampling simple directed regular graphs.
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