蛇形算法：固定边际二元矩阵的无拒绝采样器
The Snake Algorithm: A Rejection-Free Sampler for Binary Matrices with Fixed Margins
浏览论文内容
中文总结 AI 辅助
针对固定行和列和的二元矩阵均匀采样问题，提出无拒绝的马尔可夫链蒙特卡罗蛇形算法，证明其快速混合性，经对比实验显示在采样效率等方面优于多种现有算法。
中文摘要 AI 辅助
我们研究固定行和列和的二元矩阵的均匀采样问题，这是生态零模型、Rasch模型测试、网络分析和组合数学中反复出现的问题。我们提出了蛇形（Snake）算法，这是一种无拒绝的马尔可夫链蒙特卡罗采样器，它增长一条交替路径直至其首次自相交，并翻转所得的环。该链在固定边际状态空间上是可逆且不可约的，因此具有均匀的平稳分布。我们证明，在稀疏且平衡的方阵情形下，一次步骤会翻转约√n个元素，给出了每步路径长度的上界，并表明在稀疏且平衡的情形下，每个翻转元素的工作量是速率最优的，在单侧半平衡条件下，其工作量在多对数因子内接近最优。马尔可夫链比较结合最近确立的交换链的通用谱间隙界，证明对于每一对可行的边际，惰性蛇形链是快速混合的；在置换矩阵情形下，原始链具有尖锐的总变差混合时间Θ(n log n)。我们还描述了一种有向图扩展和一种等边际标签洗牌变体。针对交换（Swap）、矩形环（Rectangle Loop）、球曲线（Curveball）、顺序重要采样和有向边交换算法的数值实验显示，在移动大小、挂钟收敛速度和采样效率方面均有一致的提升。
英文摘要
We study uniform sampling of binary matrices with fixed row and column sums, a recurring problem in ecological null models, Rasch-model testing, network analysis, and combinatorics. We propose the Snake algorithm, a rejection-free Markov chain Monte Carlo sampler that grows an alternating path until its first self-intersection and flips the resulting loop. The chain is reversible and irreducible on the fixed-margin state space, hence has the uniform stationary distribution. We prove that one step flips on the order of $\sqrt{n}$ entries in sparse and balanced square regimes, give upper bounds on the per-step path length, and show that the resulting work per flipped entry is rate optimal in sparse and balanced regimes and near-optimal up to a polylogarithmic factor under a one-sided half-balanced condition. A Markov-chain comparison, combined with the recently established universal spectral-gap bound for the swap chain, proves that the lazy Snake chain is rapidly mixing for every feasible pair of margins; in the permutation-matrix case, the raw chain has the sharp total-variation mixing time $Θ(n \log n)$. We also describe a directed-graph extension and an equal-margin label-shuffling variant. Numerical experiments against Swap, Rectangle Loop, Curveball, sequential importance sampling, and a directed edge-swap algorithm show consistent gains in move size, wall-clock convergence, and sampling efficiency.
