有限马尔可夫链k-块平均核的谱划分
Spectral partitioning for $k$-block averaging kernels of finite Markov chains
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中文总结 AI 辅助
提出基于马尔可夫链核底部特征模式的谱划分算法，用于加速平均核收敛，在三类实验中验证了其收敛与统计估计的迭代改进效果。
中文摘要 AI 辅助
我们开发了用于选择状态空间划分的谱算法，该划分定义了有限、遍历且可逆马尔可夫链的平均核。对于划分$\boldsymbol{\text{O}}$，吉布斯核$G_{\boldsymbol{\text{O}}}$从平稳条件分布中在当前块内重采样；当该更新可处理时，将其与基线核$P$组合或混合可加速收敛。我们通过对$P^2$的底部非恒定特征函数，或对加性混合情况的$P$的代数最小特征函数使用加权$k$-均值来选择$\boldsymbol{\text{O}}$。对于$F(\boldsymbol{\text{O}})=\boldsymbol{\text{||}}G_{\boldsymbol{\text{O}}}P-\boldsymbol{\text{||}}_{F,\boldsymbol{\text{\text{π}}}}^2$，我们推导了精确的迹和归一化割表示，并证明$F$等于初始块标签与一次转移后状态之间的皮尔逊$\boldsymbol{\text{\text{χ}}}^2$-互信息，为该矩阵目标提供了自然的概率解释。在两块情况下，阈值扫描可精确求解相关的一维加权两均值舍入问题。对于一般$k\boldsymbol{\text{\text{≥}}}2$，加权$k$-均值对底部$\boldsymbol{\text{(}}k\boldsymbol{\text{-}}1\boldsymbol{\text{)}}$-维嵌入进行舍入，之后通过$F$对候选进行重新评分；舍入失真为子空间间的距离，可得到谱近似界。我们将该框架扩展到加性混合、有限时域目标和折扣无限时域目标。与使用顶部非恒定模式寻找低流持久簇的经典归一化谱聚类不同，我们的方法使用底部模式来促进大的归一化跨块流和块标签信息的快速损失。在受控谱图、平均场伊辛模型和贝叶斯变量选择上的实验显示，在收敛和统计估计方面有显著的每次迭代改进。
英文摘要
We develop spectral algorithms for selecting state-space partitions that define averaging kernels for finite, ergodic and reversible Markov chains. For a partition $\mathcal O$, the Gibbs kernel $G_{\mathcal O}$ resamples within the current block from the stationary conditional distribution; when this update is tractable, composing or mixing it with a baseline kernel $P$ can accelerate convergence. We select $\mathcal O$ by rounding the bottom nonconstant eigenfunctions of $P^2$, or the algebraically smallest eigenfunctions of $P$ for additive mixtures, using weighted $k$-means. For $F(\mathcal O)=\|G_{\mathcal O}P-Π\|_{F,π}^2$, we derive exact trace and normalized-cut representations and show that $F$ equals the Pearson $χ^2$-mutual information between the initial block label and the state after one transition, giving this matrix objective a natural probabilistic interpretation. In the two-block case, a threshold sweep exactly solves the associated one-dimensional weighted two-means rounding problem. For general $k \geq 2$, weighted $k$-means rounds the bottom $(k-1)$-dimensional embedding, after which candidates are rescored by $F$; the rounding distortion is a distance between subspaces that yields spectral approximation bounds. We extend the framework to additive mixtures, finite-horizon objectives, and discounted infinite-horizon objectives. In contrast to classical normalized spectral clustering, which uses top nonconstant modes to find low-flow persistent clusters, our method uses bottom modes to favor large normalized cross-block flow and rapid loss of block-label information. Experiments on a controlled-spectrum graph, a mean-field Ising model, and Bayesian variable selection show notable per-iteration improvements in convergence and statistical estimation.
发表机构
- National University of Singapore（新加坡国立大学）
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