论文标题
快速采样$β$ imemembles
Fast sampling from $β$-ensembles
论文作者
论文摘要
我们研究了合奏基质性的$β$浓度的抽样算法，其时间复杂性小于立方体。遵循Dumitriu＆Edelman（2002），我们将合奏视为随机三肌矩阵的特征值，即随机的jacobi矩阵。首先，我们提供了与三种经典赫米特（Laguerre）和雅各比（Jacobi）合奏相关的三角形模型的统一和基本处理。为此，我们使用了定义三角形矩阵的系数的连续重新分析之间的变量的简单更改。其次，我们得出了一个近似的采样器，以模拟$β$浓度，并说明了多项式电位的速度。该方法结合了jacobi矩阵上的吉布斯采样器和这些矩阵的对角线化。实际上，即使对于大型合奏，只有少数吉布斯足以使特征值的边际分布适合预期的理论分布。当可以精确模拟Gibbs采样器中的条件时，对于最大的特征值的波动，观察到相同的快速经验收敛。我们的实验结果支持Krishnapur等人的猜想。 （2016年），jacobi矩阵上的gibbs链中的$ n $ mix in $ \ mathcal {o}（\ log（n））$。
We study sampling algorithms for $β$-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of $β$-ensembles, and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size $N$ mixes in $\mathcal{O}(\log(N))$.
