用马尔可夫链采样估计协方差，效率可比独立采样。
Covariance estimation using Markov chain Monte Carlo
- 基于马尔可夫链依赖样本构造协方差估计器
- 在Poincaré不等式和谱间隙条件下，样本复杂度接近独立采样
- 适用于凸体均匀采样中的各向同性归一化，提升查询效率
我们研究了基于马尔可夫链生成的依赖样本对吉布斯分布协方差矩阵估计的复杂度。当分布π满足Poincaré不等式且链具有谱间隙时，使用MCMC可达到与独立同分布样本相当的样本复杂度，且查询复杂度可能显著更低。作为应用，我们在约束与非约束设置下对具体MCMC实例的查询复杂度给出了改进。特别地，我们为凸体上均匀采样的各向同性归一化过程提供了理论保证。
原文摘要 · Abstract (English)
We investigate the complexity of covariance matrix estimation for Gibbs distributions based on dependent samples from a Markov chain. We show that when $π$ satisfies a Poincaré inequality and the chain possesses a spectral gap, we can achieve similar sample complexity using MCMC as compared to an estimator constructed using i.i.d. samples, with potentially much better query complexity. As an application of our methods, we show improvements for the query complexity in both constrained and unconstrained settings for concrete instances of MCMC. In particular, we provide guarantees regarding isotropic rounding procedures for sampling uniformly on convex bodies.
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