系统扫描随机旋转采样器的Wasserstein混合
Wasserstein mixing of a systematic-scan random rotation sampler
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中文总结 AI 辅助
该研究分析系统扫描Kac游走采样器的混合时间，证明其收敛到Haar测度的速度远慢于推测，在特定精度下需n/log n至n阶扫描，且输出分布奇异，但可提供特定应用的有效随机化。
中文摘要 AI 辅助
我们研究了Kac游走的一种系统扫描模拟的混合时间，该模拟被提议作为随机化高维算法中Haar分布正交矩阵的快速替代品，并推测在仅对数次扫描后即可接近Haar测度。我们证明，对于全矩阵分布到Haar测度在Frobenius Wasserstein距离下的收敛，这一推测的加速并不成立。在固定归一化精度下，混合时间介于$n/\log n$阶和$n$阶扫描之间；在固定绝对Frobenius精度下，相应的界限介于$n$阶和$n\log n$阶之间。更强地，在$n/\log n$尺度以下，归一化Wasserstein距离渐近地保持在其极值。我们还证明，对于少于$n/2$次扫描，输出分布相对于Haar测度是奇异的。因此，该采样器可能提供有效的特定应用随机化，而不会表现出更快的完全Haar混合。
英文摘要
We study the mixing time of a systematic-scan analogue of Kac's walk that was proposed as a fast surrogate for Haar-distributed orthogonal matrices in randomized high-dimensional algorithms and was conjectured to approach Haar measure after only logarithmically many sweeps. We show that this conjectured speed-up does not occur for convergence of the full matrix law to Haar measure in Frobenius Wasserstein distance. At fixed normalized accuracy, the mixing time lies between order $n/\log n$ and order $n$ sweeps; at fixed absolute Frobenius accuracy, the corresponding bounds are between order $n$ and order $n\log n$. More strongly, below the scale $n/\log n$, the normalized Wasserstein distance remains asymptotically at its extremal value. We also show that the output law is singular with respect to Haar measure for fewer than $n/2$ sweeps. Thus the sampler may provide effective application-specific randomization without exhibiting the much faster full-Haar mixing.
