发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究在未知低维流形约束下的采样问题，提出基于局部凸包重构和平方距离惩罚的采样方法，证明其误差与豪斯多夫误差平方根成正比，并给出有限时间保证。
AI 中文摘要
我们研究从支撑在未知紧致 $d$ 维 $C^2$ 流形 $M\subset\mathbb{R}^D$ 上的分布进行采样的问题，该分布仅通过来自 $M$ 的独立同分布均匀点观测。我们使用自适应局部凸包估计器重构约束，并针对到重构集的平方距离惩罚的环境分布进行采样。尽管重构集可能非光滑或无法构成流形，但其豪斯多夫精度足以控制 Wasserstein 误差。我们量化了重构精度与惩罚强度之间的权衡，并表明最优调参给出的误差与豪斯多夫误差的平方根成正比。一个平面例子证明了该依赖关系对所提方案是尖锐的。对于从 $N$ 个观测重构的 $d$ 维约束，误差为 $\mathcal{O}((N/N)^{1/d})$。最后，我们证明了高斯随机游走 Metropolis--Hastings 采样的均匀几何遍历性，并将重构、逼近和混合结合成显式的有限时间保证。
英文摘要
We study sampling from a distribution supported on an unknown compact $d$-dimensional $C^2$ manifold $M\subset\mathbb{R}^D$, observed only through i.i.d. uniform points from $M$. We reconstruct the constraint using an adaptive local-convex-hull estimator and target an ambient distribution penalized by squared distance to the reconstruction. Although the reconstructed set may be nonsmooth or fail to be a manifold, its Hausdorff accuracy alone suffices to control the Wasserstein error. We quantify the tradeoff between reconstruction accuracy and penalty strength and show that optimal tuning gives an error proportional to the square root of the Hausdorff error. A planar example proves that this dependence is sharp for the proposed scheme. For a $d$ dimensional constraint reconstructed from $N$ observations, the error becomes $\mathcal{O}( (N/N)^{1/d})$. Finally, we prove uniform geometric ergodicity of a Gaussian random-walk Metropolis--Hastings sampler and combine reconstruction, approximation, and mixing into an explicit finite-time guarantee.
Comments33 pages, 4 figures
