AI 中文总结
该研究提出剖面分离框架，分析多项式和偏渐进无-U形-turn采样器的定量收敛性，给出无条件转移界，恢复高斯维度依赖并量化热启动后消除线性各向异性的时机。
AI 中文摘要
我们研究针对强对数凹目标的多项式和偏渐进无-U形-turn采样器（No-U-Turn Samplers），这类目标满足 $mI_d\preceq \nabla^2U(x)\preceq LI_d$，且 $\\|\nabla^2U(x)-\nabla^2U(y)\\|_{\mathrm F}\le \gamma L^{3/2}\\|x-y\\|$，其中 $\kappa\coloneqq L/m$。我们引入剖面分离，这是针对平稳均值U形-turn诊断量的充分符号条件，并将其与诊断量集中性、 leapfrog 保真度以及全轨道能量控制相结合，以证明在高概率认证事件下，每个翻倍实现都通过真实U形-turn达到共同终端深度。若 $T_\star$ 为选定的物理轨迹长度且 $a_\star=\sqrt m\\,T_\star$，则终端深度转移论证会产生受限电导和热启动混合，且不会松弛任何一个核。在对数热启动和精度因子范围内，多项式选择和偏渐进选择的转移界分别为 $\widetilde O\\!\left( 1+a_\star^2\kappa^2(1+\gamma)^{4/3} \right)$ 和 $\widetilde O\\!\left( 1+a_\star^4\kappa^3(1+\gamma)^2 \right)$，这些转移界是无条件的。当最大深度上限与认证深度可比时，梯度-工作量界是确定性的，否则会采取感知上限的期望和高概率形式。该框架在这些工作量核算条件下恢复了高斯维度依赖性，为非线性乘积目标提供了总体剖面和精确诊断验证，为实用树认证提供了近各向同性特化，并量化了热启动后固定度量消除线性各向异性的时机。
英文摘要
We study multinomial and biased-progressive No-U-Turn Samplers for strongly log-concave targets satisfying $mI_d\preceq \nabla^2U(x)\preceq LI_d,$ and $\|\nabla^2U(x)-\nabla^2U(y)\|_{\mathrm F}\le γL^{3/2}\|x-y\|$ with \(κ\coloneqq L/m\). We introduce profile separation, a sufficient sign condition on the stationary mean U-turn diagnostics, and combine it with diagnostic concentration, leapfrog fidelity, and whole-orbit energy control to show that on a high-probability certification event, every doubling realization reaches a common terminal depth through a genuine U-turn. If \(T_\star\) is the selected physical trajectory length and \(a_\star=\sqrt m\,T_\star\), a terminal-depth transfer argument yields restricted conductance and warm-start mixing without lazifying either kernel. Up to logarithmic warm-start and accuracy factors, the transition bounds are \[ \widetilde O\!\left( 1+a_\star^2κ^2(1+γ)^{4/3} \right) \quad\text{and}\quad \widetilde O\!\left( 1+a_\star^4κ^3(1+γ)^2 \right) \] for multinomial and biased-progressive selection, respectively. These transition bounds are unconditional. Gradient-work bounds are deterministic when the maximum-depth cap is comparable to the certified depth and otherwise take cap-aware expected and high-probability forms. The framework recovers the Gaussian dimension dependence under these work-accounting conditions, provides population-profile and exact-diagnostic verification for nonlinear product targets, a near-isotropic specialization of the practical-tree certificate, and quantifies when a fixed post-warmup metric removes linear anisotropy.
