AI 中文总结
该研究基于Girsanov定理的密度比无偏估计量，提出改进的扩散精确模拟及对数凹采样算法，优化了采样复杂度与维度依赖关系，还给出相关应用。
AI 中文摘要
我们利用Girsanov定理得到的密度比无偏估计量，在路径空间上通过拒绝采样研究扩散过程的精确模拟。将其应用于欠阻尼Langevin扩散时，可得到一种算法：在d维空间中，从条件数为κ的强对数凹且对数光滑分布中采样，在Rényi散度精度ε下，查询复杂度为\\(\widetilde O(\kappa^{2/3} d^{1/3}\\,\mathrm{polylog}(1/\varepsilon))\\)；在三阶导数有界的条件下，维度依赖关系可优化至\\(d^{1/5}\\)。这大幅优于现有最优复杂度：Metropolis调整Langevin算法的\\(\widetilde O(\kappa d^{1/2}\\,\mathrm{polylog}(1/\varepsilon))\\)，以及相同三阶导数有界条件下Metropolized哈密顿蒙特卡洛的\\(d^{1/4}\\)维度依赖。我们还给出了其在镜像Langevin扩散中的应用，以及非对数凹情形下Fisher信息界的获取方法。
英文摘要
We study exact simulation of diffusions via rejection sampling on path space using unbiased estimators of the density ratio obtained from Girsanov's theorem. When applied to the underdamped Langevin diffusion, it yields an algorithm for sampling from a strongly log-concave and log-smooth distribution with condition number $κ$, in dimension $d$, to accuracy $\varepsilon$ in Rényi divergence, in $\widetilde O(κ^{2/3} d^{1/3}\,\mathrm{polylog}(1/\varepsilon))$ queries. Under a third derivative bound, the dimension dependence improves to $d^{1/5}$. This improves substantially over the prior state-of-the-art complexity of $\widetilde O(κd^{1/2}\,\mathrm{polylog}(1/\varepsilon))$ for the Metropolis-adjusted Langevin algorithm, and over the $d^{1/4}$ dimension dependence of Metropolized Hamiltonian Monte Carlo under the same third derivative bound. We also present applications to the mirror Langevin diffusion, and for obtaining Fisher information bounds in the non-log-concave case.
