Sampling from Unknown Transition Densities of Diffusion processes

Which Sampling Densities are Suitable for Spectral Clustering on Unbounded Domains?

We consider a random geometric graph with vertices sampled from a probability measure supported on $\mathbb R^d$, and study its connectivity. We show the graph is typically disconnected, unless the sampling density has superexponential decay. In the later setting, we identify an asymptotic threshold value for the radius parameter of the graph such that, for radius values beyond the threshold, some concentration properties hold for the sampled points of the graph, while the graph is disconnected for radius values below the same threshold. Properties of point processes are well-known to be close…

Convergence of the Riemannian Langevin Algorithm

We study the Riemannian Langevin Algorithm for the problem of sampling from a distribution with density $ν$ with respect to the natural measure on a manifold with metric $g$. We assume that the target density satisfies a log-Sobolev inequality with respect to the metric and prove that the manifold generalization of the Unadjusted Langevin Algorithm converges rapidly to $ν$ for Hessian manifolds. This allows us to reduce the problem of sampling non-smooth (constrained) densities in ${\bf R}^n$ to sampling smooth densities over appropriate manifolds, while needing access only to the gradient of …

从玻尔兹曼密度进行神经采样：Wasserstein几何中的Fisher-Rao曲线

Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry

本文研究从非归一化玻尔兹曼密度采样问题，针对线性插值的质量瞬移缺陷，提出仅参数化能量并固定适当速度场的插值方法，对应Langevin动力学下KL散度的Wasserstein梯度流，并通过数值实验验证其有效性。

Sampling Matrices from Harish-Chandra-Itzykson-Zuber Densities with Applications to Quantum Inference and Differential Privacy

Given two $n \times n$ Hermitian matrices $Y$ and $Λ$, the Harish-Chandra-Itzykson-Zuber (HCIZ) distribution on the unitary group $\text{U}(n)$ is $e^{\text{tr}(UΛU^*Y)}dμ(U)$, where $μ$ is the Haar measure on $\text{U}(n)$. The density $e^{\text{tr}(UΛU^*Y)}$ is known as the HCIZ density. Random unitary matrices distributed according to the HCIZ density are important in various settings in physics and random matrix theory. However, the basic question of efficient sampling from the HCIZ distribution has remained open. We present two efficient algorithms to sample matrices from distributions th…

随机微分方程离散时间采样的极值理论与泊松统计

Extreme Value theory and Poisson statistics for discrete time samplings of stochastic differential equations

本文研究随机微分方程离散时间采样所得随机过程的极端事件分布与多次发生，通过退火传递算子谱分析，结合SDE理论与泛函分析方法，揭示其极值统计规律。

通过Hessian-free高分辨率动力学采样非对数凹密度

Sampling non-log-concave densities via Hessian-free high-resolution dynamics

通过Hessian-free高分辨率动力学实现非对数凹密度采样，理论分析显示其收敛性和收缩率优于动能兰格-恩动力学。