论文标题
通过转化的未经调整的Langevin算法进行的重尾抽样
Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm
论文作者
论文摘要
我们分析了基于在目标密度的某些转换版本上运行未经调整的langevin算法的多项式衰减重尾目标密度的采样复杂性。我们构造的特定封闭形式转换图的特定类别被证明是差异性的，并且特别适合开发有效的基于扩散的采样器。我们表征了可以获得多项式甲骨文复杂性（以维度和逆目标精度）的精确的重尾密度，并提供说明性的例子。我们强调了我们的假设与基于通过分数拉普拉斯运算符定义的非本地差异形式的功能性不平等（超级和弱的庞加莱不平等）之间的关系，用于表征某些稳定驱动的随机微分方程的重尾平衡密度。
We analyze the oracle complexity of sampling from polynomially decaying heavy-tailed target densities based on running the Unadjusted Langevin Algorithm on certain transformed versions of the target density. The specific class of closed-form transformation maps that we construct are shown to be diffeomorphisms, and are particularly suited for developing efficient diffusion-based samplers. We characterize the precise class of heavy-tailed densities for which polynomial-order oracle complexities (in dimension and inverse target accuracy) could be obtained, and provide illustrative examples. We highlight the relationship between our assumptions and functional inequalities (super and weak Poincaré inequalities) based on non-local Dirichlet forms defined via fractional Laplacian operators, used to characterize the heavy-tailed equilibrium densities of certain stable-driven stochastic differential equations.
