We study a sampling problem whose target distribution is $π\propto \exp(-f-r)$ where the data fidelity term $f$ is Lipschitz smooth while the regularizer term $r=r_1-r_2$ is a non-smooth difference-of-convex (DC) function, i.e., $r_1,r_2$ are convex. By leveraging the DC structure of $r$, we can smooth out $r$ by applying Moreau envelopes to $r_1$ and $r_2$ separately. In line of DC programming, we then redistribute the concave part of the regularizer to the data fidelity and study its corresponding proximal Langevin algorithm (termed DC-LA). We establish convergence of DC-LA to the target distribution $π$, up to discretization and smoothing errors, in the $q$-Wasserstein distance for all $q \in \mathbb{N}^*$, under the assumption that $V$ is distant dissipative. Our results improve previous work on non-log-concave sampling in terms of a more general framework and assumptions. Numerical experiments show that DC-LA produces accurate distributions in synthetic settings and reliably provides uncertainty quantification in a real-world Computed Tomography application.

翻译：本文研究一个采样问题，其目标分布为 $π\propto \exp(-f-r)$，其中数据保真项 $f$ 是Lipschitz光滑的，而正则化项 $r=r_1-r_2$ 是一个非光滑的凸差函数，即 $r_1$ 和 $r_2$ 均为凸函数。通过利用 $r$ 的凸差结构，我们可以分别对 $r_1$ 和 $r_2$ 应用Moreau包络来平滑 $r$。依据凸差规划的思路，我们将正则化项的凹部分重新分配到数据保真项，并研究其对应的近端Langevin算法（称为DC-LA）。在假设 $V$ 是距离耗散的前提下，我们建立了DC-LA在任意 $q \in \mathbb{N}^*$ 的 $q$-Wasserstein距离下收敛到目标分布 $π$ 的结果，其中收敛精度受离散化误差与平滑误差的限制。我们的结果在更一般的框架和假设下，改进了先前关于非对数凹采样的研究工作。数值实验表明，DC-LA在合成设置中能生成精确的分布，并在真实世界的计算机断层扫描应用中可靠地提供不确定性量化。