提出DC-LA算法，用于非凸正则化的高效采样。
DC-LA: Difference-of-Convex Langevin Algorithm
- 利用DC函数结构，通过Moreau包络平滑正则项
- 在距离耗散条件下实现$q$-Wasserstein距离收敛
- 适用于医学成像等真实场景的不确定性量化
我们研究目标分布为 $π/propto \ ext{exp}(-f-r)$ 的采样问题，其中数据保真项 $f$ 是Lipschitz光滑的，而正则项 $r = r_1 - r_2$ 为非光滑的差分凸（DC）函数，即 $r_1, r_2$ 均为凸函数。通过利用 $r$ 的DC结构，可分别对 $r_1$ 和 $r_2$ 应用Moreau包络实现 $r$ 的平滑化。借鉴DC规划思想，将正则项的凹部分重新分配至数据保真项，并研究其对应的近端Langevin算法（称为DC-LA）。在假设 $V$ 距离耗散的条件下，建立了DC-LA在所有 $q \in \mathbb{N}^*$ 的 $q$-Wasserstein距离下对目标分布 $π$ 的收敛性，仅受离散化与平滑误差影响。结果在更一般的框架和假设下优于此前非对数凹采样工作。数值实验表明，DC-LA在合成设置中生成准确分布，并在真实世界中的计算机断层扫描（CT）应用中提供了定性合理的不确定性量化。
原文摘要 · Abstract (English)
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 with 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 provides qualitatively reasonable uncertainty quantification in a real-world Computed Tomography application.
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