🤖 AI Summary
This work addresses the challenge of sampling from non-log-concave distributions arising from nonsmooth difference-of-convex (DC) regularized models. It introduces, for the first time, a DC decomposition within the Langevin sampling framework: the convex component is smoothed via the Moreau envelope, while the concave part is incorporated into the data fidelity term, yielding a novel proximal Langevin algorithm. This approach substantially relaxes the assumptions on the target distribution required by existing theory, thereby extending the applicability of non-log-concave sampling methods. Through Wasserstein convergence analysis, the algorithm accurately recovers the target distribution on synthetic data and demonstrates reliable uncertainty quantification in real-world CT imaging tasks.
📝 Abstract
We study a sampling problem whose target distribution is $\pi \propto \exp(-f-r)$ 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 $\pi$, 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.