DC-LA: Difference-of-Convex Langevin Algorithm

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 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 $\pi$, 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 provides qualitatively reasonable uncertainty quantification in a real-world Computed Tomography application.

In applications like reconstructing a CT scan from limited X-ray measurements, a single "best guess" image isn't enough — doctors also need to know how uncertain that guess is. This means drawing random samples from a probability distribution that combines the measurements with prior assumptions about realistic images. Many useful priors make this distribution mathematically awkward, breaking standard sampling algorithms. We designed a new algorithm, DC-LA, for a broad class of such cases. The key idea: many practical priors can be written as the difference of two simpler, well-behaved pieces. We smooth each piece separately, rearrange the terms, and then run a standard sampling procedure -- and we prove mathematically that it converges, up to some smoothing and discretization erros, to the correct distribution. Numerical experiments show that DC-LA produces accurate distributions in synthetic settings and provides qualitatively reasonable uncertainty quantification in a real-world CT application.