# DC-LA: Difference-of-Convex Langevin Algorithm

arXiv: https://arxiv.org/abs/2601.22932 · HTML: https://mlsanity.com/paper/2601.22932/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 68 references that are in the map.
- First posted on arXiv 30 Jan 2026.

## Builds on (most cited first, 30 of 32)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Provable Probabilistic Imaging using Score-Based Generative Priors](https://mlsanity.com/paper/2310.10835/) (2023)
- [Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity](https://mlsanity.com/paper/1907.11331/) (2019)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm](https://mlsanity.com/paper/2006.09270/) (2020)
- [Inverse Problems and Data Assimilation](https://mlsanity.com/paper/1810.06191/) (2018)
- [Stochastic Proximal Langevin Algorithm: Potential Splitting and Nonasymptotic Rates](https://mlsanity.com/paper/1905.11768/) (2019)
- [Fast L1-L2 minimization via a proximal operator](https://mlsanity.com/paper/1609.09530/) (2016)
- [Subgradient Langevin Methods for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2308.01417/) (2023)
- [Composite Difference-Max Programs for Modern Statistical Estimation Problems](https://mlsanity.com/paper/1803.00205/) (2018)
- [Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction](https://mlsanity.com/paper/2106.04170/) (2021)
- [DC approximation approaches for sparse optimization](https://mlsanity.com/paper/1407.0286/) (2014)
- [Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Models](https://mlsanity.com/paper/2310.03546/) (2023)
- [Learning Difference-of-Convex Regularizers for Inverse Problems: A Flexible Framework with Theoretical Guarantees](https://mlsanity.com/paper/2502.00240/) (2025)
- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)
- [From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling](https://mlsanity.com/paper/2505.14177/) (2025)
- [Efficient Learning with a Family of Nonconvex Regularizers by Redistributing Nonconvexity](https://mlsanity.com/paper/1606.03841/) (2016)
- [Non-geodesically-convex optimization in the Wasserstein space](https://mlsanity.com/paper/2406.00502/) (2024)
- [On the Pervasiveness of Difference-Convexity in Optimization and Statistics](https://mlsanity.com/paper/1704.03535/) (2017)
- [Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions](https://mlsanity.com/paper/2405.18577/) (2024)
- [The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions](https://mlsanity.com/paper/2502.03458/) (2025)
- [Further properties of the forward-backward envelope with applications to difference-of-convex programming](https://mlsanity.com/paper/1605.00201/) (2016)
- [Stochastic variance-reduced Gaussian variational inference on the Bures-Wasserstein manifold](https://mlsanity.com/paper/2410.02490/) (2024)
- [The Gaussian Latent Machine: Efficient Prior and Posterior Sampling for Inverse Problems](https://mlsanity.com/paper/2505.12836/) (2025)
- [On the stability of the stochastic gradient Langevin algorithm with dependent data stream](https://mlsanity.com/paper/2105.01422/) (2021)

## Built on by (most cited first, 2 of 2)

- [The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexity](https://mlsanity.com/paper/2608.06283/) (2026)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)

## Abstract (arXiv)

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=r1-r2 is a non-smooth difference-of-convex (DC) function, i.e., r1,r2 are convex. By leveraging the DC structure of r, we can smooth out r by applying Moreau envelopes to r1 and r2 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 ∈ mathbbN^*, 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.