Hadamard Langevin dynamics for sampling the l1-prior

TL;DR

This paper introduces Hadamard Langevin dynamics, a novel sampling method for non-smooth, nonconvex posteriors like the l1-prior, with rigorous theoretical guarantees for its well-posedness and convergence.

Contribution

It proposes a new Hadamard product-based parameterization for the l1-prior, enabling Langevin sampling without smoothing or proximal steps, and provides a theoretical foundation for its use.

Findings

Established existence and uniqueness of solutions for the continuous HLD.

Proved geometric ergodicity of the continuous dynamics.

Showed convergence of the discretized scheme as step size decreases.

Abstract

Priors with non-smooth log-densities, such as the l1-prior, are widely used in Bayesian inverse problems for their sparsity-inducing properties. Existing Langevin-based sampling methods typically rely on proximal mappings or smooth approximations, which alter the target distribution. We propose an alternative approach based on a Hadamard product parameterization of the l1-norm, leading to a smooth but nonconvex and non-globally Lipschitz potential whose marginal law exactly recovers the desired posterior. The resulting Hadamard Langevin dynamics (HLD) defines a diffusion process that is analytically distinct from proximal or mirror-type Langevin schemes. Our main contribution is a rigorous well-posedness theory for both the continuous and discrete HLD. We establish existence and uniqueness of strong solutions, geometric ergodicity of the continuous dynamics, and convergence of the discretized scheme as the step size tends to zero. These results provide the first theoretical foundation for sampling from nonconvex, nonsmooth posteriors through overparameterized Langevin dynamics.