Priors with non-smooth log densities have been widely used in Bayesian inverse problems, particularly in imaging, due to their sparsity inducing properties. To date, the majority of algorithms for handling such densities are based on proximal Langevin dynamics where one replaces the non-smooth part by a smooth approximation known as the Moreau envelope. In this work, we introduce a novel approach for sampling densities with $\ell_1$-priors based on a Hadamard product parameterization. This builds upon the idea that the Laplace prior has a Gaussian mixture representation and our method can be seen as a form of overparametrization: by increasing the number of variables, we construct a density from which one can directly recover the original density. This is fundamentally different from proximal-type approaches since our resolution is exact, while proximal-based methods introduce additional bias due to the Moreau-envelope smoothing. For our new density, we present its Langevin dynamics in continuous time and establish well-posedness and geometric ergodicity. We also present a discretization scheme for the continuous dynamics and prove convergence as the time-step diminishes.

翻译：非光滑对数密度先验因其稀疏诱导特性，已在贝叶斯逆问题（特别是成像领域）中得到广泛应用。迄今为止，处理此类密度的算法主要基于近似Langevin动力学，即通过称为Moreau包络的光滑近似替代非光滑部分。本研究提出一种基于Hadamard积参数化的新方法，用于采样具有$\ell_1$先验的密度。该方法基于Laplace先验具有高斯混合表示的特性，可视为一种过参数化形式：通过增加变量数量，构建可直接恢复原始密度的新密度。这与近似类方法存在本质区别——我们的解法是精确的，而基于近似的方法会因Moreau包络平滑引入额外偏差。针对新构建的密度，我们给出了其连续时间Langevin动力学，并建立了适定性与几何遍历性。同时提出了连续动力学的离散化方案，并证明了时间步长趋近于零时的收敛性。