We propose a solution for linear inverse problems based on higher-order Langevin diffusion. More precisely, we propose pre-conditioned second-order and third-order Langevin dynamics that provably sample from the posterior distribution of our unknown variables of interest while being computationally more efficient than their first-order counterpart and the non-conditioned versions of both dynamics. Moreover, we prove that both pre-conditioned dynamics are well-defined and have the same unique invariant distributions as the non-conditioned cases. We also incorporate an annealing procedure that has the double benefit of further accelerating the convergence of the algorithm and allowing us to accommodate the case where the unknown variables are discrete. Numerical experiments in two different tasks in communications (MIMO symbol detection and channel estimation) and in three tasks for images showcase the generality of our method and illustrate the high performance achieved relative to competing approaches (including learning-based ones) while having comparable or lower computational complexity.

翻译：我们提出了一种基于高阶朗之万扩散的线性逆问题求解方法。具体而言，我们提出了预条件化的二阶和三阶朗之万动力学，这些动力学能够从感兴趣未知变量的后验分布中进行可证明的采样，同时其计算效率优于一阶动力学以及两种动力学的非预条件化版本。此外，我们证明了两种预条件化动力学是良好定义的，并且具有与非预条件化情况相同的唯一不变分布。我们还引入了一种退火过程，该过程具有双重优势：进一步加速算法收敛，并允许我们处理未知变量为离散情况。在通信领域的两个不同任务（MIMO符号检测和信道估计）以及图像领域的三个任务中进行的数值实验展示了我们方法的通用性，并说明了相对于竞争方法（包括基于学习的方法）所实现的卓越性能，同时具有相当或更低的计算复杂度。