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，我们设计了预条件二阶和三阶朗之万动力学，能够从感兴趣未知变量的后验分布中进行可证明的采样，同时其计算效率显著高于一阶动力学及两种动力学的无条件版本。此外，我们证明了两种预条件动力学均具有良定义性，且与无条件情形保持相同的唯一不变分布。我们进一步引入退火过程，其双重优势在于既能加速算法收敛，又能处理未知变量为离散的情况。在通信领域两项不同任务（MIMO符号检测与信道估计）以及三项图像任务中的数值实验表明，该方法具有通用性，在计算复杂度相当或更低的情况下，相比现有竞争方法（包括基于学习的方法）实现了更优越的性能。