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 (MIMO symbol detection and channel estimation) 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：我们提出一种基于高阶朗之万扩散的线性逆问题求解方案。具体而言，我们提出预条件二阶与三阶朗之万动力学方法，该方法能在从感兴趣未知变量后验分布采样时被严格证明具有计算效率优势——相较于其一阶对应方法及未条件化的高阶动力学方法而言。进一步地，我们证明两种预条件动力学均具有良好定义，且与未条件化情形具有相同的唯一不变分布。同时引入退火策略，该策略具有双重优势：既能进一步加速算法收敛，又能处理未知变量为离散的情形。在两项不同任务（MIMO符号检测与信道估计）上的数值实验展示了本方法的普适性，并表明其在保持可比或更低计算复杂度的前提下，相较于竞争方法（包括基于学习的方法）实现了更优性能。