We develop a new efficient method for high-dimensional sampling called Subspace Langevin Monte Carlo. The primary application of these methods is to efficiently implement Preconditioned Langevin Monte Carlo. To demonstrate the usefulness of this new method, we extend ideas from subspace descent methods in Euclidean space to solving a specific optimization problem over Wasserstein space. Our theoretical analysis demonstrates the advantageous convergence regimes of the proposed method, which depend on relative conditioning assumptions common to mirror descent methods. We back up our theory with experimental evidence on sampling from an ill-conditioned Gaussian distribution.

翻译：我们提出了一种称为子空间朗之万蒙特卡洛的新型高效高维采样方法。该方法的主要应用是实现预条件朗之万蒙特卡洛的高效计算。为证明此新方法的实用性，我们将欧氏空间中子空间下降法的思想推广至求解瓦瑟斯坦空间上的特定优化问题。理论分析表明，所提方法在收敛性方面具有优势，其收敛机制依赖于镜像下降法中常见的相对条件性假设。我们通过对病态高斯分布进行采样的实验验证了理论结果。