We propose an adaptive MCMC method that learns a linear preconditioner which is dense in its off-diagonal elements but sparse in its parametrisation. Due to this sparsity, we achieve a per-iteration computational complexity of $O(m^2d)$ for a user-determined parameter $m$, compared with the $O(d^2)$ complexity of existing adaptive strategies that can capture correlation information from the target. Diagonal preconditioning has an $O(d)$ per-iteration complexity, but is known to fail in the case that the target distribution is highly correlated, see \citet[Section 3.5]{hird2025a}. Our preconditioner is constructed using eigeninformation from the target covariance which we infer using online principal components analysis on the MCMC chain. It is composed of a diagonal matrix and a product of carefully chosen reflection matrices. On various numerical tests we show that it outperforms diagonal preconditioning in terms of absolute performance, and that it outperforms traditional dense preconditioning and multiple diagonal plus low-rank alternatives in terms of time-normalised performance.

翻译：我们提出一种自适应MCMC方法，该方法能够学习一个线性预处理器，其非对角元素稠密但参数化稀疏。借助这种稀疏性，对于用户设定的参数$m$，我们实现了每次迭代的计算复杂度为$O(m^2d)$，而现有能捕捉目标分布相关信息的自适应策略的复杂度为$O(d^2)$。对角预处理虽然每次迭代复杂度为$O(d)$，但在目标分布高度相关时已知会失效（参见\citet[第3.5节]{hird2025a}）。我们的预处理器利用目标协方差矩阵的特征信息构建，该特征信息通过在线主成分分析法从MCMC链中推断得到，由对角矩阵与精心选择的反射矩阵乘积构成。多种数值实验表明，该方法在绝对性能上优于对角预处理，且在时间归一化性能上优于传统稠密预处理及多种对角加低秩替代方法。