This paper studies the estimation of large precision matrices and Cholesky factors obtained by observing a Gaussian process at many locations. Under general assumptions on the precision and the observations, we show that the sample complexity scales poly-logarithmically with the size of the precision matrix and its Cholesky factor. The key challenge in these estimation tasks is the polynomial growth of the condition number of the target matrices with their size. For precision estimation, our theory hinges on an intuitive local regression technique on the lattice graph which exploits the approximate sparsity implied by the screening effect. For Cholesky factor estimation, we leverage a block-Cholesky decomposition recently used to establish complexity bounds for sparse Cholesky factorization.

翻译：本文研究通过观测高斯过程在多个位置的数据来估计大规模精度矩阵及其Cholesky因子的问题。在关于精度矩阵与观测数据的一般性假设下，我们证明样本复杂度随精度矩阵及其Cholesky因子规模的增大呈多对数级别增长。这些估计任务中的关键挑战在于目标矩阵的条件数随其规模呈多项式增长。对于精度矩阵估计，我们的理论依赖于格点图上一种直观的局部回归技术，该技术利用了屏蔽效应所隐含的近似稀疏性。对于Cholesky因子估计，我们采用了一种近期用于建立稀疏Cholesky分解复杂度界限的分块Cholesky分解方法。