Sparse Cholesky Factorization by Kullback--Leibler Minimization

We propose to compute a sparse approximate inverse Cholesky factor $L$ of dense covariance matrix $\Theta$ by minimizing the Kullback--Leibler divergence between Gaussian distributions $\mathcal{N}(0, \Theta)$ and L^{-\top} L^{-1})$, subject sparsity constraint. Surprisingly, this problem has closed-form solution that can be computed efficiently, recovering popular Vecchia approximation in spatial statistics. Based on recent results factors obtained from pairwise evaluation Green's functions elliptic boundary-value problems at points $\{x_{i}\}_{1 \leq i N} \subset \mathbb{R}^{d}$, we an elimination ordering pattern allows us $\epsilon$-approximate such computational complexity $\mathcal{O}(N \log(N/\epsilon)^d)$ space \log(N/\epsilon)^{2d})$ time. To best our knowledge, is asymptotic for class problems. Furthermore, method embarrassingly parallel, automatically exploits low-dimensional structure data, perform Gaussian-process regression linear (in $N$) complexity. Motivated its optimality properties, applying joint training prediction regression, greatly improving stability cost. Finally, show how apply important setting processes with additive noise, compromising neither accuracy nor