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Dense kernel matrices $\Theta \in \mathbb{R}^{N \times N}$ obtained frompoint evaluations of a covariance function $G$ at locations $\{ x_{i} \}_{1\leq i \leq N} \subset \mathbb{R}^{d}$ arise in statistics, machine learning,and numerical analysis. For covariance functions that are Green's functions ofelliptic boundary value problems and homogeneously-distributed sampling points,we show how to identify a subset $S \subset \{ 1 , \dots , N \}^2$, with $\# S= O ( N \log (N) \log^{d} ( N /\epsilon ) )$, such that the zero fill-inincomplete Cholesky factorisation of the sparse matrix $\Theta_{ij} 1_{( i, j )\in S}$ is an $\epsilon$-approximation of $\Theta$. This factorisation canprovably be obtained in complexity $O ( N \log( N ) \log^{d}( N /\epsilon) )$in space and $O ( N \log^{2}( N ) \log^{2d}( N /\epsilon) )$ in time, improvingupon the state of the art for general elliptic operators; we further presentnumerical evidence that $d$ can be taken to be the intrinsic dimension of thedata set rather than that of the ambient space. The algorithm only needs toknow the spatial configuration of the $x_{i}$ and does not require an analyticrepresentation of $G$. Furthermore, this factorization straightforwardlyprovides an approximate sparse PCA with optimal rate of convergence in theoperator norm. Hence, by using only subsampling and the incomplete Choleskyfactorization, we obtain, at nearly linear complexity, the compression,inversion and approximate PCA of a large class of covariance matrices. Byinverting the order of the Cholesky factorization we also obtain a solver forelliptic PDE with complexity $O ( N \log^{d}( N /\epsilon) )$ in space and $O (N \log^{2d}( N /\epsilon) )$ in time, improving upon the state of the art forgeneral elliptic operators.

Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity