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

Dense kernel matrices $\Theta \in \mathbb{R}^{N \times N}$ obtained from point 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 of elliptic 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-in incomplete Cholesky factorisation of the sparse matrix $\Theta_{ij} 1_{( i, j ) \in S}$ is an $\epsilon$-approximation of $\Theta$. This factorisation can provably 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, improving upon the state of the art for general elliptic operators; we further present numerical evidence that $d$ can be taken to be the intrinsic dimension of the data set rather than that of the ambient space. The algorithm only needs to know the spatial configuration of the $x_{i}$ and does not require an analytic representation of $G$. Furthermore, this factorization straightforwardly provides an approximate sparse PCA with optimal rate of convergence in the operator norm. Hence, by using only subsampling and the incomplete Cholesky factorization, we obtain, at nearly linear complexity, the compression, inversion and approximate PCA of a large class of covariance matrices. By inverting the order of the Cholesky factorization we also obtain a solver for elliptic 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 for general elliptic operators.