We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy-minimizing functions on local patches. On a regular mesh with mesh size $h$, the localized basis functions have supports of diameter $O(h\log(1/h))$ and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter $O(h\log(1/h))$, our method achieves the optimal compression rate of the solution operator. From the perspective of the generalized finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate $O(h^k)$ for a $(2k)$th-order elliptic problem in the energy norm. From the perspective of the sparse PCA, our results show that a large set of Matérn covariance functions can be approximated by a rank-$n$ operator with a localized basis and with the optimal accuracy.
Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity
从位置$\{ x_{i} \}_{1 \leq i \leq N} \subset \mathbb{R}^{d}$处的协方差函数$G$的点评估获得的稠密核矩阵$Θ\in \mathbb{R}^{N \times N}$出现在统计、机器学习和数值分析中。对于协方差函数是格林函数的椭圆边值问题和离散分布的采样点，我们展示了如何确定一个子集$S \subset \{ 1 , \dots , N \}^2$，与$\# S = O ( N \log (N) \log^{d} ( N /ε) )$，使零填充不完全Cholesky分解的稀疏矩阵$Θ_{ij} 1_{( i, j ) \in S}$是$ε$ -近似的$Θ$。这种因式分解可以证明在空间和$O ( N \log^{2}( N ) \log^{2d}( N /ε) )$时间的复杂性$O ( N \log( N ) \log^{d}( N /ε) )$，提高了现有技术的一般椭圆算子，我们进一步提出的数值证据表明，$d$可以被视为是固有的尺寸的数据集，而不是周围的空间。该算法只需要知道的$x_{i}$的空间配置，并不需要$G$的解析表示。此外，这种因式分解直接提供了一种近似稀疏PCA，其在算子范数中具有最佳收敛速度。因此，通过仅使用子采样和不完全Cholesky分解，我们得到，在接近线性的复杂性，压缩，反演和近似PCA的一大类协方差矩阵。通…
