We propose a sparse algebra for samplet compressed kernel matrices, to enable efficient scattered data analysis. We show the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. It can be performed in cost and memory that scale essentially linearly with the matrix size $N$, for kernels of finite differentiability, along with addition and multiplication of S-formatted matrices. We prove and exploit the fact that the inverse of a kernel matrix (if it exists) is compressible in the S-format as well. Selected inversion allows to directly compute the entries in the corresponding sparsity pattern. The S-formatted matrix operations enable the efficient, approximate computation of more complicated matrix functions such as ${\bm A}^\alpha$ or $\exp({\bm A})$. The matrix algebra is justified mathematically by pseudo differential calculus. As an application, efficient Gaussian process learning algorithms for spatial statistics is considered. Numerical results are presented to illustrate and quantify our findings.

翻译：我们提出了一种针对样本紧支核矩阵的稀疏代数方法，以实现高效的散乱数据分析。研究表明，通过样本压缩核矩阵可在特定S格式下生成最优稀疏矩阵。对于有限可微的核函数，这种压缩的计算代价与内存需求与矩阵规模$N$呈近似线性关系，且支持S格式矩阵的加法与乘法运算。我们证明并利用了如下性质：核矩阵的逆矩阵（若存在）同样可在S格式下进行压缩。选择性求逆算法允许直接计算对应稀疏模式中的矩阵元素。通过S格式矩阵运算，可高效近似计算更复杂的矩阵函数，如${\bm A}^\alpha$或$\exp({\bm A})$。该矩阵代数方法在数学上由伪微分算子理论提供支撑。作为应用实例，本研究探讨了面向空间统计的高效高斯过程学习算法，并通过数值结果对所提出的方法进行说明与量化分析。