  In spatial statistics and machine learning, the kernel matrix plays a pivotal
role in prediction, classification, and maximum likelihood estimation. A
thorough examination reveals that for large sample sizes, the kernel matrix
becomes ill-conditioned, provided the sampling locations are fairly evenly
distributed. This condition poses significant challenges to numerical
algorithms used in prediction and estimation computations and necessitates an
approximation to prediction and the Gaussian likelihood. A review of current
methodologies for managing large spatial data indicates that some fail to
address this ill-conditioning problem. Such ill-conditioning often results in
low-rank approximations of the stochastic processes. This paper introduces
various optimality criteria and provides solutions for each.
