# Spatial Process Approximations: Assessing Their Necessity

arXiv: https://arxiv.org/abs/2311.03201 · HTML: https://mlsanity.com/paper/2311.03201/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 57 references that are in the map.
- First posted on arXiv 6 Nov 2023.

## Builds on (most cited first, 9 of 9)

- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Does data interpolation contradict statistical optimality?](https://mlsanity.com/paper/1806.09471/) (2018)
- [Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition](https://mlsanity.com/paper/1504.05477/) (2015)
- [A Case Study Competition Among Methods for Analyzing Large Spatial Data](https://mlsanity.com/paper/1710.05013/) (2017)
- [Approximation beats concentration? An approximation view on inference with smooth radial kernels](https://mlsanity.com/paper/1801.03437/) (2018)
- [Permutation and Grouping Methods for Sharpening Gaussian Process Approximations](https://mlsanity.com/paper/1609.05372/) (2016)
- [Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice](https://mlsanity.com/paper/1402.4281/) (2014)
- [A Comparison of Spatial Predictors when Datasets Could be Very Large](https://mlsanity.com/paper/1410.7748/) (2014)
- [Sparse nearest neighbor Cholesky matrices in spatial statistics](https://mlsanity.com/paper/2102.13299/) (2021)

## Abstract (arXiv)

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.