# Fast spatial Gaussian process maximum likelihood estimation via skeletonization factorizations

arXiv: https://arxiv.org/abs/1603.08057 · HTML: https://mlsanity.com/paper/1603.08057/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Mar 2016.

## Built on by (most cited first, 6 of 6)

- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [The Fast Kernel Transform](https://mlsanity.com/paper/2106.04487/) (2021)
- [Fast Kernel Summation in High Dimensions via Slicing and Fourier Transforms](https://mlsanity.com/paper/2401.08260/) (2024)
- [Fast Summation of Radial Kernels via QMC Slicing](https://mlsanity.com/paper/2410.01316/) (2024)
- [Turbocharging Gaussian Process Inference with Approximate Sketch-and-Project](https://mlsanity.com/paper/2505.13723/) (2025)
- [Adaptive multi-resolution Gaussian processes: Scalable exact inference with naturally data-sparse covariance matrices](https://mlsanity.com/paper/2609.30348/) (2026)

## Abstract (arXiv)

Maximum likelihood estimation for parameter-fitting given observations from a Gaussian process in space is a computationally-demanding task that restricts the use of such methods to moderately-sized datasets. We present a framework for unstructured observations in two spatial dimensions that allows for evaluation of the log-likelihood and its gradient (i.e., the score equations) in tilde O(n3/2) time under certain assumptions, where n is the number of observations. Our method relies on the skeletonization procedure described by Martinsson & Rokhlin in the form of the recursive skeletonization factorization of Ho & Ying. Combining this with an adaptation of the matrix peeling algorithm of Lin et al. for constructing H-matrix representations of black-box operators, we obtain a framework that can be used in the context of any first-order optimization routine to quickly and accurately compute maximum-likelihood estimates.