[Paper Review] Fast Direct Methods for Gaussian Processes and the Analysis of NASA Kepler Mission Data

Sivaram Ambikasaran, Daniel Foreman-Mackey+3 more|arXiv (Cornell University)|Mar 24, 2014

Gaussian Processes and Bayesian Inference21 citations

TL;DR

This paper presents an O(n log²n) direct method for Gaussian process inference by hierarchically factorizing the covariance matrix C = σ²I + K into block low-rank updates of the identity, enabling efficient matrix inversion and determinant evaluation. The approach achieves near-optimal scaling, making high-dimensional marginalization and hyperparameter adaptation feasible on a single CPU core, as demonstrated on Kepler Mission data.

ABSTRACT

Abstract—A number of problems in probability and statistics can be addressed using the multivariate normal (or multivariate Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the n-dimensional setting, however, it requires the inversion of an n × n covariance matrix, C, as well as the evaluation of its determinant, det(C). In many cases, the covariance matrix is of the form C = σ2I + K, where K is computed using a specified kernel, which depends on the data and additional parameters (called hyperparameters in Gaussian process computations). The matrix C is typically dense, causing standard direct methods for inversion and determinant evaluation to require O(n3) work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix C can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an O(n log2 n) algorithm for inversion, as discussed in Ambikasaran and Darve, 2013. More importantly, we show that this factorization enables the evaluation of the determinant det(C), permitting the direct calculation of probabilities in high dimensions under fairly broad assumption about the kernel defining K. Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels, and apply it to a real data set obtained from the Kepler Mission.

Motivation & Objective

• Address the computational infeasibility of standard O(n³) Gaussian process methods for large-scale problems.
• Develop a fast, direct algorithm for matrix inversion and determinant evaluation in multivariate Gaussian models.
• Enable practical computation of marginal likelihoods and hyperparameter adaptation in high-dimensional settings.
• Demonstrate the method's effectiveness on real-world astronomical data from the Kepler Mission.
• Achieve near-optimal O(n log²n) scaling while maintaining numerical accuracy for common kernel functions.

Proposed method

• Use hierarchical matrix factorization to represent the covariance matrix C = σ²I + K as a product of block low-rank updates of the identity matrix.
• Leverage the structure of common kernel functions to enable efficient hierarchical compression of the matrix K.
• Apply direct factorization techniques to compute the inverse and determinant of C in O(n log²n) time.
• Ensure numerical stability and accuracy by preserving the low-rank structure throughout the factorization process.
• Use the computed determinant and inverse to evaluate the marginal likelihood for hyperparameter optimization.
• Implement the algorithm using a single CPU core, avoiding reliance on distributed computing for scalability.

Experimental results

Research questions

• RQ1 Can the computational cost of Gaussian process inference be reduced from O(n³) to O(n log²n) for standard kernel functions?
• RQ2 Is it possible to compute the determinant of a dense covariance matrix efficiently using hierarchical low-rank structure?
• RQ3 Can direct matrix inversion and determinant evaluation be performed in near-linearithmic time while preserving accuracy?
• RQ4 Does the proposed method enable practical hyperparameter learning and marginal likelihood evaluation on large-scale datasets?
• RQ5 Can the algorithm be effectively applied to real-world, high-dimensional data such as that from the Kepler space telescope?

Key findings

• The proposed method reduces the computational cost of Gaussian process inference from O(n³) to O(n log²n) for common kernel functions.
• The hierarchical low-rank factorization enables accurate and efficient evaluation of both the matrix inverse and determinant.
• The algorithm allows for direct computation of the marginal likelihood, facilitating hyperparameter adaptation on large datasets.
• The method achieves high performance on a single CPU core, making it practical for problems previously considered intractable.
• The approach is successfully applied to real Kepler Mission data, demonstrating scalability and numerical robustness.
• The scheme maintains near-optimal scaling, enabling future integration with high-performance computing for even larger problems.