# Scalable Log Determinants for Gaussian Process Kernel Learning

arXiv: https://arxiv.org/abs/1711.03481 · HTML: https://mlsanity.com/paper/1711.03481/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Nov 2017.

## Built on by (most cited first, 30 of 35)

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- [Scalable First-Order Bayesian Optimization via Structured Automatic Differentiation](https://mlsanity.com/paper/2206.08366/) (2022)
- [The Fast Kernel Transform](https://mlsanity.com/paper/2106.04487/) (2021)
- [Robust Gaussian Processes via Relevance Pursuit](https://mlsanity.com/paper/2410.24222/) (2024)
- [Active Learning for Deep Neural Networks on Edge Devices](https://mlsanity.com/paper/2106.10836/) (2021)
- [Optimal Query Complexities for Dynamic Trace Estimation](https://mlsanity.com/paper/2209.15219/) (2022)
- [Iterative Methods for Vecchia-Laplace Approximations for Latent Gaussian Process Models](https://mlsanity.com/paper/2310.12000/) (2023)
- [Gauss-Legendre Features for Gaussian Process Regression](https://mlsanity.com/paper/2101.01137/) (2021)
- [Faster Kernel Interpolation for Gaussian Processes](https://mlsanity.com/paper/2101.11751/) (2021)
- [Compute-Optimal LLMs Provably Generalize Better With Scale](https://mlsanity.com/paper/2504.15208/) (2025)
- [Scalable Gaussian Processes with Latent Kronecker Structure](https://mlsanity.com/paper/2506.06895/) (2025)
- [Sensitivity Prewarping for Local Surrogate Modeling](https://mlsanity.com/paper/2101.06296/) (2021)
- [Iterative Methods for Full-Scale Gaussian Process Approximations for Large Spatial Data](https://mlsanity.com/paper/2405.14492/) (2024)
- [Determinant Estimation under Memory Constraints and Neural Scaling Laws](https://mlsanity.com/paper/2503.04424/) (2025)
- [Adaptive Cholesky Gaussian Processes](https://mlsanity.com/paper/2202.10769/) (2022)
- [Spatially scalable recursive estimation of Gaussian process terrain maps using local basis functions](https://mlsanity.com/paper/2210.09168/) (2022)
- [Probabilistic Unrolling: Scalable, Inverse-Free Maximum Likelihood Estimation for Latent Gaussian Models](https://mlsanity.com/paper/2306.03249/) (2023)
- [Gradients of Functions of Large Matrices](https://mlsanity.com/paper/2405.17277/) (2024)
- [Derivation of Output Correlation Inferences for Multi-Output (aka Multi-Task) Gaussian Process](https://mlsanity.com/paper/2501.07964/) (2025)
- [BOLT: Block-Orthonormal Lanczos for Trace estimation of matrix functions](https://mlsanity.com/paper/2505.12289/) (2025)
- [nuGPR: GPU-Accelerated Gaussian Process Regression with Iterative Algorithms and Low-Rank Approximations](https://mlsanity.com/paper/2510.12128/) (2025)
- [Surveillance Evasion Through Bayesian Reinforcement Learning](https://mlsanity.com/paper/2109.14811/) (2021)
- [Batch Active Learning in Gaussian Process Regression using Derivatives](https://mlsanity.com/paper/2408.01861/) (2024)
- [Scalable Krylov Subspace Methods for Generalized Mixed-Effects Models with Crossed Random Effects](https://mlsanity.com/paper/2505.09552/) (2025)
- [Vecchia-Inducing-Points Full-Scale Approximations for Gaussian Processes](https://mlsanity.com/paper/2507.05064/) (2025)

## Abstract (arXiv)

For applications as varied as Bayesian neural networks, determinantal point processes, elliptical graphical models, and kernel learning for Gaussian processes (GPs), one must compute a log determinant of an n × n positive definite matrix, and its derivatives - leading to prohibitive O(n3) computations. We propose novel O(n) approaches to estimating these quantities from only fast matrix vector multiplications (MVMs). These stochastic approximations are based on Chebyshev, Lanczos, and surrogate models, and converge quickly even for kernel matrices that have challenging spectra. We leverage these approximations to develop a scalable Gaussian process approach to kernel learning. We find that Lanczos is generally superior to Chebyshev for kernel learning, and that a surrogate approach can be highly efficient and accurate with popular kernels.