Adaptive multi-resolution Gaussian processes: Scalable exact inference with naturally data-sparse covariance matrices
Abstract
Gaussian processes constitute a cornerstone of probabilistic machine learning, yet scaling them to large datasets typically forces a trade-off between computational efficiency and model fidelity. This work bridges this gap by presenting an adaptive multi-resolution Gaussian process framework that is both scalable and exact. Our key innovation is constructing a naturally data-sparse covariance matrix with adaptive multi-resolution basis functions. These basis functions are directly anchored to samples, eliminating the need for auxiliary points. By shrinking the support domains of multi-resolution basis, the matrix block sizes are limited, guaranteeing sparsity. The inverse of the data-sparse covariance matrix is computed exactly and efficiently via the sparse Cholesky inverse algorithm. To further improve predictive uncertainties, we construct an augmented basis function. Theoretical analysis and numerical experiments demonstrate that our model achieves exact inference with 𝒪(n log2 n) training cost and 𝒪(logd n) prediction cost, establishing a principled framework for scalable and high-fidelity Gaussian process regression.
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BibTeX
@misc{cao2026adaptive,
title = {{Adaptive multi-resolution Gaussian processes: Scalable exact inference with naturally data-sparse covariance matrices}},
author = {Yanchuang Cao and Jun Liu and Tengchao Yu and Heng Yong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30348},
eprint = {2609.30348},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30348},
}
RIS
TY - UNPB
TI - Adaptive multi-resolution Gaussian processes: Scalable exact inference with naturally data-sparse covariance matrices
AU - Yanchuang Cao
AU - Jun Liu
AU - Tengchao Yu
AU - Heng Yong
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30348
UR - https://arxiv.org/abs/2609.30348
AB - Gaussian processes constitute a cornerstone of probabilistic machine learning, yet scaling them to large datasets typically forces a trade-off between computational efficiency and model fidelity. This work bridges this gap by presenting an adaptive multi-resolution Gaussian process framework that is both scalable and exact. Our key innovation is constructing a naturally data-sparse covariance matrix with adaptive multi-resolution basis functions. These basis functions are directly anchored to samples, eliminating the need for auxiliary points. By shrinking the support domains of multi-resolution basis, the matrix block sizes are limited, guaranteeing sparsity. The inverse of the data-sparse covariance matrix is computed exactly and efficiently via the sparse Cholesky inverse algorithm. To further improve predictive uncertainties, we construct an augmented basis function. Theoretical analysis and numerical experiments demonstrate that our model achieves exact inference with 𝒪(n log^2 n) training cost and 𝒪(log^d n) prediction cost, establishing a principled framework for scalable and high-fidelity Gaussian process regression.
ER -