Block Vecchia Approximation for Scalable and Efficient Gaussian Process Computations

2024-10-06

arXiv.org

AI Summary

High computational complexity and memory overhead of Gaussian processes (GPs) on large-scale irregular spatial data, coupled with the inefficiency and redundancy of conventional Vecchia approximations, hinder scalable spatial inference. To address this, we propose a blockwise Vecchia approximation that replaces univariate conditional decompositions with multivariate ones, substantially reducing both computational cost and memory usage. We introduce a K-means clustering–driven observation blocking strategy and empirically discover that random ordering significantly improves approximation accuracy under large block counts. Furthermore, we design a GPU-accelerated batched linear algebra framework supporting Cholesky decomposition, matrix inversion, and other core operations. Evaluated on million-point 3D wind velocity data, our method achieves >10× speedup and drastic memory reduction over exact GP inference while preserving high predictive accuracy—enabling efficient, scalable spatial inference for massive geospatial datasets.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Gaussian Processes (GPs) are vital for modeling and predicting irregularly-spaced, large geospatial datasets. However, their computations often pose significant challenges in large-scale applications. One popular method to approximate GPs is the Vecchia approximation, which approximates the full likelihood via a series of conditional probabilities. The classical Vecchia approximation uses univariate conditional distributions, which leads to redundant evaluations and memory burdens. To address this challenge, our study introduces block Vecchia, which evaluates each multivariate conditional distribution of a block of observations, with blocks formed using the K-means algorithm. The proposed GPU framework for the block Vecchia uses varying batched linear algebra operations to compute multivariate conditional distributions concurrently, notably diminishing the frequent likelihood evaluations. Diving into the factor affecting the accuracy of the block Vecchia, the neighbor selection criterion is investigated, where we found that the random ordering markedly enhances the approximated quality as the block count becomes large. To verify the scalability and efficiency of the algorithm, we conduct a series of numerical studies and simulations, demonstrating their practical utility and effectiveness compared to the exact GP. Moreover, we tackle large-scale real datasets using the block Vecchia method, i.e., high-resolution 3D profile wind speed with a million points.

Problem

• Gaussian Processes
• Large-scale Geospatial Data
• Vecchia Approximation

Innovation

• Block Vecchia method
• GPU optimization
• K-means clustering