Bigraphical Matérn-Whittle (BMW) Processes for Fast Inference of Big Multivariate Spatial Data on General Domains

2026-09-01

AI Summary

本文提出了一种双图马尔滕-惠特尔过程，用于解决大规模多变量空间数据在任意拓扑结构上的联合建模问题，通过两个图来保留变量级解释并实现高效推断。

Abstract

Large spatial data sets now record many correlated variables at many thousands of locations, often on domains where Euclidean distance misrepresents proximity. The central difficulty is modelling the cross-variable dependence jointly while retaining variable-level interpretation. We introduce the bigraphical Mat'ern-Whittle process, a multivariate Gaussian process that resolves this with two graphs. A spatial graph generates the Mat'ern structure of each variable through a fractional power of a graph Laplacian, so the process is valid on any topology, with per-variable range, smoothness and amplitude. A directed acyclic variable graph encodes the scientific structure: we prove that each absent edge yields an exact conditional independence between the corresponding fields. We further prove that the operator determinant does not involve the cross-dependence coefficients, which keeps matrix-free likelihood evaluation and Bayesian learning of the variable graph tractable at scale. Estimation requires only sparse matrix-vector products and scales to tens of millions of space-variable pairs. In simulations the method recovered parameters and graphs accurately, remained robust under misspecification, and halved held-out prediction error on a non-convex domain. In a spatial transcriptomics section with 19,809 cells and 1,122 genes, fitted in 75 minutes on a laptop, borrowing across the learned gene graph reduced held-out prediction error by 50 to 91 percent. Theoretical challenges, such as the achievable efficiency of estimating the variance of the nugget, are also explored.

Problem

• multivariate spatial data
• cross-variable dependence
• variable-level interpretation
• non-Euclidean domain

Innovation

• Bigraphical Matérn-Whittle Processes
• Spatial Graphs
• Directed Acyclic Variable Graphs
• Sparse Matrix-Vector Products
• Scalable Bayesian Learning