提出可解释的三维各向异性高斯过程核，显式建模主尺度与旋转方向。
Interpretable Machine Learning for Spatial Science: A Lie-Algebraic Kernel for Rotationally Anisotropic Gaussian Processes
- 用李代数指数映射构建旋转参数，实现无约束推断同时保证协方差矩阵正定
- 在旋转各向异性合成数据上后验能恢复真实度量，预测性能优于轴对齐基线
- 适合需要几何可解释性的空间科学建模，如材料密度、地质结构分析
许多三维空间场呈现各向异性，其快速与缓慢变化方向未必与坐标轴对齐。标准具有自动相关性确定（ARD）的高斯过程核仅能捕捉轴对齐各向异性，而通用对称正定（SPD）度量虽可表示旋转各向异性，却无法直接参数化主长度尺度和方向。本文引入一种可解释的旋转各向异性高斯过程核，通过三个主长度尺度和显式的SO(3)旋转参数化三维SPD协方差度量。旋转由轴角向量表示，并通过李代数指数映射到SO(3)，在推断中使用无约束欧氏坐标，始终保证生成有效SPD度量。该构造覆盖与通用全SPD参数化相同的三维SPD协方差度量族，但显式揭示了尺度与方向：其可解释性强，便于先验设定与后验总结。我们采用马尔可夫链蒙特卡洛（MCMC）进行贝叶斯推断，分析对称性与弱辨识区域。在具有旋转各向异性的合成数据上，后验能恢复生成度量，预测性能优于轴对齐的ARD基线，且匹配通用全SPD基线表现。当真实情况为轴对齐时，后验质量集中于单位旋转，预测性能与ARD相当。在实验室制造的纳米砖材料密度数据集上，推断出的度量揭示了轴对齐核无法捕捉的旋转各向异性。
原文摘要 · Abstract (English)
Many three-dimensional spatial fields are anisotropic, with directions of rapid and slow variation that need not align with the coordinate axes. Standard Gaussian process kernels with Automatic Relevance Determination (ARD) capture only axis-aligned anisotropy, while generic full symmetric positive definite (SPD) metrics can represent rotated anisotropy but do not parameterise principal length-scales and directions directly. We introduce an interpretable rotationally anisotropic GP kernel that parameterises a three-dimensional SPD covariance metric using three principal length-scales and an explicit SO(3) rotation. The rotation is represented by an axis-angle vector and mapped to SO(3) via the Lie-algebra exponential map, giving unconstrained Euclidean coordinates for inference while always inducing a valid SPD metric. The construction spans the same family of three-dimensional SPD covariance metrics as a generic full-SPD parameterisation, but exposes the geometry differently: length-scales and orientation are explicit, interpretable, and directly available for prior specification and posterior summaries. We perform Bayesian inference on these quantities using Markov Chain Monte Carlo (MCMC), and characterise the resulting symmetries and weakly identified regimes. On synthetic data with rotated anisotropy, the posterior recovers the generating metric and improves prediction relative to an axis-aligned ARD baseline, while matching the predictive performance of a generic full SPD baseline. When the ground truth is axis-aligned, posterior mass concentrates near the identity rotation and predictive performance matches ARD. On a material-density dataset from a laboratory-fabricated nano-brick, the inferred metric reveals rotated anisotropy that is not captured by axis-aligned kernels.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。