New Lie-algebraic kernel unlocks rotated anisotropy for 3D spatial GP models

Many 3D spatial fields exhibit anisotropy where directions of rapid and slow variation don't align with coordinate axes – think geological strata, material density gradients, or atmospheric dispersion. Standard Gaussian process kernels with Automatic Relevance Determination (ARD) capture only axis-aligned anisotropy, missing real-world rotated patterns. Generic full symmetric positive definite (SPD) covariance metrics can model rotated anisotropy but lack interpretable parameters for principal length-scales and orientation.

Warrior and Chakrabarty's solution: a novel GP kernel that parameterizes a 3D SPD metric using explicit principal length-scales and an SO(3) rotation represented as an axis-angle vector. The rotation is mapped to SO(3) via the Lie-algebra exponential map, giving unconstrained Euclidean coordinates compatible with Bayesian inference (MCMC). This kernel spans the same family as full SPD metrics but exposes length-scales and rotation directly – making them available for prior specification and posterior summaries. On synthetic rotated anisotropy, the posterior recovers the true metric and outperforms ARD prediction while matching a full-SPD baseline. When ground truth is axis-aligned, posterior mass concentrates near identity rotation. Applied to a density dataset from a laboratory-fabricated nano-brick, the kernel reveals a rotated anisotropy pattern invisible to axis-aligned kernels, demonstrating practical value for materials science and geophysics.