The authors introduce a Gaussian process kernel for 3D spatial data that explicitly parameterizes rotated anisotropy: three principal length-scales plus an SO(3) rotation encoded as an axis–angle vector. Standard ARD kernels only capture axis-aligned stretching; generic symmetric-positive-definite parameterizations can represent arbitrary rotations but don't expose the principal directions. Here the rotation is mapped via the Lie-algebra exponential, giving unconstrained Euclidean coordinates for inference while guaranteeing a valid covariance metric. Bayesian MCMC on synthetic and real nano-material density data shows the method recovers the true generating metric, improves prediction over ARD when the field is rotated, and matches ARD when the field is axis-aligned.
Main takeaways:
- Exposes three principal length-scales and an explicit SO(3) rotation as interpretable parameters, unlike black-box full-SPD matrices
- Uses axis–angle Lie-algebra exponential to keep the rotation unconstrained during optimization while always producing a valid rotation matrix
- Posterior inference via MCMC; the paper characterizes symmetries and weakly identified regimes (e.g., when two length-scales are similar)
- On synthetic rotated-anisotropic data, recovers the ground-truth metric and outperforms axis-aligned ARD; matches ARD when data is axis-aligned
- Applied to a nano-brick material-density dataset, the inferred metric reveals rotated principal directions