Theory and methods for signature kernels in extremely high-dimensional path regimes

Establish theoretical and algorithmic foundations for computing and applying signature kernels to piecewise-linear paths with very high dimensionality (on the order of hundreds to thousands of dimensions) arising from global weather grids, a regime not covered by existing literature, including analysis of stability and scaling when path dimensionality d is large relative to path length l.

Background

The numerical instability appendix explains that existing theoretical and empirical results for signature methods have been studied for comparatively low-dimensional paths (up to d≈8) and for regimes where path length l is much larger than dimension d.

The authors emphasize that their weather forecasting setting involves very high-dimensional paths (e.g., d=256 for training patches and d=2048 for global evaluation), which falls outside the scope of current literature. Formalizing theory and reliable methods for this regime would provide principled guidance on stability, scaling, and accuracy of signature kernels in realistic meteorological applications.