Theoretical determination of optimal pre-signature scaling factor

Determine theoretically justified optimal values for the pre-signature scaling factor α used to rescale inputs prior to computing the standard signature for use with the signature kernel in spatio-temporal weather forecasting, taking into account that different depth-k signature terms cannot be adjusted separately under this scaling scheme.

Background

In the discussion of signature augmentations, the paper considers rescaling strategies to address the rapid decay of higher-order signature terms, noting that depth-k terms are of order O(1/k!). One approach is pre-signature scaling, which multiplies inputs by a scalar α, making depth-k terms of size O(α^k/k!).

The authors explicitly note that there is no theoretical justification for choosing optimal α values and highlight a structural limitation: different depth-k terms cannot be adjusted independently under pre-signature scaling. Establishing a principled method for selecting α would directly address this gap and could improve numerical stability and representation fidelity when applying the signature kernel to high-dimensional weather paths.