Theoretical foundations for lead–lag augmentation in signature methods

Establish rigorous theoretical results explaining why lead–lag augmentation of discrete time series (i.e., appending a lagged copy of the sequence as an additional coordinate, optionally with time augmentation) improves the effectiveness of signature-based feature mappings and signature kernels, by characterizing the properties of the augmentation that enhance capture of temporal dependencies and specifying conditions under which such augmentation is beneficial.

Background

The paper embeds discrete financial time series as continuous paths and augments them with time and a lead–lag channel prior to computing signature features or signature kernels. Empirically, such augmentation is widely used and reported to help capture temporal dependencies.

However, the authors note that there is no existing theoretical explanation for the observed empirical effectiveness of the lead–lag augmentation. They speculate that the augmentation enables the signature to encode temporal dependencies more effectively, but a formal justification is lacking.