Learning with Expected Signatures: Theory and Applications

Lorenzo Lucchese; Mikko S. Pakkanen; Almut E. D. Veraart

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arXiv · 2505.20465

Learning with Expected Signatures: Theory and Applications

Lorenzo Lucchese·Mikko S. Pakkanen·Almut E. D. Veraart

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Abstract

The expected signature maps a collection of data streams to a lower dimensional representation, with a remarkable property: the resulting feature tensor can fully characterize the data generating distribution. This "model-free" embedding has been successfully leveraged to build multiple domain-agnostic machine learning (ML) algorithms for time series and sequential data. The convergence results proved in this paper bridge the gap between the expected signature's empirical discrete-time estimator and its theoretical continuous-time value, allowing for a more complete probabilistic interpretation of expected signature-based ML methods. Moreover, when the data generating process is a martingale, we suggest a simple modification of the expected signature estimator with significantly lower mean squared error and empirically demonstrate how it can be effectively applied to improve predictive performance.

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BibTeX RIS

Lorenzo Lucchese, Mikko S. Pakkanen, Almut E. D. Veraart. 2025-05-26. Learning with Expected Signatures: Theory and Applications. https://arxiv.org/abs/2505.20465

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