Learning with Expected Signatures: Theory and Applications

• URL: http://arxiv.org/abs/2505.20465v1
• Date: Mon, 26 May 2025 19:01:20 GMT
• Title: Learning with Expected Signatures: Theory and Applications
• Authors: Lorenzo Lucchese, Mikko S. Pakkanen, Almut E. D. Veraart,
• Abstract summary: This paper bridge the gap between the expected signature's empirical discrete-time estimator and its theoretical continuous-time value.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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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