Learning with Expected Signatures: Theory and Applications

Lorenzo Lucchese, Mikko S. Pakkanen, Almut E. D. Veraart
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:40995-41055, 2025.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-lucchese25a, title = {Learning with Expected Signatures: Theory and Applications}, author = {Lucchese, Lorenzo and Pakkanen, Mikko S. and Veraart, Almut E. D.}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {40995--41055}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/lucchese25a/lucchese25a.pdf}, url = {https://proceedings.mlr.press/v267/lucchese25a.html}, 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.} }
Endnote
%0 Conference Paper %T Learning with Expected Signatures: Theory and Applications %A Lorenzo Lucchese %A Mikko S. Pakkanen %A Almut E. D. Veraart %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-lucchese25a %I PMLR %P 40995--41055 %U https://proceedings.mlr.press/v267/lucchese25a.html %V 267 %X 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.
APA
Lucchese, L., Pakkanen, M.S. & Veraart, A.E.D.. (2025). Learning with Expected Signatures: Theory and Applications. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:40995-41055 Available from https://proceedings.mlr.press/v267/lucchese25a.html.

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