Learning with Expected Signatures: Theory and Applications
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908382610325504 |
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| author | Lucchese, Lorenzo Pakkanen, Mikko S. Veraart, Almut E. D. |
| author_facet | Lucchese, Lorenzo Pakkanen, Mikko S. Veraart, Almut E. D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20465 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Learning with Expected Signatures: Theory and Applications Lucchese, Lorenzo Pakkanen, Mikko S. Veraart, Almut E. D. Machine Learning Probability Statistics Theory 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. |
| title | Learning with Expected Signatures: Theory and Applications |
| topic | Machine Learning Probability Statistics Theory |
| url | https://arxiv.org/abs/2505.20465 |