How Fast Do Signatures Learn? Statistical Theory and Applications for Path Regression

[2607.17865] How Fast Do Signatures Learn? Statistical Theory and Applications for Path Regression Skip to main content Search Submit Donate Log in Mathematics > Statistics Theory arXiv:2607.17865 (math) [Submitted on 20 Jul 2026 ( v1 ), last revised 22 Aug 2026 (this version, v2)] Title: How Fast Do Signatures Learn? Statistical Theory and Applications for Path Regression Authors: Blanka Horvath , Wen Su , Wu Su , Binnan Wang , Ruixun Zhang View PDF HTML (experimental) Abstract: Many prediction and decision-making problems in operations research involve path-valued covariates -- data that evolve over time -- for which path signatures have become a canonical feature representation. Their use is justified by a universal approximation theorem, but this is an existence result: it guarantees that a finite-level signature can approximate any continuous path functional, without quantifying how fast the approximation error decreases as the truncation level grows. This paper develops approximation and statistical theory for signature-based path regression. We establish an \(L^2\) approximation rate for smooth functionals of Itô diffusions and show that it is minimax optimal. We then propagate the truncation error through three statistical learning procedures -- Signature-OLS, Signature-LASSO, and Signature-Logistic -- and establish their consistency. Three real-data applications show that signatures provide informative finite-dimensional representations of path-valued covariates and can improve prediction relative to handcrafted features, in the context of finance -- foreign exchange realized volatility forecasting from intraday price paths; energy -- battery end-of-life prediction from early diagnostic current-voltage pulse paths; and medicine -- epileptic seizure detection from short electroencephalogram windows. Comments: 82 pages, 8 main figures Subjects: Statistics Theory (math.ST) ; Methodology (stat.ME); Machine Learning (stat.ML) Cite as: arXiv:2607.17865 [math.ST] (or arXiv:2607.17865v2 [math.ST] for this version) https://doi.org/10.48550/arXiv.2607.17865 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Wu Su [ view email ] [v1] Mon, 20 Jul 2026 12:07:37 UTC (507 KB) [v2] Sat, 22 Aug 2026 01:39:47 UTC (213 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.ST < prev | next > new | recent | 2026-07 Change to browse by: math stat stat.ME stat.ML stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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