How Fast Do Signatures Learn? Statistical Theory and Applications for Path Regression
Blanka Horvath, Wen Su, Wu Su, Binnan Wang, Ruixun Zhang
math.ST, stat.ME, stat.ML, stat.TH
Submitted: 2026-08-22
Updated: 2026-08-25
Comments: 82 pages, 8 main figures
License: http://creativecommons.org/licenses/by/4.0/
The gist: 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
Terminology
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 squared approximation rate for smooth functionals of It o 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.
Sources
- A Primer on the Signature Method in Machine Learning
- Signature moments to characterize laws of stochastic processes
- Deep Signature Transforms
- Learning from the past, predicting the statistics for the future, learning an evolving system
- Neural Rough Differential Equations for Long Time Series
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