ArXiv

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

2026-07-20 · 12:07 UTC ·Blanka Horvath, Wen Su, Wu Su... ·1 min read
Authors
Blanka Horvath, Wen Su, Wu Su...
Categories
math.ST, stat.ME, stat.ML
arXiv
https://arxiv.org/abs/2607.17865v1
PDF
https://arxiv.org/pdf/2607.17865v1

Brief

Signature-based path regression: the authors obtain an explicit L^2 approximation rate for smooth functionals of Itô diffusions and show it is minimax optimal, then propagate truncation error through Signature-OLS, Signature-LASSO, and Signature-Logistic to prove estimator consistency. Empirical experiments on finance, energy, and EEG tasks demonstrate predictive gains over handcrafted features.

Why it matters

Derives an L^2 approximation rate for smooth functionals of Itô diffusions and proves this rate is minimax optimal, quantifying how signature truncation error decays (addresses the gap left by the universal approximation existence result).

Key details

  • Propagates signature truncation error through three learning procedures—Signature-OLS, Signature-LASSO, and Signature-Logistic—and establishes consistency for all three estimators.
  • Validates practical utility on three real-data tasks—foreign-exchange realized-volatility forecasting from intraday price paths, battery end-of-life prediction from early diagnostic current–voltage pulse paths, and epileptic seizure detection from short EEG windows; paper is 82 pages with 8 figures (arXiv:2607.17865v1).
Source evidence

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.

Comment: 82 pages, 8 main figures