Concentration of Truncated Signatures of Gaussian Rough Paths

TL;DR

This paper develops a comprehensive concentration theory for truncated signatures of Gaussian rough paths, providing tail decay rates, variance formulas, and concentration inequalities relevant for machine learning and stochastic analysis.

Contribution

It introduces new non-asymptotic concentration results for signature features, combining rough path theory with Gaussian analysis, and explores implications for statistical learning.

Findings

Level-k signature coordinates have optimal exponential tail decay.

Dimension-free concentration inequalities are established for truncated signatures.

Explicit variance formulas and sharp constants for Brownian and fractional Brownian motions.

Abstract

This paper establishes a comprehensive concentration theory for truncated signatures of Gaussian rough paths. The signature of a path, defined as the collection of all iterated integrals, provides a complete description of its geometric structure and has emerged as a powerful tool in machine learning and stochastic analysis. Despite growing applications in finance, healthcare, and engineering, the non-asymptotic concentration properties of signature features remain largely unexplored. We prove that level-$k$ signature coordinates exhibit optimal $\exp(-c t^{2/k})$ tail decay and establish dimension-free concentration inequalities for the full truncated signature vector. Our results reveal a fundamental trade-off: higher truncation levels capture more complex path properties but exhibit heavier tails. For Brownian motion and fractional Brownian motion with Hurst parameter $H > 1/4$, we derive explicit variance formulas and sharp constants. The technical contributions combine rough path theory with Gaussian analysis, leveraging Wiener chaos decomposition, hypercontractivity, and the algebraic structure of tensor algebras. We further establish concentration for log-signatures, lead-lag transformations, and provide sample complexity bounds for statistical learning with signature features. This work bridges advanced probability theory with practical applications, offering both theoretical guarantees and computational methods for signature-based approaches in sequential data analysis.