Varieties of Signature Tensors

Abstract

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.

arXiv comment52 pages, 1 figure, 6 tables; to appear in Forum of Mathematics, Sigma

References in corpus (5)

• A theory of regularity structures
• Uniqueness for the signature of a path of bounded variation and the reduced path group
• Characteristic functions of measures on geometric rough paths
• Moment Varieties of Gaussian Mixtures
• Algebraic Identifiability of Gaussian Mixtures

Cited by in corpus (7)

• Tropical time series, iterated-sums signatures and quasisymmetric functions
• Smooth rough paths, their geometry and algebraic renormalization
• The Rough Veronese variety
• Toric geometry of path signature varieties
• A quadratic identity in the shuffle algebra and an alternative proof for de Bruijn's formula
• Generalized Euler-Maclaurin formula and Signatures
• Discrete signature varieties