Computing Path Signature Varieties in Macaulay2

2025/06/02 by Améndola, Carlos, Saliby, Angelo El, Lotter, Felix +1
#14Q15 #15A69 #60L10 #60L70 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.01429

Abstract

The signature of a path is a non-commutative power series whose coefficients are given by certain iterated integrals over the path coordinates. This series almost uniquely characterizes the path up to translation and reparameterization. Taking only fixed degree parts of these series yields signature tensors. We introduce the Macaulay2 package PathSignatures to simplify the study of these interesting objects for piecewise polynomial paths. It allows for the creation and manipulation of parametrized families of paths and provides methods for computing their signature tensors and their associated algebraic varieties.

Citations

• Rank and symmetries of signature tensors
• An algebraic geometry of paths via the iterated-integrals signature
• Signatures of paths transformed by polynomial maps
• Learning Paths from Signature Tensors
• Rough paths, Signatures and the modelling of functions on streams

Related

• Rectifiable paths with polynomial log-signature are straight lines
• Signatures of paths transformed by polynomial maps
• Computation of some transcendental integrals from path signatures
• An algebraic geometry of paths via the iterated-integrals signature
• Inversion of signature for paths of bounded variation