SignatureTensors.jl

SignatureTensors.jl is a Julia package for computing and manipulating signature tensors of paths and membranes, built on top of the OSCAR computer algebra system.

Version: 0.5 — No support is guaranteed.

Overview

Path signatures are fundamental objects in rough path theory that capture the essential geometry of sequential data. This package provides a general algebraic framework for computing truncated signatures and working with them symbolically, leveraging the full power of OSCAR's ring arithmetic, Gröbner bases, and Lie theory.

The package supports:

• Path signatures — iterated-integrals signatures of piecewise linear, polynomial, spline, axis, and monomial paths over arbitrary OSCAR rings.
• Membrane signatures — two-parameter (id-)signatures of piecewise bilinear and polynomial membranes, extending the one-parameter theory.
• Tensor learning — recovery of paths from their signature tensors via polynomial systems and Gröbner bases, including the efficient algorithm from [1].
• Lie group barycenters — computation of Fréchet means on the free nilpotent Lie group $G_{d,k}$, with several algorithm options including BCH-based and polynomial map approaches.
• Algebraic operations — group multiplication, inverse, logarithm, exponential, and graded projections on truncated tensor algebra elements.

All constructions work over arbitrary OSCAR rings, making them compatible with symbolic computation over $\mathbb{Q}$, polynomial rings, rational function fields, and more.

Installation

] activate .
] instantiate

Then load the package alongside OSCAR:

using Oscar
using SignatureTensors

Quick Start

using Oscar, SignatureTensors

# Define a truncated tensor algebra: dimension d=2, truncation level k=3
d, k = 2, 3
T = TruncatedTensorAlgebra(QQ, d, k)

# Signature of the canonical axis path
C=sig(T, :axis)

# Signature of a polynomial path t ↦ (t + 2t², 3t + 4t²)
S=sig(T, :pwln, coef = QQ.[1 2; 3 4])

# Path recovery from a signature tensor
recover(S,Co=C)

Documentation

• Documentation
• • Types
  • Methods
  • Signature Constructors
  • Tensor Learning & Path Recovery
  • Barycenters
  • Element Operations

Authors

Funded by the Deutsche Forschungsgemeinschaft (DFG) — CRC/TRR 388 "Rough Analysis, Stochastic Dynamics and Related Fields" — Project A04 and B01, 516748464.