2026-08-14 –, Room 3
We introduce SignatureTensors.jl, a new package for computing signature tensors of paths and membranes. By leveraging the symbolic computation framework provided by OSCAR, the package implements flexible algebraic structures for truncated
tensor signatures, and provides efficient constructors for path signatures. Furthermore, it features implementations of Lie group barycenters and optimized algorithms for learning from signature tensors. We illustrate the package’s versatility with practical applications in geometric statistics, feature extraction,
spline interpolation, and computational algebraic geometry.
Path signatures are fundamental objects in rough path theory and serve as a noncommutative feature that captures the essential geometry of sequential data. Their utility has expanded across diverse fields including mathematical finance, machine learning, or topological data analysis.
Recently, a tangible link to algebraic geometry was established through the study of signature varieties associated with specific families of paths.
This viewpoint proved particularly useful for studying the problem of learning paths from their signature tensors.
For this purpose, a practical and easily extendable package within a
modern computer algebra system is required, providing access to multivariate arrays, Lie theory,
non-commutative polynomials, Gröbner bases, and other structures. We introduce SignatureTensors.jl, a new package that leverages the symbolic computation capabilities of OSCAR, a modern open source computer algebra system written in Julia.
The package provides a general framework for computing and manipulating path signatures using algebraic
and symbolic methods while seamlessly interacting with the OSCAR ecosystem.
In this talk, we will provide a brief introduction to signatures and their implementation within our package. We present efficient algorithms to compute signatures for (piecewise) polynomial paths. Furthermore, we provide an implementation of the recently introduced two-parameter signature of membranes. The package supports several operations on signatures such as group multiplication, the logarithm, or the geometric group barycenter. A key advantage is that our constructions work over arbitrary OSCAR rings and thus combine with common symbolic computation techniques.
We conclude with two illustrative examples of the package in action. First, we present efficient tensor learning arising in rough analysis, where recovering a path from its signature can be formalized by stabilizers with respect to congruence group actions. The second example focuses on understanding the image of the two-parameter signature, when restricted to piecewise bilinear and polynomial membranes.
Our long-term goal is for SignatureTensors.jl to become a foundational tool for theoretical research on signature tensors, while providing a flexible base framework for interdisciplinary application, such as time series, image analysis, spatial data, and more.
This is a joint work with Leonard Schmitz (TU Berlin) and the code is available at https://github.com/leonardSchmitz/signature-tensors-in-OSCAR
I’m a PhD student with Professor Carlos Améndola at the research group of Algebraic and Geometric Methods in Data Analysis at Technische Universität Berlin. I am part of the DFG Collaborative Research Center Rough Analysis, Stochastic Dynamics and Related Fields CRC/TRR 388 as a research assistant for project B01: Statistical Learning from Path Observations.
My academic interests lie in the field of statistics, particularly in areas such as Algebraic Statistics and Topological Data Analysis, focusing on the application of algebra and geometry to statistical problems. I am especially interested in tackling problems that involve integrating multiple branches of mathematics. Additionally, I have strong interests in Spatial Statistics, Time Series, and Computational Statistics.
Previously, I obtained my Bachelor's degree in Mathematical Engineering and my Master's degree in Mathematics from Universidad Técnica Federico Santa María.