Decomposing Tensor Spaces via Path Signatures

This notebook contains code accompanying the paper:

Carlos Améndola, Francesco Galuppi, Ángel David Ríos Ortiz, Pierpaola Santarsiero, Tim Seynnaeve. Decomposing tensor spaces via path signatures.

ARXIV: https://arxiv.org/abs/2308.11571

ABSTRACT: The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature variety. We show that the parametrization of this variety induces a natural decomposition of tensor spaces via representation theory, and connect this to the study of path invariants. We also examine the question of determining what is the tensor rank of a signature tensor.

• 4. Decomposition of the Thrall modules
• 5. Idempotents
• 6. Invariants of paths
• 7. The universal variety and tensor rank

The first three pages are based on Jupyter notebooks in SageMath which can be downloaded here:

• section_4.ipynb

• section_5.ipynb

• section_6.ipynb

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Project page created: 22/08/2023

Project contributors: Carlos Améndola, Francesco Galuppi, Ángel David Ríos Ortiz, Pierpaola Santarsiero, Tim Seynnaeve

Corresponding author of this page: Tim Seynnaeve, tim.seynnaeve@kuleuven.be

Software used: SageMath (Version 9.5), Macaulay2 (Version 1.21)

License for code of this project page: MIT License (https://spdx.org/licenses/MIT.html)

License for all other content of this project page: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

Last updated 23/08/2023.