Decomposing Tensor Spaces via Path Signatures
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.
The first three pages are based on Jupyter notebooks in SageMath which can be downloaded here:
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, firstname.lastname@example.org
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.