Rank and symmetries of signature tensors

Francesco Galuppi; Pierpaola Santarsiero

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arXiv · 2407.20405

Rank and symmetries of signature tensors

Francesco Galuppi·Pierpaola Santarsiero

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Abstract

The signature of a path is a sequence of tensors which allows to uniquely reconstruct the path. In this paper we propose a systematic study of basic properties of signature tensors, starting from their rank, symmetries and conciseness. We prove a sharp upper bound on the rank of signature tensors of piecewise linear paths. We show that there are no skew-symmetric signature tensors of order three or more, and we also prove that specific instances of partial symmetry can only happen for tensors of order three. Finally, we give a simple geometric characterization of paths whose signature tensors are not concise.

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Francesco Galuppi, Pierpaola Santarsiero. 2024-07-29. Rank and symmetries of signature tensors. https://arxiv.org/abs/2407.20405

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