Learning Paths from Signature Tensors
Max Pfeffer, Anna Seigal, and Bernd Sturmfels
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Submission date: 10. Sep. 2018
published in: SIAM journal on matrix analysis and applications, 40 (2019) 2, p. 394-416
DOI number (of the published article): 10.1137/18M1212331
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Link to arXiv: See the arXiv entry of this preprint.
Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their signature tensors of order three. We establish identifiability results and recovery algorithms for piecewise linear paths, polynomial paths, and generic dictionaries. A detailed analysis of the relevant condition numbers is presented. We also compute the shortest path with a given signature tensor.