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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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.