Signatures of paths transformed by polynomial maps
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Signatures of paths transformed by polynomial maps Laura Colmenarejo1 · Rosa Preiß2 Received: 5 February 2020 / Accepted: 24 February 2020 © The Managing Editors 2020
Abstract We characterize the signature of piecewise continuously differentiable paths transformed by a polynomial map in terms of the signature of the original path. For this aim, we define recursively an algebra homomorphism between two shuffle algebras on words. This homomorphism does not depend on the path and behaves well with respect to composition and homogeneous maps. We also study this map as a halfshuffle homomorphism and give a generalization of our main theorem in terms of Zinbiel algebras. Keywords Signature tensors · Iterated integrals · Tensor algebra · Shuffle product · Polynomial maps · Zinbiel algebras Mathematics Subject Classification 60L10 · 13P25 · 17D99
1 Introduction In the 1950s, Chen introduced the iterated-integral signature of a piecewise continuously differentiable path, which up to a natural equivalence relation, determines the initial path. In general, the signature of a path can be seen as a multidimensional time series. When the terminal time is fixed, the signature of a path can be seen as tensors and the calculation of the signature becomes a standard problem in data science. In Pfeffer et al. (2019), study the inverse problem: given partial information from a signature, can we recover the path? They consider signature tensors of order three under
R. Preiß is currently supported by European Research Council through CoG-683164.
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Laura Colmenarejo [email protected] https://sites.google.com/view/l-colmenarejo/home Rosa Preiß [email protected] http://page.math.tu-berlin.de/∼preiss/
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Department of Mathematics and Statistics, UMass Amherst, Amherst, USA
2
Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
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Beitr Algebra Geom
linear transformations and establish identifiability results and recovery algorithms for piecewise linear paths, polynomial paths, and generic dictionaries. Coming from stochastic analysis, the signatures are becoming more relevant in other areas, such as algebraic geometry and combinatorics, and we would like to highlight some recent work. For instance, in Diehl and Reizenstein (2019), Diehl and Reizenstein offer a combinatorial approach to the understanding of invariants of multidimensional time series based on their signature. Another reference is Améndola et al. (2019), in which Améndola et al. look at the varieties of signatures of tensors for both deterministic and random paths, focusing on piecewise linear paths and polynomials paths, among others. Answering one of their questions, in Galuppi (2019), Galuppi looks at rough paths, for which their signature variety shows surprising analogies with the Veronese variety. In stochastic analysis, the study of the signatures of paths arises in the theory of rough paths, where Friz and Victoir (2010); Friz and Hairer (2014) are textbook references. Iterated integrals and the non-commutative
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