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- ItemTime-Warping Invariants of Multidimensional Time Series(Dordrecht [u.a.] : Springer Science + Business Media B.V., 2020) Diehl, Joscha; Ebrahimi-Fard, Kurusch; Tapia, NikolasIn data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants. We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties. © 2020, The Author(s).
- ItemThe geometry of the space of branched rough paths(Chichester : Wiley, 2020) Tapia, Nikolas; Zambotti, LorenzoWe construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between these two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker–Campbell–Hausdorff formula, on a constructive version of the Lyons–Victoir extension theorem and on the Hairer–Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.
- ItemThe geometry of controlled rough paths(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Varzaneh, Mazyar Ghani; Riedel, Sebastian; Schmeding, Alexander; Tapia, NikolasWe prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns well-known maps such as the rough integration map and the Itô-Lyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.
- ItemStability of deep neural networks via discrete rough paths(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Friz, Peter; Tapia, NikolasUsing rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks. In particular we derive stability bounds in terms of the total p-variation of trained weights for any p ≥ 1.
- ItemShifted substitution in non-commutative multivariate power series with a view towards free probability(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Ebrahimi-Fard, Kurusch; Patras, Frédéric; Tapia, Nikolas; Zambotti, LorenzoWe study a particular group law on formal power series in non-commuting parameters induced by their interpretation as linear forms on a suitable non-commutative and non- cocommutative graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu?s theory of free probability.
- ItemGeneralized iterated-sums signatures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Diehl, Joscha; Ebrahimi-Fard, Kurusch; Tapia, NikolasWe explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Király and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.
- ItemThe geometry of the space of branched rough paths(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Tapia, Nikolas; Zambotti, LorenzoWe construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker-Campbell-Hausdorff formula, on a constructive version of the Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.
- ItemTropical time series, iterated-sum signatures and quasisymmetric functions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Diehl, Joscha; Ebrahimi-Fard, Kurusch; Tapia, NikolasDriven by the need for principled extraction of features from time series, we introduce the iterated-sums signature over any commutative semiring. The case of the tropical semiring is a central, and our motivating, example, as it leads to features of (real-valued) time series that are not easily available using existing signature-type objects.
- ItemTime-warping invariants of multidimensional time series(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Diehl, Joscha; Kurusch, Ebrahimi-Fard; Tapia, NikolasIn data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants.We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties.
- ItemIterated-sums signature, quasi-symmetric functions and time series analysis(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Diehl, Joscha; Ebrahimi-Fard, Kurusch; Tapia, NikolasWe survey and extend results on a recently defined character on the quasi-shuffle algebra. This character, termed iterated-sums signature, appears in the context of time series analysis and originates from a problem in dynamic time warping. Algebraically, it relates to (multidimensional) quasisymmetric functions as well as (deformed) quasi-shuffle algebras.