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We've compiled a comprehensive collection of standard references, alongside recent articles, preprints, and online lectures, all dedicated to exploring signatures in rough paths theory and its diverse applications.
Talks given:
Talks given:
"International Conference on Scientific Computation and Differential Equations" (SciCADE24), July 15 - 19, 2024, National University of Singapore, Singapore
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Joscha Diehl (Univ. Greifswald, Germany)
Title: A multiplicative surface signature through its Magnus expansion
Abstract: tba
Conference "Signatures of paths and images", June 10-14, 2024, Conference center Hotel Lysebu, Oslo, Norway.
Thursday, June 13, 2024
Joscha Diehl (Univ. Greifswald, Germany)
Title: tba
Abstract: tba
Conference "Signatures of paths and images", June 10-14, 2024, Conference center Hotel Lysebu, Oslo, Norway.
Wednesday, June 12, 2024
Fabian Harang (BI Oslo & CAS, Oslo)
Title: On the Signature of an Image
Abstract: An analytic and algebraic understanding of iterated integral signatures associated to continuous paths has played a central role in in a wide range of mathematical areas, such as the construction of stochastic integration for non-martingales with rough paths theory, to formal representations and expansions of solutions to (partial) differential equations. In recent years, the signature has proven to be an efficient feature map for machine learning tasks, where the learning task is related to time series data, or data streams. In contrast to time series data, image data can naturally be seen as two-parameter fields taking values in multidimensional space, and in recent years there has been some research into the extension of the path signature to multi-parameter fields (see e.g. Chouk/Gubinelli 14, Lee/Oberhauser (21 and 23)). In this talk, I will propose a new extension of the path signature to two-parameter fields motivated by expansions of solutions to certain hyperbolic PDEs with multiplicative noise. The algebraic structure of this object turns out to be rather complicated and I will discuss our current understanding of the challenges with going from 1 to 2 parameters, and provide some interesting observations related to a Chen type relation and a Shuffle type relation. At last, I will briefly discuss the universality of the 2D signature, providing a universal approximation theorem, and discuss some open problems. This talk is based on joint work (arXiv: 2403.00130) with Joscha Diehl, Kurusch Ebrahimi-Fard, and Samy Tindel, and is part of the Signatures for Images project for 2023/2024 at CAS
Conference "Signatures of paths and images", June 10-14, 2024, Conference center Hotel Lysebu, Oslo, Norway.
Tuesday, June 11, 2024
Ilya Chevyrev (Univ. of Edinburgh & CAS, Oslo)
Title: Higher signatures
Abstract: The signature of a path, which is the collection of the path’s iterated integrals, is a central object of study in rough path theory and has found applications in machine learning. In this talk, I will present a notion of 2-dimensional signature (i.e. a signature for images) that enjoys many of the properties of the classical path signatures, including invariance under reparametrisation and Chen’s identity. The framework is inspired by ideas of Kapranov in 2015 and relates closely to recent work of Oberhauser and Lee 2023. Of interest, our construction relies on a non-commutative sewing type lemma which has the potential for the study of signatures of “rough surfaces”. Based on joint work with Joscha Diehl, Kurusch Ebrahimi-Fard, and Nikolas Tapia.
UiO, Computational Mathematics, Section 4 seminar, Spring 2024
Thursday, April 18, 2024, 14:15-15:00
Kurusch Ebrahimi-Fard (NTNU Trondheim & CAS, Oslo)
Title: Remarks on the Magnus expansion
Abstract: In 1954, Wilhelm Magnus introduced an infinite Lie series to represent the solution of a first-order operator/matrix-valued homogeneous linear differential equation. This expansion, later named the Magnus expansion or series, has become a crucial tool utilized in various fields, including physics, chemistry, and engineering. Over the past 25 years, the Magnus expansion has undergone significant mathematical developments, uncovering intricate connections between algebra, combinatorics, and geometry. In this presentation, we will explore the Magnus expansion from the perspective of crossed morphism. If time permits, we will also discuss some recent applications in the context of Chen's signature extended to membranes (instead of paths).
Conference "Mathematics of data streams: signatures, neural differential equations, and diffusion models", April 8-13, 2024, Alfried Krupp Wissenschaftskolleg, Greifswald, Germany.
Friday, April 12, 2024
Kurusch Ebrahimi-Fard (NTNU Trondheim & CAS, Oslo)
Title: Log-signature of a surface
Abstract: We will discuss the concept of log-signature in the context of surface holonomy. Based on joint work with I. Chevyrev, J. Diehl and N. Tapia, and is part of the Signatures for Images project for 2023/2024 at CAS.
Conference "Mathematics of data streams: signatures, neural differential equations, and diffusion models", April 8-13, 2024, Alfried Krupp Wissenschaftskolleg, Greifswald, Germany.
Friday, April 12, 2024
Fabian Harang (BI Oslo & CAS, Oslo)
Title: On the Signature of an Image
Abstract: An analytic and algebraic understanding of iterated integral signatures associated to continuous paths has played a central role in in a wide range of mathematical areas, such as the construction of stochastic integration for non-martingales with rough paths theory, to formal representations and expansions of solutions to (partial) differential equations. In recent years, the signature has proven to be an efficient feature map for machine learning tasks, where the learning task is related to time series data, or data streams. In contrast to time series data, image data can naturally be seen as two-parameter fields taking values in multi-dimensional space, and in recent years there has been some research into the extension of the path signature to multi-parameter fields (see e.g. Chouk/Gubinelli 14, Lee/Oberhauser (21 and 23)). In this talk, I will propose a new extension of the path signature to two-parameter fields motivated by expansions of solutions to certain hyperbolic PDEs with multiplicative noise. The algebraic structure of this object turns out to be rather complicated and I will discuss our current understanding of the challenges with going from 1 to 2 parameters, and provide some interesting observations related to a Chen type relation and a Shuffle type relation. At last, I will briefly discuss the universality of the 2D signature, providing a universal approximation theorem, and discuss some open problems. This talk is based on joint work (arXiv: 2403.00130) with Joscha Diehl, Kurusch Ebrahimi-Fard, and Samy Tindel, and is part of the Signatures for Images project for 2023/2024 at CAS.
Online lectures and talks:
Online lectures and talks:
References:
References:
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Springer, 2021, doi: 10.1007/978-3-030-77845-3Article:
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"Controlling rough paths"
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"Differential equations driven by rough paths: an approach via discrete approximation"
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Applied Mathematics Research eXpress, Volume 2008, 2008, abm009, doi: 10.1093/amrx/abm009Article:
"Ramification of rough paths"
M. Gubinelli
Journal of Differential Equations, 248 (4) (2010), 693-721, doi: 10.1016/j.jde.2009.11.015Article:
"A Chen model for mapping spaces and the surface product"
G. Ginot, T. Tradler, M. Zeinalian
Annales Scientifiques de l'École Normale Supérieure, Serie 4, 43, 2010, 811-881, doi: 10.24033/asens.2134Article:
"A short survey on pre-Lie algebras"
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"A theory of regularity structures"
M. Hairer
Inventiones mathematicae 198, 269–504 (2014), doi: 10.1007/s00222-014-0505-4Article:
"Geometric versus non-geometric rough paths"
M. Hairer, D. Kelly
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , 51 (1) (2015), 207-251, doi: 10.1214/13-AIHP564Article:
"Expected signature of Brownian motion up to the first exit time from a bounded domain"
T. Lyons, H. Ni
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"Paracontrolled distributions and singular PDEs"
M. Gubinelli, P. Imkeller, N. Perkowski
Forum of Mathematics, Pi , Volume 3 , 2015 , doi: 10.1017/fmp.2015.2Article:
"Deformations of shuffles and quasi-shuffles"
L. Foissy, F. Patras, J.-T. Thibon
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, 209-237, doi: 10.5802/aif.3010Article:
"General rough integration, Lévy rough paths and a Lévy– Kintchine type formula"
P. Friz, A. Shekhar
Annals of Probability, 45(4): 2707–2765, 2017. doi: 10.1214/16-AOP1123Article:
"Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra"
F. Fauvet, F. Menous
Annales Scientifiques de l École Normale Supérieure 50(1), 39-83, 2017, doi: 10.24033/asens.2315Article:
"Algebraic renormalisation of regularity structures"
Y. Bruned, M. Hairer, L. Zambotti
Inventiones mathematicae, 215 (3) (2019), 1039-1156, doi: 10.1007/s00222-018-0841-xArticle:
"A rough path perspective on renormalization"
Y. Bruned, I. Chevyrev, P.K. Friz, R. Preiß
Journal of Functional Analysis, Volume 277, Issue 11, 2019, 108283, doi: 10.1016/j.jfa.2019.108283Article:
"Kernels for sequentially ordered data"
F. J. Kiraly, H. Oberhauser
Journal of Machine Learning Research, 20(31): 1-45, 2019. onlineArticle:
"An isomorphism between branched and geometric rough paths"
H. Boedihardjo, I. Chevyrev
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 55 (2) (2019), 1131-1148, doi: 10.1214/18-aihp912Article:
"Canonical RDEs and general semimartingales as rough paths"
I. Chevyrev, P. K. Friz.
Annals of Probability, 47(1):420–463, 2019. doi: 10.1214/18-AOP1264Article:
"Planarly branched rough paths and rough differential equations on homogeneous spaces"
C. Curry, K. Ebrahimi-Fard, D. Manchon, H.Z. Munthe-Kaas
Journal of Differential Equations, Volume 269, Issue 11, 2020, 9740-9782, doi: 10.1016/j.jde.2020.06.058Article:
"The geometry of the space of branched rough paths"
N. Tapia, L. Zambotti
Proceedings of the London Mathematical Society, Volume 121, Issue 2, 2020, 220-251, doi: 10.1112/plms.12311Article:
"Pre- and Post-Lie Algebras: The Algebro-Geometric View"
G. Fløystad, H. Munthe-Kaas
In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. doi: 10.1007/978-3-030-01593-0_12Article:
"Signature Cumulants, Ordered Partitions, and Independence of Stochastic Processes"
P. Bonnier, H. Oberhauser
Bernoulli, Volume 26, Number 4 (2020), 2727-2757, doi: 10.3150/20-BEJ1205Article:
"Nonlinear system identification for multivariable control via discrete-time Chen–Fliess series"
W. Steven Gray, G.S. Venkatesh, Luis A. Duffaut Espinosa
Automatica, Volume 119, 2020, 109085, doi: 10.1016/j.automatica.2020.109085Article:
"Time-warping invariants of multidimensional time series"
J. Diehl, K. Ebrahimi-Fard, N. Tapia
Acta Applicandae Mathematicae 170(1) (2020), 265-29, doi: 10.1007/s10440-020-00333-xArticle:
"Persistence Paths and Signature Features in Topological Data Analysis"
I. Chevyrev, V. Nanda, H. Oberhauser
IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(1): 192–202, 2020. doi: 10.1109/TPAMI.2018.2885516Article:
"Discrete-Time Signatures and Randomness in Reservoir Computing"
C. Cuchiero, L. Gonon, L. Grigoryeva, J.-P. Ortega, J. Teichmann
IEEE Transactions on Neural Networks and Learning Systems, Volume: 33, Issue: 11, 2022, 6321-6330, doi: 10.1109/TNNLS.2021.3076777Article:
"Non-geometric rough paths on manifolds"
J. Armstrong, D. Brigo, T. Cass, E. Rossi Ferrucci
Journal of the London Mathematical Society, Volume 106, Issue 2, 2022, 756-817, doi: 10.1112/jlms.12585Article:
"Unified Signature Cumulants and Generalized Magnus Expansions"
P. K. Friz, P. Hager, N. Tapia
Forum of Mathematics, Sigma, Volume 10, 2022, doi: 10.1017/fms.2022.20Article:
"Constructing general rough differential equations through flow approximations"
A. Lejay
Electronic Journal of Probability 27 (2022), article no. 7, 1–24. doi: 10.1214/21-EJP717"2-d signature of images and texture classification"
S. Zhang, G. Lin, S. Tindel
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 478:20220346,
doi: 10.1098/rspa.2022.0346Article:
"Functional linear regression with truncated signatures"
A. Fermanian
Journal of Multivariate Analysis, Volume 192, 2022, 105031, doi: 10.1016/j.jmva.2022.105031Article:
"Tropical time series, iterated-sums signatures and quasisymmetric functions"
J. Diehl, K. Ebrahimi-Fard, N. Tapia
SIAM Journal on Applied Algebra and Geometry, Vol 6, Issue 4, (2022), 563-599, doi: 10.1137/20M138004Article:
"Smooth Rough Paths, Their Geometry and Algebraic Renormalization"
C. Bellingeri, P. K. Friz, S. Paycha, R. Preiß
Vietnam Journal of Mathematics, volume 50, 719-761 (2022), doi: 10.1007/s10013-022-00570-7Article:
"Forests, cumulants, martingales"
P. K. Friz, J. Gatheral, R. Radoičić
Annals of Probability, 50(4): 1418-1445 (2022), doi: 10.1214/21-AOP1560Article:
"Signature Moments to Characterize Laws of Stochastic Processes"
I. Chevyrev, H. Oberhauser
Journal of Machine Learning Research 23(176): 1-42, 2022. onlineArticle:
"The Moving-Frame Method for the Iterated-Integrals Signature: Orthogonal Invariants"
J. Diehl, R. Preiß, M. Ruddy, N. Tapia
Foundations of Computational Mathematics (2022), doi: 10.1007/s10208-022-09569-5Article:
"Stability of Deep Neural Networks via discrete rough paths"
Ch. Bayer, P. K. Friz, N. Tapia
SIAM Journal on Mathematics of Data Science 5(1), 2023, pp 50-76, doi: 10.1137/22M1472358Article:
"The structure group for quasi-linear equations via universal enveloping algebras"
P. Linares, F. Otto, M. Tempelmayr
Communications of the American Mathematical Society 3 (2023), 1-64, doi: 10.1090/cams/16Article:
"Generalized iterated-sums signatures"
J. Diehl, K. Ebrahimi-Fard, N. Tapia
Journal of Algebra 632, 2023, 801-824, doi: 10.1016/j.jalgebra.2023.06.007
Recent preprints:
Recent preprints:
"Rough sheets"
K. Chouk, M. Gubinelli, arXiv:1406.7748"A Primer on the Signature Method in Machine Learning"
I. Chevyrev, A. Kormilitzin, arXiv:1603.03788"Areas of areas generate the shuffle algebra"
J. Diehl, T. Lyons, R. Preiß, J. Reizenstein, arXiv:2002.02338"A Generalised Signature Method for Multivariate Time Series Feature Extraction"
J. Morrill, A. Fermanian, P. Kidger, T. Lyons, arXiv:2006.00873"Feature Engineering with Regularity Structures"
I. Chevyrev, A. Gerasimovics, H. Weber, arXiv:2108.05879"A Topological Approach to Mapping Space Signatures"
C. Giusti, D. Lee, V. Nanda, H. Oberhauser, arXiv:2202.00491"Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation"
F. Bechtold, F. A. Harang, N. Rana, arXiv:2206.05360"Hopf and pre-Lie algebras in regularity structures"
I. Chevyrev, arXiv:2206.14557"Signature Methods in Machine Learning"
T. Lyons, A. D. McLeod, arXiv:2206.14674"A tree-free approach to regularity structures: The regular case for quasi-linear equations"
P. Linares, F. Otto, arXiv:2207.10627"Post-Lie algebras in Regularity Structures"
Y. Bruned, F. Katsetsiadis, arXiv:2208.00514"Two-parameter sums signatures and corresponding quasisymmetric functions"
J. Diehl, L. Schmitz, arXiv:2210.14247"A structure theorem for streamed information"
C. Salvi, J. Diehl, T. Lyons, R. Preiss, J. Reizenstein, arXiv:2212.00134"Singular SPDEs on Homogeneous Lie Groups"
A. Mayorcas, H. Singh, arXiv:2301.05121"Learning the Dynamics of Sparsely Observed Interacting Systems"
L. Bleistein, A. Fermanian, A.-S. Jannot, A. Guilloux, arXiv:2301.11647"Signature SDEs from an affine and polynomial perspective"
Ch. Cuchiero, S. Svaluto-Ferro, J. Teichmann, arXiv:2302.01362"New directions in the applications of rough path theory"
A. Fermanian, T. Lyons, J. Morrill, C. Salvi, arXiv:2302.04586"The insertion method to invert the signature of a path"
A Fermanian, J. Chang, T. Lyons, G. Biau, arXiv:2304.01862"The Signature Kernel"
D. Lee, H. Oberhauser, arXiv:2305.04625"Post-Lie algebras of derivations and regularity structures"
J.-D. Jacques, L. Zambotti, arXiv:2306.02484"A multiparameter Stochastic Sewing lemma and the regularity of local times associated to Gaussian sheets"
F. Bechtold, F. A. Harang, H. Kern, arXiv:2307.11527...
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