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VARIETIES OF SIGNATURE TENSORS

Part of: Stochastic analysis Computational aspects in algebraic geometry

Published online by Cambridge University Press:  04 April 2019

CARLOS AMÉNDOLA ,
PETER FRIZ  and
BERND STURMFELS
CARLOS AMÉNDOLA
Affiliation:
Technische Universität München, Germany; carlos.amendola@tum.de
PETER FRIZ
Affiliation:
Technische Universität Berlin and WIAS Berlin, Germany; friz@math.tu-berlin.de
BERND STURMFELS
Affiliation:
MPI for Mathematics in the Sciences, Leipzig and UC Berkeley, Germany; bernd@mis.mpg.de

Abstract

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The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.

MSC classification

Primary: 14Q15: Higher-dimensional varieties 60H99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e10
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Améndola, C., ‘Algebraic statistics of Gaussian mixtures’, PhD Dissertation, TU Berlin, 2017.Google Scholar
Améndola, C., Faugère, J.-C. and Sturmfels, B., ‘Moment varieties of Gaussian mixtures’, J. Algebr. Stat. 7 (2016), 14–28.Google Scholar
Améndola, C., Ranestad, K. and Sturmfels, B., ‘Algebraic identifiability of Gaussian mixtures’, Int. Math. Res. Not. IMRN 21 (2018), 6556–6580.Google Scholar
Bates, D., Hauenstein, J., Sommese, A. and Wampler, C., Numerically Solving Polynomial Systems with Bertini, Software, Environments and Tools, 25 (SIAM, Philadelphia, 2013).Google Scholar
Boedihardjo, H., Geng, X., Lyons, T. and Yang, D., ‘The signature of a rough path: uniqueness’, Adv. Math. 293 (2016), 720–737.Google Scholar
Brambilla, M. and Ottaviani, G., ‘On the Alexander–Hirschowitz theorem’, J. Pure Appl. Algebra 212 (2008), 1229–1251.Google Scholar
Chen, K.-T., ‘Iterated integrals and exponential homomorphisms’, Proc. Lond. Math. Soc. (3) 4 (1954), 502–512.Google Scholar
Chen, K.-T., ‘Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula’, Ann. of Math. 65 (1957), 163–178.Google Scholar
Chen, K.-T., ‘Integration of paths – a faithful representation of paths by noncommutative formal power series’, Trans. Amer. Math. Soc. 89 (1958), 395–407.Google Scholar
Chevyrev, I. and Kormilitzin, A., ‘A primer on the signature method in machine learning’. Preprint, 2016, arXiv:1603.03788.Google Scholar
Chevyrev, I. and Lyons, T., ‘Characteristic functions of measures on geometric rough paths’, Ann. Probab. 44 (2016), 4049–4082.Google Scholar
Chow, W.-L., ‘Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung’, Math. Ann. 117 (1940/1941), 98–105.Google Scholar
Cox, D., Little, J. and O’Shea, D., Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edn, Undergraduate Texts in Mathematics (Springer, New York, 1997).Google Scholar
De Lathauwer, L., De Moor, B. and Vandewalle, J., ‘A multilinear singular value decomposition’, SIAM J. Matrix Anal. Appl. 21 (2000), 1253–1278.Google Scholar
Diehl, J. and Reizenstein, J., “Invariants of multidimensional time series based on their iterated-integral signature’, J. Acta. Appl. Math. (2018), https://doi.org/10.1007/s10440-018-00227-z.Google Scholar
Drton, M., Sturmfels, B. and Sullivant, S., Lectures on Algebraic Statistics, Oberwolfach Seminars, 39 (Birkhäuser, Basel, 2009).Google Scholar
Eisenbud, D., Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150 (Springer, New York, 1995).Google Scholar
Friz, P. and Hairer, M., A Course on Rough Paths, Universitext (Springer, Cham, 2014), With an introduction to regularity structures.Google Scholar
Friz, P. and Shekhar, A., ‘General rough integration, Lévy rough paths and a Lévy–Kintchine type formula’, Ann. Probab. 45 (2017), 2707–2765.Google Scholar
Friz, P. and Victoir, N., Multidimensional Stochastic Processes as Rough Paths, Theory and Applications (Cambridge University Press, Cambridge, UK, 2010).Google Scholar
Grayson, D. and Stillman, M., ‘Macaulay2, a software system for research in algebraic geometry’, available at www.math.uiuc.edu/Macaulay2/.Google Scholar
Hairer, M., ‘A theory of regularity structures’, Invent. Math. 198 (2014), 269–504.Google Scholar
Hairer, M. and Kelly, D., ‘Geometric versus non-geometric rough paths’, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), 207–251.Google Scholar
Hambly, B. and Lyons, T., ‘Uniqueness for the signature of a path of bounded variation and the reduced path group’, Ann. of Math. 171 (2010), 109–167.Google Scholar
Kolda, T. and Bader, B., ‘Tensor decompositions and applications’, SIAM Rev. 51 (2009), 455–500.Google Scholar
Lalonde, P. and Ram, A., ‘Standard Lyndon bases of Lie algebras and enveloping algebras’, Trans. Amer. Math. Soc. 347 (1993), 1821–1831.Google Scholar
Lancaster, P. and Rodman, L., ‘Canonical forms for symmetric/skew-symmetric real matrix pairs under strict equivalence and congruence’, Linear Algebra Appl. 406 (2005), 1–76.Google Scholar
Landsberg, J. M., Tensors: Geometry and Applications, Graduate Studies in Mathematics, 128 (American Mathematical Society, Providence, RI, 2012).Google Scholar
Lyons, T., ‘Differential equations driven by rough signals’, Rev. Mat. Iberoam. 14 (1998), 215–310.Google Scholar
Lyons, T., ‘Rough paths, signatures and the modelling of functions on streams’, inProc. International Congress of Mathematicians (Kyung Moon Publishers, Seoul, 2014), 163–184.Google Scholar
Lyons, T., Caruana, M. and Lévy, T., Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics, 1908 (Springer, Berlin, 2007).Google Scholar
Lyons, T. and Qian, Z., System Control and Rough Paths (Oxford University Press, Oxford, UK, 2002).Google Scholar
Lyons, T. and Sidorova, N., ‘Sound compression – a rough path approach’, inProceedings 4th Intern. Symp. Information and Communication Technologies (Cape Town, 2005), 223–229.Google Scholar
Lyons, T. and Xu, W., ‘Hyperbolic development and inversion of signature’, J. Funct. Anal. 272 (2017), 2933–2955.Google Scholar
Lyons, T. and Xu, W., ‘Inverting the signature of a path’, J. Eur. Math. Soc. (JEMS) 20 (2018), 1655–1687.Google Scholar
Melançon, G. and Reutenauer, C., ‘Lyndon words, free algebras and shuffles’, Canadian J. Math. 16 (1989), 577–591.Google Scholar
Oxley, J. G., Matroid Theory (Oxford University Press, PSA, 2006).Google Scholar
Radford, D. E., ‘A natural ring basis for the shuffle algebra and an application to group schemes’, J. Algebra 58 (1979), 432–453.Google Scholar
Reutenauer, C., Free Lie Algebras, London Mathematical Society Monographs, New Series, 7 (Oxford University Press, New York, 1993).Google Scholar
Thompson, R. C., ‘Pencils of complex and real symmetric and skew matrices’, Linear Algebra Appl. 147 (1991), 323–371.Google Scholar