×

Toric geometry of path signature varieties. (English) Zbl 1467.14149

For a map \(X:[0,1]\rightarrow {\mathbb R}^d\), and its projection \(X_i\) on the \(i\)-th coordinate, for a fixed \(k\in{\mathbb N}\), K. T. Chen defined in [Trans. Am. Math. Soc. 89, 395–407 (1958; Zbl 0097.25803)] the \(k\)-th signature of \(X\) as the \(k\)-tensor \(\sigma^{(k)}(X)\in ({\mathbb R}^d)^{\otimes k}\) whose \((i_1,\ldots,i_k)\)-th entry is the iterated integral \[\int_0^1\int_0^{t_k}\cdots \int_0^{t_3}\int_0^{t_2} \dot{X}_{i_1}(t_1)\cdots \dot{X}_{i_k}(t_k)\, dt_1\cdots dt_k\] and \(\sigma^{(0)}(X)=1\). The sequence \(\sigma(X)=(\sigma^{(k)}(X): k\geq 0)\) is the signature of the path \(X\) and there is a truncated version of it \(\sigma^{\leq m}(X)=(\sigma^{(k)}(X): 0\leq k\leq m)\). For smooth \(X\), the iterated integrals do not give any extra information not derived from \(X\), however when \(X\) is not smooth, there can exist sequences of smooth paths \(X^n, Y^n\) both point-wise converging to \(X\) and with corresponding sequences of iterated integrals also converging but to different limits; these limits are no longer iterated integrals, but they satisfy the so-called Chen identities [loc. cit.]
Fixing the class of paths and an integer \(k\), the \(k\)-th signature \(\sigma^{(k)}\) is an algebraic map into \(({\mathbb R}^d)^{\otimes k}\) and the (Zariski) closure of its image is called the \(k\)-th signature variety. In the paper under review, the authors consider signature varieties for two classes of paths. First, they consider the signature variety of rough paths, which are first defined as the Zariski closure of the image of rough paths of order \(m\) in the tensor space \(({\mathbb R}^d)^{\otimes k}\), but since this is only a semi-algebraic subset, they complexify first and then take the projectivization in \(({\mathbb C}^d)^{\otimes k}\). The corresponding signature variety has some similarities with the classical Veronese variety and is known as the rough Veronese variety \({\mathcal R}_{d,k,m}\). One such similarity considers the fact that the classical Veronese variety is the image of the map given by degree \(k\) monomials in the usual grading, and it is known that the rough Veronese variety \({\mathcal R}_{d,k,m}\) is the closure of the image of a weighted projective space by a map given by all monomials of weighted degree \(k\). Looking for more similarities, and since the classical Veronese variety is defined by quadrics, the corresponding property for the rough Veronese variety is not true by a counterexample in Proposition 28 of [F. Galuppi, Linear Algebra Appl. 583, 282–299 (2019; Zbl 1432.14041)]. The first main result of the paper under review is that, in general, does not even exist a bound to the degree of the generators of the ideal defining the rough Veronese variety (Proposition 2.9). On the other hand, the second main result (Proposition 2.11) shows that \({\mathcal R}_{d,k,m}\) is defined by quadrics outside of a coordinate linear subspace of large codimension. Using toric geometry, the authors characterize the cases in which \({\mathcal R}_{d,k,m}\) is an embedding of the weighted projective space and conditions that make it (projectively) normal and examples when it is not. Lastly, they obtain formulas for the dimension and degree of \({\mathcal R}_{d,k,m}\) in Proposition 2.21.
The second family of signature varieties that the authors study correspond to the class of axis-parallel paths, for which they obtain a combinatorial parametrization in Lemma 3.3 and using this description they show that the variety is toric in some cases.

MSC:

14Q15 Computational aspects of higher-dimensional varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
60H99 Stochastic analysis
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Améndola, Carlos; Friz, Peter; Sturmfels, Bernd, Varieties of signature tensors, Forum Math. Sigma, 7, e10 (2019) · Zbl 1411.14068
[2] Assarf, Benjamin; Gawrilow, Ewgenij; Herr, Katrin; Joswig, Michael; Lorenz, Benjamin; Paffenholz, Andreas; Rehn, Thomas, Computing convex hulls and counting integer points with polymake, Math. Program. Comput., 9, 1, 1-38 (2017) · Zbl 1370.90009
[3] Bruns, Winfried; Ichim, Bogdan; Römer, Tim; Sieg, Richard; Söger, Cristof, Normaliz. Algorithms for rational cones and affine monoids, Available at · Zbl 1203.13033
[4] Colmenarejo, Laura; Galuppi, Francesco; Michałek, Mateusz, Code available at L. Colmenarejo’s webpage
[5] Chen, Kuo-Tsai, Iterated integrals and exponential homomorphism, Proc. Lond. Math. Soc., 4, 502-512 (1954) · Zbl 0058.25603
[6] Chen, Kuo-Tsai, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. Math. (2), 65, 163-178 (1957) · Zbl 0077.25301
[7] Chen, Kuo-Tsai, Integration of paths - a faithful representation of paths by noncommutative formal power series, Trans. Am. Math. Soc., 89, 95-407 (1958) · Zbl 0097.25803
[8] Chevyrev, Ilya; Kormilitzin, Andrey, A primer on the signature method in machine learning (2016)
[9] Chevyrev, Ilya; Nanda, Vidit; Oberhauser, Harald, Persistence paths and signature features in topological data analysis, IEEE Trans. Pattern Anal. Mach. Intell., 42, 1, 192-202 (2020)
[10] Chevyrev, Ilya; Oberhauser, Harald, Signature moments to characterize laws of stochastic processes (2018)
[11] Colmenarejo, Laura; Preiß, Rosa, Signatures of paths transformed by polynomial maps, Beitr. Algebra Geom. (2020) · Zbl 1455.13051
[12] Cox, David A.; Little, John B.; Schenck, Henry K., Toric Varieties, Graduate Studies in Mathematics, vol. 124 (2011), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1223.14001
[13] Diehl, Joscha; Reizenstein, Jeremy, Invariants of multidimensional time series based on their iterated-integral signature, Acta Appl. Math., 164, 83-122 (2019) · Zbl 1428.62391
[14] Friz, Peter; Gassiat, Paul; Lyons, Terry, Physical Brownian motion in a magnetic field as a rough path, Trans. Am. Math. Soc., 367, 11, 7939-7955 (2015) · Zbl 1390.60257
[15] Friz, Peter; Hairer, Martin, A Course on Rough Paths (2014), Springer · Zbl 1327.60013
[16] Friz, Peter; Victoir, Nicolas, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications (2010), Cambridge University Press · Zbl 1193.60053
[17] Fulton, William, Introduction to Toric Varieties, Annals of Mathematics Studies, vol. 131 (1993), Princeton University Press: Princeton University Press Princeton, NJ, The William H. Roever Lectures in Geometry · Zbl 0813.14039
[18] Galuppi, Francesco, The rough Veronese variety, Linear Algebra Appl., 583, 282-299 (2019) · Zbl 1432.14041
[19] Gyurkó, Lajos Gergely; Lyons, Terry; Kontkowski, Mark; Field, Jonathan, Extracting information from the signature of a financial data stream (2013)
[20] Hain, Richard, Iterated Integrals and Algebraic Cycles: Examples and Prospects, 55-118 (2002), World Scientific Publishing: World Scientific Publishing NJ, (Chapter 4) · Zbl 1065.14012
[21] Katthän, Lukas; Michałek, Mateusz; Miller, Ezra, When is a polynomial ideal binomial after an ambient automorphism?, Found. Comput. Math., 1-23 (2017) · Zbl 1429.13010
[22] Lazarsfeld, Robert, Positivity in Algebraic Geometry, I and II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (2004), Springer-Verlag: Springer-Verlag Berlin, Classical setting: line bundles and linear series · Zbl 1066.14021
[23] Lyons, Terry; Caruana, Michael; Lévy, Thierry, Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics, vol. 1908 (2007), Springer: Springer Berlin, Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6-24, 2004 · Zbl 1176.60002
[24] Lyons, Terry; Sidorova, Nadia, Sound compression - a rough path approach, (Proceedings of the 4th International Symposium on Information and Communication Technologies, Cape Town (2005)), 223-229
[25] Pfeffer, Max; Seigal, Anna; Sturmfels, Bernd, Learning paths from signature tensors, SIAM J. Matrix Anal. Appl., 40, 2, 394-416 (2019) · Zbl 1469.14116
[26] Reutenauer, Christophe, Free Lie Algebras (1993), Oxford University Press · Zbl 0798.17001
[27] Barkley Rosser, J.; Schoenfeld, Lowell, Approximate formulas for some functions of prime numbers, Ill. J. Math., 6, 64-94 (1962) · Zbl 0122.05001
[28] Sturmfels, Bernd, Gröbner Bases and Convex Polytopes, University Lecture Series, vol. 8 (1996), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0856.13020
[29] The Sage Developers, SageMath, the Sage mathematics software system (version x.y.z) (2018)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.