Invariants of multidimensional time series based on their iterated-integral signature

Diehl, Joscha; Reizenstein, Jeremy

doi:10.48550/arxiv.1801.06104

Cited by 3 publications

(6 citation statements)

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“…2 by multiplying each of its k sides with the d × m matrix X. This is the tensor analogue to the congruence action on matrix space seen in (15). The x i j are homogeneous coordinates on the projective space P dm−1 over an algebraically closed field K that contains R. The matrix-tensor multiplication described above defines a rational map of degree k:…”

Section: Polynomial Maps Intomentioning

confidence: 99%

“…Now let k = 4. The varieties P 2,4,2 and L 2,4,2 are orbit closures for the GL(d, K)-action on P 15 . We can use invariant theory to show that these orbits are different.…”

Section: Polynomial Maps Intomentioning

confidence: 99%

“…We can use invariant theory to show that these orbits are different. According to Diehl and Reizenstein [15,Remark 14], the space of SL(d, K)-invariants linear forms on (K 2 ) ⊗4 has dimension 2 and is spanned by…”

Section: Polynomial Maps Intomentioning

confidence: 99%

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Varieties of Signature Tensors

Améndola

Friz

Sturmfels

2019

Forum of Mathematics, Sigma

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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 here examined 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.

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Section: Polynomial Maps Intomentioning

confidence: 99%

“…Now let k = 4. The varieties P 2,4,2 and L 2,4,2 are orbit closures for the GL(d, K)-action on P 15 . We can use invariant theory to show that these orbits are different.…”

Section: Polynomial Maps Intomentioning

confidence: 99%

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Varieties of Signature Tensors

Améndola

Friz

Sturmfels

2019

Forum of Mathematics, Sigma

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“…Coming from stochastic analysis, the signatures are becoming more relevant in other areas, such as algebraic geometry or combinatorics, and we would like to highlight some recent work. For instance, in [DR18], J. Diehl and J. Reizenstein offer a combinatorial approach to the understanding of invariants of multidimensional time series based on their signature. Another reference is [AFS18], in which C. Améndola, P. Friz, and B. Sturmfels look at the varieties of signatures of tensors for both deterministic and random paths, focusing on piecewise linear paths and polynomials paths, among others.…”

Section: Introductionmentioning

confidence: 99%

Signatures of paths transformed by polynomial maps

Colmenarejo

Preiß

2020

Beitr Algebra Geom

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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. It allows us to describe the relation between the coefficients of the signature of a piecewise continuously differentiable path transformed by a polynomial map and the coefficients of the signature of the initial path.

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“…Signature tensors were introduced by Chen [7], and they play an important role in stochastic analysis [12,21]. We refer to [22,23] for the recovery problem, and to [10,13,17,18] for algorithms and applications. Our point of departure is the approach to signature tensors via algebraic geometry that was proposed in [1].…”

Section: Introductionmentioning

confidence: 99%

Learning Paths from Signature Tensors

Pfeffer¹,

Seigal²,

Sturmfels³

2019

SIAM J. Matrix Anal. Appl.

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Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor.2010 Mathematics Subject Classification. 14Q15, 15A72, 65K10.

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Invariants of multidimensional time series based on their iterated-integral signature

Cited by 3 publications

References 0 publications

Varieties of Signature Tensors

Varieties of Signature Tensors

Signatures of paths transformed by polynomial maps

Learning Paths from Signature Tensors

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