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2011

Inversion of signature for paths of bounded variation

Lyons, Terry, Xu, Weijun

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We develop two methods to reconstruct a path of bounded variation from its signature.

The first method gives a simple and explicit expression of any axis path in terms of its signature, but it does not apply directlty to more general ones.
The second method, based on an approximation scheme, recovers any tree-reduced path from its signature as the limit of a uniformly convergent sequence of lattice paths.

Inversion of signature for paths of bounded variation · Around

Built on

K-T Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Annals of Mathematics
1957
Earlier work this paper cites.
K-T Chen, Integration of paths - a faithful representation of paths by noncommutative formal power series, Transactions of the A.M.S
1958
Earlier work this paper cites.
K-T Chen, Iterated path integrals, Bulletin of the A.M.S
1977
Earlier work this paper cites.
L.C.Evans, R.F.Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1991
1991
Earlier work this paper cites.

Similar

T.J.Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana
1998
Cited alongside, same era.
R.P.Stanley, Enumerative Combinatorics, Vol.1, second edition, Cambridge Studies in Advanced Mathematics
2000
Cited alongside, same era.
T.J.Lyons, Z.Qian, System Control and Rough Paths, Oxford Mathematical Monographs
2002
Cited alongside, same era.
T.J.Lyons, M.J.Caruana, T.Lévy, Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics
2006
Cited alongside, same era.

Then

B.M.Hambly, T.J.Lyons, Some notes on trees and paths, unpublished manuscript, available at http://arxiv.org/abs/0809.1365, 2008
Original
2008
Later among the works it cites.
B.M.Hambly, T.J.Lyons, Uniqueness for the signature of a path of bounded variation and the reduced path group, Annals of Mathematics
2010
Later among the works it cites.
Y.Jiang, W.Xu, On number of turns in reduced random lattice paths, preprint, available at http://arxiv.org/abs/1007.3507, 2010
Original
2010
Later among the works it cites.

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