Uniqueness for the signature of a path of bounded variation and the reduced path group by Hambly, B, Lyons, T

Effective algorithms for inverting the signature of a path by Chang, J

“…It has been proved in the literature that the terms in the signature of a path are bounded above, and it would then be interesting to consider whether a lower bound exists for the signature of a bounded-variation path. It has also been proved that the signature of a bounded-variation path is unique up to some modifications, then a natural question is reconstructing the path from its signature, i.e. …”

A high order solver for signature kernels by Lemercier, M, Lyons, T

“…The kernel of two bounded variation paths (such as piecewise linear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. …”

Rough paths, signatures and the modelling of functions on streams by Lyons, T

“…Among paths of bounded variation with given Signature there is always a unique shortest representative. …”

The signature of a rough path: Uniqueness by Boedihardjo, H, Geng, X, Lyons, T, Yang, D

“…Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. …”

General signature kernels by Cass, T, Lyons, T, Xu, X

“…Suppose that γ and σ are two continuous bounded variation paths which take values in a finite-dimensional inner product space V. …”

Weighted signature kernels by Cass, T, Lyons, T, Xu, X

“…Suppose that γ and σ are two continuous bounded variation paths which take values in a finite-dimensional inner product space V. …”

Inverting the signature of a path by Xu, W

“…Finally, we prove a uniform estimate for the signatures of paths of bounded variations, and obtain a reconstruction theorem via that uniform estimate. …”

The adaptive patched cubature filter and its implementation by Lee, W, Lyons, T

“…The algorithm can be performed by integrating along a number of carefully selected bounded variation paths. The iterated application of the KLV method has a tendency for the number of particles to increase. …”

Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length by Chang, J, Lyons, T, Ni, H

“…Lévy, Differential equations driven by rough paths, Springer, 2007) that the norm of the <i>n</i>-th term of the normalised signature of a bounded-variation path is bounded above by 1. In this article, we discuss the super-multiplicativity of the norm of the signature of a path with finite length, and prove by Fekete's lemma the existence of a non-zero limit of the <i>n</i>-th root of the norm of the <i>n</i>-th term in the normalised signature as <i>n</i> approaches infinity.…”