On the integration of weakly geometric rough paths

Cass, Tony; Driver, Bruce K.; Lim, Nengli; Litterer, Christian

doi:10.2969/jmsj/06841505

Cited by 16 publications

(26 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will use 1 to denote the identity element with respect to ⊗ in T ((V )). [20] and Corollary 3.9 in [7]) Let x ∈ W GΩ p (V ). There exists a unique continuous path S (x) 0,· :…”

Section: )mentioning

confidence: 98%

The signature of a rough path: Uniqueness

Boedihardjo

Geng

Lyons

et al. 2016

Advances in Mathematics

128

View full text Add to dashboard Cite

In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly-Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.where x : [0, T ] → R d is a path with bounded variation and X 1 , . . . , X d are formal non-commutative indeterminates. After proving a homomorphism property of the map S ([8], see (2.1) below), he gave an argument [10] that the map S restricted to appropriate classes of paths is, up to translation and reparametrisation, injective. Hambly and Lyons [16], motivated by the application of the map S in rough path theory, posed the following problem:How to characterise the kernel of the map S?Hambly and Lyons [16] proved that for a bounded variation path x, S (x) = 1 if and only if x is tree-like. They conjectured that the result extends to weakly geometric rough paths, a fundamental class of control paths for which controlled differential equations can be defined. Their result directly implies that the space *

show abstract

“…We will use 1 to denote the identity element with respect to ⊗ in T ((V )). [20] and Corollary 3.9 in [7]) Let x ∈ W GΩ p (V ). There exists a unique continuous path S (x) 0,· :…”

Section: )mentioning

confidence: 98%

The signature of a rough path: Uniqueness

Boedihardjo

Geng

Lyons

et al. 2016

Advances in Mathematics

128

View full text Add to dashboard Cite

show abstract

“…If ϕ is injective and non-linear, one expects to obtain more efficient learning from signatures of paths in BV(W) since more non-linearities are provided by S •ϕ • ι • (B) ∈ T(W) than by S •ι • ∈ T(V). 8 For the choice M = ∞, τ = 0, ∆ = 0 (with l = 0) and ϕ = id : V → V, the corresponding Φ π recovers Φ, and k π (B, B ) = Φ π (B), Φ π (B ) recovers k. With slight abuse of notation, we write Φ for Φ π and k for k π for the remainder of the article. 7 The low-rank approximation algorithm in [22] reduces this cost further to O(lcM) time and O(M) memory.…”

Section: Hyperparametersmentioning

confidence: 99%

“…7 The low-rank approximation algorithm in [22] reduces this cost further to O(lcM) time and O(M) memory. 8 Generically W will be very high-or infinite-dimensional which prevents the direct calculation of S •ϕ • ι • . But if ϕ is the feature map of a kernel on V, (i.e.…”

Section: Hyperparametersmentioning

confidence: 99%

Persistence Paths and Signature Features in Topological Data Analysis

Chevyrev

Nanda

Oberhauser

2020

IEEE Trans. Pattern Anal. Mach. Intell.

View full text Add to dashboard Cite

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations -barcode to path, path to tensor series -results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.Persistence paths and signature features. Notwithstanding their usefulness for certain tasks, barcodes are notoriously unsuitable for standard statistical inference because 1 arXiv:1806.00381v2 [stat.ML]

show abstract

“…Fortunately, such

boldY

can be constructed in a canonical way. A construction based on ordered shuffles already appears in for the study of rough integration against weakly geometric rough paths, and also in for the uniqueness problem. Here, for completeness, we present an equivalent construction that does not rely on ordered shuffles and is more convenient to use from a combinatorial point of view.…”

Section: Extended Signatures For Truncated Signature Pathsmentioning

confidence: 99%

“…Fortunately, such Y can be constructed in a canonical way. A construction based on ordered shuffles already appears in [5,17] for the study of rough integration against weakly geometric rough paths, and also in [4] for the uniqueness problem.…”

Section: The Signature Of Truncated Signature Pathsmentioning

confidence: 99%

Reconstruction for the signature of a rough path

Geng

2017

Proc. London Math. Soc.

View full text Add to dashboard Cite

Recently, it was proved that the group of rough paths modulo tree‐like equivalence is isomorphic to the corresponding signature group through the signature map S (a generalized notion of taking iterated path integrals). However, the proof of this uniqueness result does not contain any information on how to ‘see’ the trajectory of a (tree‐reduced) rough path from its signature, and a constructive understanding on the uniqueness result (in particular, on the inverse of S) has become an interesting and important question. The aim of this paper is to reconstruct a rough path from its signature in an explicit and universal way.

show abstract

On the integration of weakly geometric rough paths

Abstract: We close a gap in the theory of integration for weakly geometric rough paths in the in…nite-dimensional setting. We show that the integral of a weakly geometric rough path against a su¢ciently regular one form is, once again, a weakly geometric rough path.

Cited by 16 publications

References 17 publications

The signature of a rough path: Uniqueness

The signature of a rough path: Uniqueness

Persistence Paths and Signature Features in Topological Data Analysis

Reconstruction for the signature of a rough path

Contact Info

Product

Resources

About