Path Signatures on Lie Groups

Lee, Darrick

doi:10.48550/arxiv.2007.06633

Cited by 4 publications

(5 citation statements)

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“…The condition of metrisability is stated as an assumption in a number of recent works [3,11,14] and our results would therefore seem to preclude the use of quotient topology in these cases. A careful reading of the reference [10] underpinning these results, however, shows the property of complete regularity and not metrisability is the key assumption.…”

Section: Introductionmentioning

confidence: 73%

“…and there exists a constant c > 0 such that for any 1 ≤ n < m we have As mentioned in the introduction, several recent references, see for example [3,11,14], state metrisability as an assumption. This premise may be traced to [7], which itself derives from an application of Theorems 2.4, 3.1, and 4.6 of Giles in [10].…”

Section: Complete Metrisability Of Candidate Topologies?mentioning

confidence: 99%

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Topologies on unparameterised path space

Cass¹,

Turner²

2022

Preprint

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The signature of a path, introduced by K.T. Chen [5] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [12] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [15], guarantees uniform approximation on compact sets of a continuous function by a linear functional of the signature.In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the tensor algebra and then requiring S to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlying path space, and (3) the metric topology associated to d ([γ] , [σ]) := ||γ * − σ * || 1 using the (constant-speed) treereduced representatives γ * and σ * of the respective equivalence classes. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ-compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G * of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces.We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, welldefinedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [15].

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Section: Introductionmentioning

confidence: 73%

Section: Complete Metrisability Of Candidate Topologies?mentioning

confidence: 99%

Topologies on unparameterised path space

Cass¹,

Turner²

2022

Preprint

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“…• The path signature is defined for paths valued in a Banach space [26], and our coordinate-free definitions here could be extended to the setting of Banach spaces. • Many properties of the path signature can be extended to paths valued in Lie groups [23] and thus it is natural to ask if one can replace the co-domain V by a non-linear space X. Indeed, the mapping space monomials derived from the cubical mapping space construction in Appendix A.3 can also be defined on an arbitrary manifold X, where we must choose a set ω 1 , .…”

Section: Universal and Characteristic Propertiesmentioning

confidence: 99%

A Topological Approach to Mapping Space Signatures

Giusti¹,

Lee²,

Nanda³

et al. 2022

Preprint

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A common approach for describing classes of functions and probability measures on a topological space X is to construct a suitable map Φ from X into a vector space, where linear methods can be applied to address both problems. The case where X is a space of paths [0, 1] → R n and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where X is a space of maps [0, 1] d → R n for any d ∈ N, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to d ≥ 2. The key ingredient to our approach is topological; in particular, our starting point is a generalisation of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.

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“…In the terminology of statistical learning this says that ϕ is a universal and characteristic feature map for the set Seq(V ) of sequences. For more background and extensions to non-compact sets of sequences or paths, we refer to [34,16] and [6, Section 3.2]; for a more geometric picture see [39].…”

Section: The Sequence Feature Mapmentioning

confidence: 99%

Capturing Graphs with Hypo-Elliptic Diffusions

Tóth¹,

Lee²,

Hacker³

et al. 2022

Preprint

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Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves according to a diffusion equation defined using the graph Laplacian. We extend this approach by leveraging classic mathematical results about hypo-elliptic diffusions. This results in a novel tensor-valued graph operator, which we call the hypo-elliptic graph Laplacian. We provide theoretical guarantees and efficient low-rank approximation algorithms. In particular, this gives a structured approach to capture long-range dependencies on graphs that is robust to pooling. Besides the attractive theoretical properties, our experiments show that this method competes with graph transformers on datasets requiring long-range reasoning but scales only linearly in the number of edges as opposed to quadratically in nodes.* Equal contribution; order determined by random coin flip. 1 An algebra is a vector space where one can multiply elements; e.g. the set of n × n matrices with matrix multiplication. This multiplication can be non-commutative; e.g. A • B = B • A for general matrices A, B.

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Path Signatures on Lie Groups

Cited by 4 publications

References 0 publications

Topologies on unparameterised path space

Topologies on unparameterised path space

A Topological Approach to Mapping Space Signatures

Capturing Graphs with Hypo-Elliptic Diffusions

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