Related papers: The Surface Signature and Rough Surfaces
This paper establishes a comprehensive concentration theory for truncated signatures of Gaussian rough paths. The signature of a path, defined as the collection of all iterated integrals, provides a complete description of its geometric…
Contrary to previous approaches bringing together algebraic geometry and signatures of paths, we introduce a Zariski topology on the space of paths itself, and study path varieties consisting of all paths whose iterated-integrals signature…
Over the past decade, the importance of the 1D signature which can be seen as a functional defined along a path, has been pivotal in both path-wise stochastic calculus and the analysis of time series data. By considering an image as a…
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 path signature, having enjoyed recent success in the machine learning community, is a theoretically-driven method for engineering features from irregular paths. On the other hand, graph neural networks (GNN), neural architectures for…
An ant-like observer confined to a two-dimensional surface traversed by stripes would wonder whether this striped landscape could be devised in such a way as to appear to be the same wherever they go. Differently stated, this is the problem…
While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a…
The theory of linear transports along paths in vector bundles, generalizing the parallel transports generated by linear connections, is developed. The normal frames for them are defined as ones in which their matrices are the identity…
In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors of different order that encode information of the path in a compact form. When the path varies, such signatures…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
We introduce a proper notion of 2-dimensional signature for images. This object is inspired by the so-called rough paths theory, and it captures many essential features of a 2-dimensional object such as an image. It thus serves as a…
The signature transform, defined by the formal tensor series of global iterated path integrals, is a homomorphism between the path space and the tensor algebra that has been studied in geometry, control theory, number theory as well as…
Rough sheets are two-parameter analogs of rough paths. In this work the theory of integration over functions of two parameters is extended to cover the case of irregular functions by developing an appropriate notion of rough sheet. The main…
The motivation for this paper stems \cite{CR} from the need to construct explicit isomorphisms of (possibly nontrivial) principal $G$-bundles on the space of loops or, more generally, of paths in some manifold $M$, over which I consider a…
Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come.…
We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an efficient and intrinsic theory of rough differential…
Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map…
The signature of a $p$-weakly geometric rough path summarises a path up to a generalised notion of reparameterisation. The quotient space of equivalence classes on which the signature is constant yields unparameterised path space. The study…
We introduce a nonabelian surface holonomy that is constructed from a one-form gauge potential that takes values in a loop algebra of the $U(N)$ gauge group. The surface holonomy parallel transports a nonabelian string. Although it is not…
We study signature tensors of paths from a geometric viewpoint. The signatures of a given class of paths parametrize an algebraic variety inside the space of tensors, and these signature varieties provide both new tools to investigate paths…