Related papers: Branched Signature Model
Paths of persistence diagrams provide a summary of the dynamic topological structure of a one-parameter family of metric spaces. These summaries can be used to study and characterize the dynamic shape of data such as swarming behavior in…
The universal limit theorem is a central result in rough path theory, which has been proved for: (i) rough paths with roughness $\frac{1}{3}< \alpha \leq \frac{1}{2}$; (ii) geometric rough paths with roughness $0< \alpha \leq 1$; (iii)…
We bring the theory of rough paths to the study of non-parametric statistics on streamed data. We discuss the problem of regression where the input variable is a stream of information, and the dependent response is also (potentially) a…
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…
A signed tree model of a graph $G$ is a compact binary structure consisting of a rooted binary tree whose leaves are bijectively mapped to the vertices of $G$, together with 2-colored edges $xy$, called transversal pairs, interpreted as…
In the context of stochastic portfolio theory we introduce a novel class of portfolios which we call linear path-functional portfolios. These are portfolios which are determined by certain transformations of linear functions of a…
We develop a rough-path framework for two-parameter rough differential equations on rectangular and simplicial domains, motivated by the signature kernel and Schwinger--Dyson kernel equations. The theory is formulated in spaces of jointly…
Central to rough path theory is the signature transform of a path, an infinite series of tensors given by the iterated integrals of the underlying path. The signature poses an effective way to capture sequentially ordered information,…
The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential…
Many finance, physics, and engineering phenomena are modeled by continuous-time dynamical systems driven by highly irregular (stochastic) inputs. A powerful tool to perform time series analysis in this context is rooted in rough path theory…
Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them…
The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry.…
The sequential data observed in earth science can be regarded as paths in multidimensional space. To read the path effectively, it is useful to convert it into a sequence of numbers called the signature, which can faithfully describe the…
Implicit surface representations, such as signed-distance functions, combined with deep learning have led to impressive models which can represent detailed shapes of objects with arbitrary topology. Since a continuous function is learned,…
Signature kernels, inner products of path signatures, underpin several machine learning algorithms for multivariate time series analysis. For bounded variation paths, signature kernels were recently shown to solve a Goursat PDE. However,…
The signature of a path, introduced by K.T. Chen [5] in $1954$, has been extensively studied in recent years. The $2010$ paper [12] of Hambly and Lyons showed that the signature is injective on the space of continuous finite-variation paths…
We prove that the derived direct image of the constant sheaf with field coefficients under any proper map with smooth source contains a canonical summand. This summand, which we call the geometric extension, only depends on the generic…
The expected signature kernel arises in statistical learning tasks as a similarity measure of probability measures on path space. Computing this kernel for known classes of stochastic processes is an important problem that, in particular,…
Signature, lying at the heart of rough path theory, is a central tool for analysing controlled differential equations driven by irregular paths. Recently it has also found extensive applications in machine learning and data science as a…
Signature tensors of paths are a versatile tool for data analysis and machine learning. Recently, they have been applied to persistent homology, by embedding barcodes into spaces of paths. Among the different path embeddings, the…