Related papers: On the signature of an image
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…
We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of…
In the last decade, the concept of path signature has achieved significant success in data science applications. It offers a powerful set of features that effectively capture and describe the characteristics of paths or sequential data.…
Quasisymmetric functions have recently been used in time series analysis as polynomial features that are invariant under, so-called, dynamic time warping. We extend this notion to data indexed by two parameters and thus provide warping…
We provide an introduction to the signature method, focusing on its theoretical properties and machine learning applications. Our presentation is divided into two parts. In the first part, we present the definition and fundamental…
The signature of a path is an essential object in the theory of rough paths. The signature representation of the data stream can recover standard statistics, e.g. the moments of the data stream. The classification of random walks indicates…
The signature is a representation of a path as an infinite sequence of its iterated integrals. Under certain assumptions, the signature characterizes the path, up to translation and reparameterization. Therefore, a crucial question of…
In this paper, we introduce the branched signature model, motivated by the branched rough path framework of [Gubinelli, Journal of Differential Equations, 248(4), 2010], which generalizes the classical geometric rough path. We establish a…
Recent image-to-image (I2I) translation algorithms focus on learning the mapping from a source to a target domain. However, the continuous translation problem that synthesizes intermediate results between two domains has not been…
We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric H\"older continuous rough paths can be approximated…
A common approach for describing classes of functions and probability measures on a topological space $\mathcal{X}$ is to construct a suitable map $\Phi$ from $\mathcal{X}$ into a vector space, where linear methods can be applied to address…
Given values of a piecewise smooth function $f$ on a square grid within a domain $\Omega$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of…
It is shown that if $\gamma$ is a path of finite $p$ variation ($1\leq p< 2$) in a euclidean vector space and $f,g,h$ are Lipschitz functions on the trace of $\gamma$ then $s\mapsto F(s)=\int_\gamma f^sg dh$ defines an entire holomorphic…
Sequential and temporal data arise in many fields of research, such as quantitative finance, medicine, or computer vision. A novel approach for sequential learning, called the signature method and rooted in rough path theory, is considered.…
Parallel transport, or path development, provides a rich characterization of paths which preserves the underlying algebraic structure of concatenation. The path signature is universal among such maps: any (translation-invariant) parallel…
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…
Path signatures are powerful nonparametric tools for time series analysis, shown to form a universal and characteristic feature map for Euclidean valued time series data. We lift the theory of path signatures to the setting of Lie group…
We provide an introduction to the topic of path signatures as means of feature extraction for machine learning from data streams. The article stresses the mathematical theory underlying the signature methodology, highlighting the conceptual…
Landmark-based human action recognition in videos is a challenging task in computer vision. One key step is to design a generic approach that generates discriminative features for the spatial structure and temporal dynamics. To this end, we…
We develop two methods to reconstruct a path of bounded variation from its signature. The first method gives a simple and explicit expression of any axis path in terms of its signature, but it does not apply directlty to more general ones.…