论文标题
绘制空间签名的拓扑方法
A Topological Approach to Mapping Space Signatures
论文作者
论文摘要
在拓扑空间$ \ MATHCAL {X} $上描述功能类和概率度量类别的常用方法是，从$ \ Mathcal {x} $中构造合适的地图$φ$，以在矢量空间中，可以将线性方法应用于解决这两个问题。 $ \ MATHCAL {X} $是路径$ [0,1] \ to \ Mathbb {r}^n $和$φ$的情况，是路径签名映射在随机分析和相关字段中引起了很多关注。在本文中，我们为$ \ mathcal {x} $是映射$ [0,1]^d \ to \ mathbb {r}^n $的$ \ mathcal {x} $开发了一个广义的$φ$。我们方法的关键要素是拓扑；特别是，我们的起点是将K-T Chen的路径空间Cochain结构的概括为立方映射空间的设置。
A common approach for describing classes of functions and probability measures on a topological space $\mathcal{X}$ is to construct a suitable map $Φ$ from $\mathcal{X}$ into a vector space, where linear methods can be applied to address both problems. The case where $\mathcal{X}$ is a space of paths $[0,1] \to \mathbb{R}^n$ 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 $\mathcal{X}$ is a space of maps $[0,1]^d \to \mathbb{R}^n$ for any $d \in \mathbb{N}$, and show that the map $Φ$ generalizes many of the desirable algebraic and analytic properties of the path signature to $d \ge 2$. 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.
