标题： 迭代积分签名的共轭、环和闭合不变量 显示英文标题

标题： Conjugation, loop and closure invariants of the iterated-integrals signature

摘要： 给定一个闭合环路形状的特征集，自然会问该集合中哪些特征在路径的起始点移动时不会改变。 例如，在二维情况下，路径所包围的面积不依赖于起始点。 在本文中，我们对所有被称为给定路径的迭代积分的特征进行了这样的环路不变量的表征。 此外，我们将它们与共轭不变量联系起来，当将（树约简）路径视为一个群时，共轭不变量是一个典型的研究对象，其中乘法由连接给出。 最后，闭合不变量是这一背景下第三类不变量，当研究分段线性轨迹时尤其相关，例如由时间序列的线性插值给出的轨迹。 关键词：不变特征；路径连接；组合项链；交错代数；自由李代数；带符号面积；带符号体积；树状等价。 显示英文摘要

摘要： Given a feature set for the shape of a closed loop, it is natural to ask which features in that set do not change when the starting point of the path is moved. For example, in two dimensions, the area enclosed by the path does not depend on the starting point. In the present article, we characterize such loop invariants among all those features known as interated integrals of a given path. Furthermore, we relate these to conjugation invariants, which are a canonical object of study when treating (tree reduced) paths as a group with multiplication given by the concatenation. Finally, closure invariants are a third class in this context which is of particular relevance when studying piecewise linear trajectories, e.g. given by linear interpolation of time series. Keywords: invariant features; concatenation of paths; combinatorial necklaces; shuffle algebra; free Lie algebra; signed area; signed volume; tree-like equivalence.