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ABSTRACT

The signature of a path is a sequence of iterated integrals along the path, an element taking value in the tensor algebra. Kuo-Tsai Chen developed these objects in a series of works starting from 1950's. An early theorem of Wei-Liang Chow also has interesting implications in this context. Recently, these objects have been used in machine learning, for example to distinguish features of time-series data. We can ask the following three questions regarding the signature: 1. What elements in the tensor algebra are in the image of the signature map? 2. Given a signature, is the path(s) corresponding to it unique? 3. If unique, then how can we reconstruct the path from its signature? For the first question, there are some natural necessary conditions, but whether they are sufficient is unclear. There have been good progress towards the latter two questions, and I will report on these progress.

空间中路径的 signature 变换由该路径的迭代积分给出，取值于张量代数。其始于上世纪 50 年代陈国才的一系列工作。周炜良早年的一个定理在其中也有重要推论。 近年来，signature 也系统地出现在数据分析中，主要被用于识别曲线类数据的特征。关于 signature，我们可以问三个基本的问题： 1. signature 变换的像是什么？即张量代数中什么样的元素可以经由路径的 signature 变换得到。 2. signature 变换是否是单射？即给定某个 signature，其对应的原路径是否唯一。 3. 反问题：如果唯一，那么给定 signature，如何（近似地）找出其对应的原路径。 关于第一个问题，有一些很自然的必要条件，但它们的充分性未知。对于后两个问题，近年来有了不错的进展。我将汇报一下这些进展。

Tencent meeting ID: 

221 594 197

Zoom meeting ID: 

63454179757 (Password: 941885) 

https://zoom.com.cn/j/63454179757