论文标题
通过Sussmann的轨道定理将沉浸式的Submanifold拟合到数据
Fitting an immersed submanifold to data via Sussmann's orbit theorem
论文作者
论文摘要
本文介绍了一种拟合有限维欧几里得空间的沉浸式亚策略的方法。从环境空间到所需的子手机的重建映射是作为编码器的组成而实现的，该编码器将每个点映射到（正或负）时间的元组以及由有限多个矢量场的组成从固定初始点开始的，这是有限的许多矢量场给出的解码器。编码器为流量提供时间。通过经验风险最小化获得编码器二进制图，并且相对于给定的Encoder-Decoder映射的最小预期重建误差给出了高度风险的高概率。拟议的方法是对苏斯曼的轨道定理的基本使用，该定理保证了重建图的图像确实包含在沉浸式的子手机中。
This paper describes an approach for fitting an immersed submanifold of a finite-dimensional Euclidean space to random samples. The reconstruction mapping from the ambient space to the desired submanifold is implemented as a composition of an encoder that maps each point to a tuple of (positive or negative) times and a decoder given by a composition of flows along finitely many vector fields starting from a fixed initial point. The encoder supplies the times for the flows. The encoder-decoder map is obtained by empirical risk minimization, and a high-probability bound is given on the excess risk relative to the minimum expected reconstruction error over a given class of encoder-decoder maps. The proposed approach makes fundamental use of Sussmann's orbit theorem, which guarantees that the image of the reconstruction map is indeed contained in an immersed submanifold.
