Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear, univariate functions of the distance to hyperplanes, sleeve functions are based on the squared distance to lower-dimensional manifolds. The present work is a first step to study general sleeve functions by starting with sleeve functions based on finite-length curves. To capture these curve-based sleeve functions, we propose and study a two-step method, where first the outer univariate function - the profile - is recovered, and second the underlying curve is represented by a polygonal chain. Introducing a concept of well-separation, we ensure that the proposed method always terminates and approximate the true sleeve function with a certain quality. Investigating the local geometry, we study an inexact version of our method and show its success under certain conditions.

中文翻译：

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高维中基于曲线的套筒函数的逼近

袖函数是完善的脊函数的推广，这些脊函数在偏微分方程理论、医学成像、统计学和神经网络中发挥着重要作用。脊函数是到超平面的距离的非线性单变量函数，套筒函数基于到低维流形的平方距离。目前的工作是从基于有限长度曲线的袖子函数开始研究一般袖子函数的第一步。为了捕捉这些基于曲线的套筒函数，我们提出并研究了一种两步法，首先恢复外部单变量函数 - 轮廓 - 其次，底层曲线由多边形链表示。引入井分离的概念，我们确保所提出的方法总是终止并以一定的质量逼近真实的袖子函数。调查局部几何，我们研究了我们方法的不精确版本，并在某些条件下展示了它的成功。