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 approximates 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.

套筒函数是成熟岭函数的推广，在偏微分方程、医学成像、统计学和神经网络理论中发挥着重要作用。其中岭函数是到超平面距离的非线性单变量函数，而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套筒函数开始研究一般套筒函数的第一步。为了捕获这些基于曲线的套筒函数，我们提出并研究了一种两步方法，首先恢复外部单变量函数（轮廓），其次，用多边形链表示基础曲线。引入良好分离的概念，我们确保所提出的方法始终终止并以一定的质量逼近真实的套筒函数。通过研究局部几何形状，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其成功。