The problem of extending a function $f$ defined on a training data
$\mathcal{C}$ on an unknown manifold $\mathbb{X}$ to the entire manifold and a
tubular neighborhood of this manifold is considered in this paper. For
$\mathbb{X}$ embedded in a high dimensional ambient Euclidean space
$\mathbb{R}^D$, a deep learning algorithm is developed for finding a local
coordinate system for the manifold {\bf without eigen--decomposition}, which
reduces the problem to the classical problem of function approximation on a low
dimensional cube. Deep nets (or multilayered neural networks) are proposed to
accomplish this approximation scheme by using the training data. Our methods do
not involve such optimization techniques as back--propagation, while assuring
optimal (a priori) error bounds on the output in terms of the number of
derivatives of the target function. In addition, these methods are universal,
in that they do not require a prior knowledge of the smoothness of the target
function, but adjust the accuracy of approximation locally and automatically,
depending only upon the local smoothness of the target function. Our ideas are
easily extended to solve both the pre--image problem and the out--of--sample
extension problem, with a priori bounds on the growth of the function thus
extended.

本文提出了一种基于深度学习的算法，用于在未知流形中函数的扩展问题上。该算法使用多层神经网络和局部坐标系统来实现函数近似，同时保证输出的误差范围与目标函数的导数数量成正比，并在不需要目标函数的光滑性的情况下自动调整精度。