使用深度网络高效表示低维流形
Efficient Representation of Low-Dimensional Manifolds using Deep Networks
摘要：
我们考虑深度神经网络表示位于高维空间中的低维流形附近的数据的能力。我们证明，深度网络可以有效地提取这些数据的内在低维坐标。我们首先证明了深度网络的前两层可以精确地嵌入位于单调链（一种特殊类型的分段线性流形）上的点，将它们映射到低维欧氏空间。值得注意的是，网络可以使用几乎最佳数量的参数来做到这一点。我们还表明，该网络的项目附近的点到流形上，然后嵌入他们几乎没有错误。然后，我们将这些结果推广到更一般的流形。
英文摘要：
We consider the ability of deep neural networks to represent data that lies near a low-dimensional manifold in a high-dimensional space. We show that deep networks can efficiently extract the intrinsic, low-dimensional coordinates of such data. We first show that the first two layers of a deep network can exactly embed points lying on a monotonic chain, a special type of piecewise linear manifold, mapping them to a low-dimensional Euclidean space. Remarkably, the network can do this using an almost optimal number of parameters. We also show that this network projects nearby points onto the manifold and then embeds them with little error. We then extend these results to more general manifolds.
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