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 lowdimensional 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.
For manifolds, REF show how deep networks can efficiently represent low-dimensional manifolds and that these networks are almost optimal, but they do not discuss limitations of shallow networks on the same problem.
Efficient Representation of Low-Dimensional Manifolds using Deep Networks
ArXiv
ArXiv
Computer Science Mathematics