Efficient Representation of Low-Dimensional Manifolds using Deep Networks
This addresses the challenge of efficient data representation in high-dimensional spaces, but it appears incremental as it builds on known concepts of manifolds and networks.
The paper tackled the problem of representing data near low-dimensional manifolds with deep networks, showing that they can efficiently extract intrinsic coordinates using an almost optimal number of parameters.
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.