Effects of Data Geometry in Early Deep Learning
This work addresses the problem of understanding neural network expressivity on structured data for researchers in deep learning theory, but it is incremental as it builds on existing theoretical advances.
The paper studied how randomly initialized deep neural networks with piece-wise linear activations partition data manifolds into linear regions, deriving bounds on boundary density and distance to provide insights into expressivity on non-Euclidean data. Experiments on toy problems and the MetFaces dataset showed that the number of linear regions varies across manifolds and differs from Euclidean space.
Deep neural networks can approximate functions on different types of data, from images to graphs, with varied underlying structure. This underlying structure can be viewed as the geometry of the data manifold. By extending recent advances in the theoretical understanding of neural networks, we study how a randomly initialized neural network with piece-wise linear activation splits the data manifold into regions where the neural network behaves as a linear function. We derive bounds on the density of boundary of linear regions and the distance to these boundaries on the data manifold. This leads to insights into the expressivity of randomly initialized deep neural networks on non-Euclidean data sets. We empirically corroborate our theoretical results using a toy supervised learning problem. Our experiments demonstrate that number of linear regions varies across manifolds and the results hold with changing neural network architectures. We further demonstrate how the complexity of linear regions is different on the low dimensional manifold of images as compared to the Euclidean space, using the MetFaces dataset.