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Effects of Data Geometry in Early Deep Learning
Saket Tiwari · George Konidaris

Thu Dec 01 02:00 PM -- 04:00 PM (PST) @ Hall J #432

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 piecewise 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.

Author Information

Saket Tiwari (Brown University)

I am a 4th Year PhD student working on the theory of DL and Deep RL. I look at DNNs from the lens of data geometry.

George Konidaris (Brown University)

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