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On the Number of Linear Regions of Deep Neural Networks
Guido F Montufar · Razvan Pascanu · Kyunghyun Cho · Yoshua Bengio

Tue Dec 09 04:00 PM -- 08:59 PM (PST) @ Level 2, room 210D

We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map portions of each layer's input-space to the same output. In this way, deep models compute functions that react equally to complicated patterns of different inputs. The compositional structure of these functions enables them to re-use pieces of computation exponentially often in terms of the network's depth. This paper investigates the complexity of such compositional maps and contributes new theoretical results regarding the advantage of depth for neural networks with piecewise linear activation functions. In particular, our analysis is not specific to a single family of models, and as an example, we employ it for rectifier and maxout networks. We improve complexity bounds from pre-existing work and investigate the behavior of units in higher layers.

Author Information

Guido F Montufar (UCLA)
Razvan Pascanu (Google DeepMind)
Kyunghyun Cho (Genentech / NYU)

Kyunghyun Cho is an associate professor of computer science and data science at New York University and a research scientist at Facebook AI Research. He was a postdoctoral fellow at the Université de Montréal until summer 2015 under the supervision of Prof. Yoshua Bengio, and received PhD and MSc degrees from Aalto University early 2014 under the supervision of Prof. Juha Karhunen, Dr. Tapani Raiko and Dr. Alexander Ilin. He tries his best to find a balance among machine learning, natural language processing, and life, but almost always fails to do so.

Yoshua Bengio (Mila / U. Montreal)

Yoshua Bengio (PhD'1991 in Computer Science, McGill University). After two post-doctoral years, one at MIT with Michael Jordan and one at AT&T Bell Laboratories with Yann LeCun, he became professor at the department of computer science and operations research at Université de Montréal. Author of two books (a third is in preparation) and more than 200 publications, he is among the most cited Canadian computer scientists and is or has been associate editor of the top journals in machine learning and neural networks. Since '2000 he holds a Canada Research Chair in Statistical Learning Algorithms, since '2006 an NSERC Chair, since '2005 his is a Senior Fellow of the Canadian Institute for Advanced Research and since 2014 he co-directs its program focused on deep learning. He is on the board of the NIPS foundation and has been program chair and general chair for NIPS. He has co-organized the Learning Workshop for 14 years and co-created the International Conference on Learning Representations. His interests are centered around a quest for AI through machine learning, and include fundamental questions on deep learning, representation learning, the geometry of generalization in high-dimensional spaces, manifold learning and biologically inspired learning algorithms.

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