Tuesday, April 30, 2019
04/30/2019 - 4:00pm
Abstract: Explicit encoding of group actions in data representation is desired for convolutional neural networks (CNNs) to successfully handle global deformations in input signals. In this talk, we introduce group-equivariant deep CNNs where the convolutional filters are jointly decomposed over steerable bases on the space and the group geometry simultaneously. This decomposition significantly reduces the model size and computational complexity while preserving network performance, and it also serves to regularize the convolutional filters by the truncation of bases expansion. The stability of the equivariant representation with respect to input variations is proved theoretically and also demonstrated on computer vision tasks where the datasets involve in-plane and out-of-plane object rotations. The work provides a general approach to achieve group equivariant features in deep CNNs with representation stability and computational efficiency.
04/30/2019 - 4:15pm
The second real homology group of a hyperbolic 3-manifold M has a norm, called the Thurston norm, whose unit ball is a rational polyhedron detecting information about the topology of M. A collection of faces of this polyhedron called fibered faces organize all possible fibrations of M over the circle. The cone over a fibered face has a nice description, due to Fried, as the dual of the so-called cone of homology directions of a certain flow. In this relatively self-contained defense, we will show that this cone of homology directions is generated by a canonical family of curves living in one of Agol’s veering triangulations. This gives a new characterization of the cone over a fibered face. If time permits, we will discuss some related results.