Tuesday, October 23, 2018
10/23/2018 - 4:00pm
In this work, we present a Deep Learning based approach for visual correspondence estimation, by deriving a Deep spectral graph matching network. We formulate the state-of-the-art unsupervised Spectral Graph Matching (SGM) approach, as part of an end-to-end supervised deep learning network. Thus allowing to utilize back-propagation to learn optimal image features, as well as algorithm parameters. For that, we present a transformation layer that converts the learned image feature, within a pair of images, to an affinity matrix used to solve the matching problem via a new metric loss function. The proposed scheme is shown to compare favorably with contemporary state-of-the-art matching schemes when applied to annotated data obtained from the PASCAL, ILSVRC, KITTI and CUB-2011 datasets.
10/23/2018 - 4:00pm
The rank of a group is the minimal cardinality of a generating set. While simple to define, this quantity is notoriously difficult to calculate, and often uncomputable, even for well behaved groups. In this talk I will explain general conditions that may be used to show many hyperbolic group extensions have rank equal to the rank of the kernel plus the rank of the quotient and, further, that any minimal generating set is Nielsen equivalent to one in a standard form. This builds on work of Souto on fundamental groups of fibered hyperbolic three manifolds and of Scott-Swarup who in this setting proved that infinite-index finitely generated subgroups of the fiber are quasi-convex in the ambient group. As an application, we prove that if $g_1,\dotsc,g_k$ are independent, atoroidal, fully irreducible outer automorphisms of the free group $F_n$, then there is a power $m$ so that the subgroup generated by $f_1^m,\dotsc,f_k^m$ gives rise to a hyperbolic extension of $F_n$ of rank $n+k$. Joint work with Sam Taylor.
10/23/2018 - 4:15pm
We give another proof of the endoscopic fundamental lemma (theorem of Laumon-Ngo) for unitary Lie algebras in characteristic 0. We study the compatibility between Fourier transform and transfer and we prove that the compatibility in the Jacquet-Rallis setting, after taking limits using germ expansions, will imply the compatibility in the endoscopic setting for unitary group.