Monday, January 28, 2019
01/28/2019 - 4:00pm
We generalize the scattering transform to graphs and consequently construct a convolutional neural network on graphs. We further use it to form a graph generative model. We show that under certain conditions, any feature generated by such a network is approximately invariant to permutations and stable to signal and graph manipulations. Numerical results demonstrate competitive performance on relevant datasets for the tasks of community detection, link prediction and graph generation.
01/28/2019 - 4:15pm
We establish an analogue of Ratner’s orbit closure theorem for an action of any connected subgroup generated by unipotent elements in SO(n,1) in the space Gamma\SO(n,1), assuming that the core of the associated hyperbolic manifold M=Gamma\H^d is compact with totally geodesic boundary. For dimension 3, this was proved earlier by McMullen-Mohammadi-Oh. In a higher dimensional case, the possibility of accummulation on closed orbits of intermediate groups causes serious new issues, but in the end, all orbit closures of unipotent flows are homogeneous in the correct sense. Our result implies that
1. the closure of any horocycle in M is a properly immersed submanifold
2. the closure of any geodesic plane intersecting the core of M is a properly immersed submanifold;
3. any infinite sequence of maximal properly immersed geodesic planes intersecting the core of M becomes dense in M.
(This talk is based on joint work with Minju Lee).