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
The celebrated Ratner’s orbit closure theorem proved around 1990 says that in a homogeneous space of finite volume, the closures of orbits of a subgroup generated by unipotent flows is homogeneous. Searching for analogs of Ratner’s theorem is a major challenge in various settings.
We will discuss a new result on the classification of orbit closures for the action of a connected unipotent subgroup or more generally of any connected subgroup generated by unipotent flows, on a homogeneous space of infinite volume, which arise as the frame bundle of a hyperbolic manifold M whose core is a compact submanifold with totally geodesic boundary. 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).