Tuesday, December 3, 2019
12/03/2019 - 4:00pm
The recent success of generative adversarial networks and variational learning suggests that training a classifier network may work well in addressing the classical two-sample testing problem. Neural network approaches, compared to kernel approaches, can have the computational advantage that the algorithm better scales to large samples once the model is trained. In this talk, we introduce the network-logit test, which uses the logit of a trained neural network classifier evaluated on the two finite samples as the test statistic. Theoretically, the testing power to differentiate two smooth densities is proved given that the network is sufficiently parametrized, and when the two densities lie on or near to low-dimensional manifolds embedded in possibly high-dimensional space, the needed network complexity is reduced to only depending on the intrinsic manifold geometry. In experiments, the method demonstrates better performance than previous neural network tests which use the classification accuracy as the test statistic, and can compare favorably to certain kernel maximum mean discrepancy (MMD) tests on synthetic and hand-written digits datasets. We will also discuss limitations of the method and open questions. Joint work with Alexander Cloninger.
12/03/2019 - 4:15pm
Geodesic currents are measures introduced by Bonahon in 1986 that realize a suitable closure of the space of closed curves on a surface. Bonahon proved that intersection number and hyperbolic length for curves extend to geodesic currents. Since then, many other functions defined on the space of curves have been extended to currents, such as negatively curved lengths, lengths from singular flat structures or stable lengths for surface groups. In this talk, we explain how a function defined on the space of curves satisfying some simple conditions can be extended continuously to geodesic currents. The most important of these conditions is that the function decreases under smoothing of essential crossings.
12/03/2019 - 4:15pm
Abstract: We will discuss a mod p version of the geometric Satake isomorphism, which provides a geometrization of the mod p Satake isomorphism. Along the way we will introduce perverse mod p sheaves and the Riemann-Hilbert correspondence of Emerton-Kisin. We will also apply Frobenius splitting techniques to prove the irreducibility of certain perverse mod p sheaves.