Generative Modeling based on Deep Neural Networks and Optimal Transport

Predictive Modeling tasks deal with high-dimensional data, and the curse of dimensionality is an obstacle to using some standard approaches. In many applications, real-world data occupy only a small part of high-dimensional observation space whose intrinsic dimension is essentially lower than the dimension of the space. A popular model for such data is a Manifold model under which the data lie on or near an unknown low-dimensional Data manifold (DM) embedded in an ambient high-dimensional space. Predictive Modeling tasks studied under this assumption are the manifold learning problems whose general goal is to discover a low-dimensional structure of high-dimensional manifold valued data from a given dataset. If dataset points are sampled according to an unknown probability measure on the DM, we face statistical problems about manifold valued data. In the talk, we will briefly review statistical problems regarding high-dimensional manifold valued data and their solutions. We plan to discuss some approaches to constructing generative models based on deep neural networks and optimal transport that allow modeling the distribution of data "living" on the manifold.