Statistical Problems of Manifold Learning for Predictive Modeling

Predictive Modeling tasks deal with high-dimensional data, and curse of dimensionality is an obstacle to the use of many methods for their solutions. In many applications, real-world data occupy only a very small part of high-dimensional observation space whose intrinsic dimension is essentially lower than dimension of the space. Popular model for such data is a Manifold one in accordance with which 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 referred to as the manifold learning ones whose general goal is discovering 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 with statistical problems about manifold valued data. In the talk we will provide a short review of statistical problems regarding high-dimensional manifold valued data and their solutions.