Manifold Modeling in Machine Learning

Predictive Modeling problems deal with high-dimensional data; however, the curse of dimensionality presents 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 that of the space. A popular model for such data is the manifold model, according to which the data lie on an unknown low-dimensional manifold (Data Manifold) embedded in the ambient high-dimensional space. Predictive modeling problems, which are studied under this assumption, are called manifold estimation problems. The general goal of such problems is to identify the low-dimensional structure of multidimensional data from a given dataset. If dataset points are sampled according to an unknown probability measure on the data manifold, there is a need to model manifolds when solving various machine learning problems. We provide a short survey of such problems and outline some approaches to their solution.