High-dimensional Density Estimation for Data Mining Tasks

Consider a problem of estimating an unknown high dimensional density whose support lies on unknown low-dimensional data manifold. This problem arises in many data mining tasks, and the paper proposes a new geometrically motivated solution for the problem in manifold learning framework, including an estimation of an unknown support of the density. Firstly, tangent bundle manifold learning problem is solved resulting in transforming high dimensional data into their low-dimensional features and estimating the Riemannian tensor on the Data manifold. After that, an unknown density of the constructed features is estimated with the use of appropriate kernel approach. Finally, with the use of estimated Riemannian tensor, the final estimator of the initial density is constructed.