The project extends the investigator's work on localized approximation on the torus, unit interval, and sphere to the case of data-dependent manifolds. Much of the current work on data-dependent manifolds is focussed on gaining an understanding of the geometry underlying the data. The investigator has started to develop tools that don't rely on geometric structure to approximate functions defined on data-dependent manifolds. This project brings the theory for data-dependent manifolds to the level of completion achieved in the case of the known manifolds. The approximations are defined universally, and can be based on either spectral or nonlocal, scattered data, i.e., data where one has no control on the location of the points where the target function is sampled. The error in this globally defined approximation adjusts itself optimally at different points on the manifold according to the smoothness of the target function in the vicinity of those points. In particular, the project includes (1) the development of approximation theory on certain graphs, (2) a complete characterization of the approximation properties of analogues of radial basis function networks on the manifolds in terms of the smoothness of the target functions, and (3) a study of the local approximation properties of spectral and pseudo-spectral methods to solve pseudo-differential equations on the manifold. An integral part of this research is to develop efficient algorithms to implement and test the theory.
Many modern applications involve answering queries based on unstructured data sets. For example, based on a data set consisting of hand-written digits, scanned as images, one wants to determine which digit a new unseen image represents, if any. In the past few years, new and powerful tools have been proposed in order to impose a geometrical structure on such data sets. In this project the investigator aims to go beyond geometrical considerations, developing a theory that focusses on modeling of specific queries as function approximation. The theory helps to develop efficient algorithms and test them. Potential areas of applications include crystallography, geophysics, biomathematics, semi-supervised learning, document analysis, face recognition, hyperspectral image processing, and cataloguing of galaxies.
