RUI: Multiscale and Modeling of Scattered Data

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In many practical applications, one needs to study modeling and analysis of data based on smooth manifolds such as a sphere, and more generally, metric measure spaces. Examples include document analysis, face recognition, semi-supervised learning, image processing, cataloguing of galaxies, pattern analysis of brain potentials, and the study of brain tumors. The proposer will continue his work on approximation by analogues of neural and radial basis function (RBF) networks, sometimes redefined to take advantage of the geometry of the data, for modeling such data. He will also continue his work on multiscales for the analysis of the data. He will develop theoretical results as well as efficient algorithms based on these theories. The findings of the research will be disseminated, as usual, through articles in refereed journals and conference proceedings, as well as presentations in colloquia and conferences.While classical techniques from approximation theory require a judicious choice of the sites where the data is collected, many practical applications do not allow such a choice. One of the novelties of the research is to deal with data collected at arbitrary sites. Another novelty is to utilize global data, such as coefficients in an orthogonal expansion, to study local features of the functional relationship underlying the data.