CAREER: Multiscale methods for high-dimensional data, graphs and dynamical systems

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MaggioniDMS-0847388 This award is funded under the American Recovery andReinvestment Act of 2009 (Public Law 111-5). The investigatoraddresses fundamental problems arising from the mathematicalanalysis of multiscale geometries of sets, and multiscaledecompositions of function spaces on such sets. The complexityof the geometry of these sets, which are embedded inhigh-dimensional spaces, are noisy, and often have differentdimensions at different locations, is analyzed at multipleresolutions. He develops robust mathematical and statisticalquantitative multiscale methods of analysis, yielding estimatorsof intrinsic dimensionality that are extremely robust to noiseand succeed with high probability even for small sample size. Hedevelops algorithms for multiscale processing of high-dimensionalpoint clouds much as is done for low-dimensional functions, bymeans of a geometric multiscale transform. He studies andapproximates functions on data by constructing data-drivenmultiscale dictionaries, generalizing approaches familiar inlow-dimensional Euclidean spaces. He develops efficientalgorithms for computing these multiscale decompositions anddictionary expansions. Many important application problems directly require theanalysis of high-dimensional data sets, dynamical systems, orlarge graphs, or they are posed on a framework ofhigh-dimensional data sets, dynamical systems, or large graphs.Such problems arise for instance in different disciplines andreal-world situations where instruments or simulations producelarge amounts of data. The investigator develops novel methodsfor the analysis of high-dimensional data sets and of statisticalmodels with a large number of variables, under the assumptionthat the number of free parameters (which in some senserepresents the intrinsic dimensionality or complexity) of thedata is small compared to that of the ambient space. He furtherapplies these methods to algorithms that are able to learnfunctions on such data from examples, or learn models of complexdynamical systems (e.g. molecular dynamics, protein folding,neuronal activity, etc.), or learn behavior patterns in socialnetworks. This Career award is funded by the Applied Mathematicsprogram in the Division of Mathematical Sciences/Directorate ofMathematical and Physical Sciences and by the Office ofCyberinfrastructure.