Exploiting Special Structures in High-Dimensional Data Classification

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ABSTRACTProposed research develops new practical methodology and algorithms aswell as theoretical results for high-dimensional data classification. Themain issue is that when the number of measured variables exceeds by farthe number of observations, estimating the full covariance matrixaccurately is impossible. The investigator has previously shown thatignoring the dependence completely in such a situation is a better option,but it means discarding a lot of information. This problem will beresolved by developing new sparse covariance estimators which will onlyretain important dependence information, and studying their behaviortheoretically in the context of discriminant analysis. New practical,computationally efficient algorithms for computing these estimators fromdata will be developed on the basis of clustering and graph partitioningmethods and compared to existing regularization techniques. Theoretically,asymptotically optimal or near optimal classification performance isexpected to be demonstrated. Another way to reduce the size of theproblem and get to the underlying structure is to reduce the datadimension using recently developed nonlinear manifold projection methods,which aim to discover a nonlinear low-dimensional embedding preservingmost of the information contained in the data. Using these methodsrequires estimating intrinsic data dimension and neighborhood scale on amanifold, and new rigorous estimators for both are proposed, along with ananalysis of their statistical properties. Careful estimation of these twoparameters will improve on the current mostly heuristic methods used inmachine learning and increase applicability of manifold projection methodsfor high-dimensional data classification.This proposal addresses the new challenges posed by the massive amounts ofdata collected in the modern world by developing new theoretical andpractical tools for dealing with high-dimensional data, particularly withthe situation when the number of measurements taken for one observation islarge relative to the number of observations. New sparse estimators ofdependence structure in such data are developed, which only contain theinformation relevant for data classification. The new estimators can alsobe used in any problem where large covariance matrices need to beestimated from limited amount of data, and hence will have an impact on awide range of modern applications, such as classification and analysis ofgene expression data, analysis of complex chemical and physicalexperiments, remote sensing, and medical imaging, among others.