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All scientific and social disciplines are faced with an ever-increasing demand to analyze datasets that are unprecedented in scale (amount of data and its dimensionality) as well as degree of corruption (noise, outliers, missing and indirect observations). Extraction of meaningful information from such big and dirty datasets requires achieving the competing goals of computational efficiency and statistical optimality (optimal accuracy for a given amount of data). My research goal is to understand the fundamental tradeoffs between these two quantities, and design algorithms that can learn and leverage inherent structure of data in the form of clusters, graphs, subspaces and manifolds to achieve such tradeoffs.