CIF: Medium: Collaborative Research: New Approaches to Robustness in High-Dimensions

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Rapid development of large-scale data collection technology hasignited research into high-dimensional machine learning. Forinstance, the problem of designing recommender systems, such as thoseused by Amazon, Netflix and other on-line companies, involvesanalyzing large matrices that describe users' behavior in pastsituations. In sociology, researchers are interested in fittingnetworks to large-scale data sets, involving hundreds or thousands ofindividuals. In medical imaging, the goal is to reconstructcomplicated phenomena (e.g., brain images; videos of a beating heart)based on a minimal number of incomplete and possibly corruptedmeasurements. Motivated by such applications, the goal of thisresearch is to develop and analyze models and algorithms forextracting relevant structure from such high-dimensional data sets ina robust and scalable fashion.The research leverages tools from convex optimization, signalprocessing, and robust statistics. It consists of three main thrusts:(1) Model restrictiveness: Successful methods for high-dimensionaldata exploit low-dimensional structure; however, many real-worldproblems fall outside the scope of existing models. This proposalsignificantly extends the basic set-up by allowing for multiplestructures, leading to computationally efficient algorithms whileeliminating negative effects of model mismatch. (2) Non-ideal data:Missing data are prevalent in real-world problems, and can cause majorbreakdowns in standard algorithms for high-dimensional data. Thesecond thrust devises relaxations and greedy approaches for thesenon-convex problems. (3) Arbitrary Outliers: Gross errors can arisefor various reasons, including fault-prone sensors and manipulativeagents. The third thrust proposes efficient and randomized algorithmsto address arbitrary outliers.