The recent explosion of large amounts of high dimensional data in science, engineering, and society demands new technologies to recover sparse signals from high dimensional noisy observations. To address it, this project aims to develop efficient and robust methods for analyzing high-dimensional data, which have wide applications in signal processing, communication, computational biology, machine learning, image/video coding, sensor networks, social science, etc.
Recently, compressed sensing (CS) has attracted a good deal of attention from computer science, engineering, and statistics communities. However, the CS recovery only makes sense when the features are weakly correlated. A single rogue outlier may break down the reconstruction completely. The computational procedures cannot meet the challenge of ultrahigh dimensional problems in terms of statistical accuracy, algorithmic stability and computation expediency. To address these challenges, the investigators develop novel nonconvex regularization techniques to attain prediction accuracy and model parsimony for coherent statistical models that go much beyond Gaussianity. Theoretical analysis of its performance in estimation, prediction, and sparsity recovery is conducted. A class of simple algorithms feasible for solving essentially any nonconvex penalized generalized linear models is developed, together with a randomization technique of nonmarginal feature screening for ultra-high dimensional data. Furthermore, the investigators explicitly study the critical effects of outliers and develop a robust CS for handling high leverage points and gross outliers. A unified framework that applies to small-sample-size-high-dimension problems is provided for simultaneous variable selection and outlier identification under Gaussian/non-Gaussian noise. Finally, this project involves rich motivating examples and widespread applications in various areas including spectral analysis, network topology and dynamics modeling, graphical models, computational biology, machine learning, and image compression, as an essential component.
