With the advent of ubiquitous sensing (multi-modal sensors, imaging systems and cameras, etc.), various complex social networks, and the deluge of health-care data (DNA sequences, micro-arrays, etc.), society is now officially in the era of Big Data. In such a setting, the ability to systematically and efficiently derive structured models, and recover reliable and actionable information, from barrages of high dimensional data will have far-reaching impact on engineering challenges and on everyday life. Unfortunately, the data is often noisy, inaccurate, or partially missing. This research will develop a comprehensive theory to assess the performance of a very wide class of algorithms designed for this purpose which are based on convex programming techniques. Such performance guarantees will assist practitioners in a wide array of applications in signal processing, machine learning, statistics and data analysis.
Recent years has witnessed some spectacular theoretical and algorithmic advances in convex optimization and compressed sensing that have changed how large noisy data sets are handled. Despite these successes, key challenges remain, including the need for a comprehensive theory that accurately predicts the performance of the algorithms and goes beyond the customary ?order-wise? performance guarantees. The investigators will pursue an ambitious research program to give exact performance evaluations for a wide variety of convex-optimization-based signal recovery methods, including the classical LASSO and its variants. The framework can deal with a wide array of signal-to-noise ratios, different measurement matrix ensembles, and a variety of cost functions and signal structures. The techniques draw upon a host of ideas in high-dimensional geometry, statistics, and signal processing and are the culmination of a flurry of activity by several different research communities.
