Convex optimization for inverse problems

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Convex optimization is a branch of mathematical optimization that has become a cornerstone of scientific computing. It is essential to a range of current scientific and engineering applications, including machine learning, imaging, signal processing, and control systems. Advances in this area thus have the potential for immediate and wide-ranging impact across many areas of science and engineering as well as corresponding fields of industry.******Many of these applications can be framed as inverse problems, whose aim is to determine information about a model from a set of measurements (e.g., estimating parameters or recovering information about an object from limited data). There are many mathematical problems within the broad framework of optimization, including convex and nonconvex formulations. The proposed research focuses on convex optimization for three principal reasons (among many):******1. There has been tremendous progress over the last decade in developing theory that guarantees accuracy and usefulness of results. In many important cases, these guarantees are much stronger for convex formulations than for nonconvex formulations. A well-known example is compressed sensing; more recently, there is evidence that convolutional neural networks--crucial to the success of image classification and natural language processing--can be "convexified" to yield versions that have verifiable statistical properties and use training algorithms with guaranteed outcomes.******2. A major criticism of some convex formulations--notably those that involve spectral optimization--is that they lead to big-data problems so huge that they challenge our very best algorithms. This has led to work on nonconvex recovery algorithms that are often effective in practice, but whose statistical recovery guarantees may not be very strong. I am motivated by the need to produce algorithms for difficult convex problems that are both as efficient and as scalable as any other approach currently being used.******3. Because convexity underpins much of the field of mathematical optimization, innovations made for the class of problems discussed in this proposal are likely to illuminate other areas of optimization and its applications.******This proposal describes a broad five-year research program to develop fundamental theoretical tools in optimization and innovative algorithms for solving inverse problems of practical interest. The research will address issues in algorithm development, analysis, and software implementation. These projects are well suited for training students in numerical optimization, which is an area of high demand in the information technology industry.