CAREER: Nonlinear Models and Regularization for Infinite-Dimensional Inverse Problems

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In recent years, numerous data-driven applications have produced significant improvements in the quality of life across society. The resulting deluge of big data has given rise to computational and algorithmic challenges that are not addressed by traditional statistical paradigms. Data science is now providing new perspectives on how to tackle these challenges, particularly with respect to model-based inference and data acquisition. Yet, in many applications, there exists a nontrivial gap between the mathematical modeling of a physical phenomenon and the model approximation used to facilitate computations. This research project seeks to narrow this gap through a disciplined approach that combines new signal models and new optimization problem formulations that would lead to improved numerical algorithms. The project is expected to have an impact on many applications in signal processing, imaging science, and statistics. The principal investigator will mentor students at all levels through various outreach activities, and will proactively encourage participation from underrepresented groups.This research addresses fundamental questions in important data science applications which are described by infinite-dimensional models, such as in super-resolution imaging and non-parametric density estimation. In the first phase, a sampling theory will be developed together with provably robust and efficient algorithms for a class of piecewise polynomials, such a framework being sufficiently flexible to cover a variety of practical applications. Learning this model from limited observations will be formulated as a regularized optimization problem; its non-asymptotic theory will be established by leveraging insights from geometric functional analysis, high-dimensional probability, and convex optimization. In the second phase, by leveraging these piecewise polynomial models, an optimization theory will be established to solve a set of selected infinite-dimensional inverse problems without incurring the distortion traditionaly due to discretization. The effectiveness of the developed methods will be demonstrated over a set of imaging data measurements.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.