RI:Small:NSF-BSF: Computational and Statistical Tradeoffs in Inverse Problems using Deep Learning

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In many real life scenarios one may want to measure some information about a certain object, but cannot measure directly, but only indirectly. For example, if a doctor wants to see the lungs of a patient he cannot do it by just looking at them, but needs to use a special machine such as a CT scanner. This scanner emits radiation on the patient and then measures the reflected radiation to gain information about the shape of the lungs. There are many other examples often encountered in photography, telecommunication, or navigation and more that share the same challenge of inferring signals of interest from indirect, noisy measurements: these are known as inverse problems. While many techniques exist for reconstructing the original object from its measurements, one of the main challenges is to do it in a timely manner. This research aims at providing novel computationally efficient techniques for solving inverse problems. In particular, it explores the ability of deep neural networks to accelerate and improve traditional techniques. This research focuses on both the theoretical aspects of such methods, needed for example to certify that the reconstructed image from a patient CT scan using a deep neural network is close to the underlying original scan, and their applications to tomography and seismic imaging. This research provides transformative multidisciplinary educational opportunities, developed in the Center for Data Science, New York University, bringing together signal processing, statistics and machine learning in new graduate level courses. Thanks to the wide range of relevant applications, this research also provides unique outreach activities to K12 and highschool students, as well as minority students through the BSF collaboration.Inverse problems occur in many fields ranging from signal processing to machine learning and are relevant to many domains such as medicine (e.g., getting an image in computer tomography) and physics (e.g., calculating the density of the earth from measurements of its gravity fields). Many solvers have been developed for these type of problems, leveraging specific high-dimensional statistical models for the signals of interest. However, there are three main challenges that persist in current existing solutions. First, many of them are computationally demanding and therefore not applicable to many applications that require a solution in a timely manner. Second, many inverse problems pose non-convex optimization problems that do not have an efficient solution at all. Third, current methodology do not make optimal use of available training examples, which, depending on the application, ranges from tens of instances to millions. This project provides novel theoretical foundations for the neural network based solver on general inverse problems, extends this theory to include non-linear problems that do not admit convex solutions, as well as graph-structured problems, and demonstrates its efficiency on challenging applications including low-dose Computed Tomography, Seismic spike de- convolution, and Quantum State Tomography. It specifically builds on the combination of two recent tools developed by the PI and his Israeli collaborator, which provide complimentary insights on the mechanisms underpinning the neural network acceleration. This novel theoretical framework enables a series of important extensions and generalizations, such as non-linear inverse problems, partially known measuring operators, and distributed optimization. As part of this research, several educational outreach activities are conducted to make neural networks more accessible to the broad student community, including K12, highschool and minority students.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.