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

在许多现实生活场景中，人们可能想要测量关于某个对象的某些信息，但不能直接测量，而只能间接测量。例如，如果医生想要查看病人的肺部，他不能只看肺，而需要使用特殊的机器，如CT扫描仪。这种扫描仪向患者发射辐射，然后测量反射的辐射，以获得有关肺部形状的信息。还有许多在摄影、电信或导航等领域经常遇到的其他例子，它们都面临着从间接的、有噪声的测量中推断感兴趣的信号的相同挑战：这些被称为逆问题。虽然有许多技术可以根据测量结果重建原始对象，但主要挑战之一是及时完成这项工作。这项研究旨在为反问题的求解提供新的计算效率的技术。特别是，它探索了深度神经网络加速和改进传统技术的能力。这项研究集中在这些方法的理论方面，例如，需要证明使用深度神经网络的患者CT扫描的重建图像接近于基本的原始扫描，以及它们在层析成像和地震成像中的应用。这项研究提供了变革性的多学科教育机会，由纽约大学数据科学中心开发，将信号处理、统计学和机器学习结合到新的研究生水平课程中。由于相关应用的广泛，这项研究还通过BSF的合作为K12和高中生以及少数族裔学生提供了独特的推广活动。从信号处理到机器学习的许多领域都会出现相反的问题，并与许多领域相关，如医学(例如，在计算机断层扫描中获得图像)和物理学(例如，通过测量地球的重力场来计算地球的密度)。已经为这些类型的问题开发了许多解算器，利用感兴趣信号的特定高维统计模型。然而，在当前的现有解决方案中仍然存在三个主要挑战。首先，它们中的许多都需要计算，因此不适用于许多需要及时解决方案的应用程序。其次，许多反问题提出了根本没有有效解的非凸优化问题。第三，目前的方法没有最大限度地利用现有的训练实例，根据应用的不同，训练实例的数量从几十个到几百万个不等。该项目为基于神经网络的一般反问题求解器提供了新的理论基础，将该理论扩展到包括不允许凸解的非线性问题以及图结构问题，并展示了其在低剂量计算机层析成像、地震尖峰去卷积和量子状态层析成像等挑战性应用中的有效性。它特别建立在PI和他的以色列合作者最近开发的两个工具的组合上，这两个工具提供了对支撑神经网络加速的机制的补充见解。这一新的理论框架允许一系列重要的扩展和推广，如非线性反问题、部分已知的测量算子和分布式优化。作为这项研究的一部分，开展了几项教育推广活动，以使更广泛的学生社区，包括K12，高中和少数族裔学生更容易接触到神经网络。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.