Convex optimization for inverse problems

凸优化是数学优化的一个分支，已成为科学计算的基石。它对于当前的一系列科学和工程应用是必不可少的，包括机器学习、成像、信号处理和控制系统。因此，这一领域的进展有可能对许多科学和工程领域以及相应的工业领域产生直接和广泛的影响。*这些应用中的许多可以被框定为逆问题，其目的是从一组测量中确定关于模型的信息(例如，估计参数或从有限的数据中恢复关于物体的信息)。在优化的广泛框架内有许多数学问题，包括凸公式和非凸公式。建议的研究集中在凸优化上，主要有三个原因：*1.在过去的十年里，在发展保证结果的准确性和有用性的理论方面取得了巨大的进步。在许多重要的情况下，这些保证对于凸公式比对于非凸公式要强得多。一个著名的例子是压缩传感；最近，有证据表明，卷积神经网络--对图像分类和自然语言处理的成功至关重要--可以被“凸化”，以产生具有可验证的统计特性的版本，并使用具有保证结果的训练算法。*2.一些凸公式--尤其是那些涉及频谱优化的公式--的一个主要批评是，它们导致了如此巨大的大数据问题，以至于它们挑战了我们最好的算法。这导致了对非凸恢复算法的研究，这些算法在实践中通常是有效的，但其统计恢复保证可能不是很强。我的动机是需要为困难的凸性问题产生算法，这些算法与目前使用的任何其他方法一样高效和可扩展。*3.由于凸性是数学优化领域的大部分基础，本提案中讨论的这类问题的创新可能会启发其他优化领域及其应用。*本提案描述了一个广泛的五年研究计划，以开发优化的基本理论工具和解决实际感兴趣的逆问题的创新算法。这项研究将解决算法开发、分析和软件实现方面的问题。这些项目非常适合培养学生的数值优化，这是一个信息技术行业的高需求领域。

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.