Convex optimization for inverse problems

反问题的凸优化

• 批准号: RGPIN-2017-04461
• 负责人: Friedlander, Michael
• 金额: $ 3.64万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=651182
• 关键词: Convex; optimization; inverse; problems

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.

凸优化是数学优化的一个分支，它已成为科学计算的基石。它对当前的一系列科学和工程应用至关重要，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值（例如，估计参数或从有限的数据恢复关于对象的信息）。在最优化的广泛框架内有许多数学问题，包括凸和非凸公式。建议的研究集中在凸优化的三个主要原因（在许多）：**1。在过去的十年里，在发展保证结果准确性和有用性的理论方面取得了巨大的进展。在许多重要的情况下，这些保证是更强的凸配方比非凸配方。一个众所周知的例子是压缩感知;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“转化”，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.对一些凸公式的一个主要批评-特别是那些涉及光谱优化的公式-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。这导致了非凸恢复算法的工作，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问题产生算法，这些算法与目前使用的任何其他方法一样有效和可扩展。3.由于凸性是数学优化大部分领域的基础，因此针对本提案中讨论的这类问题所做的创新可能会阐明优化及其应用的其他领域。*该提案描述了一个广泛的五年研究计划，以开发优化和创新算法的基本理论工具，用于解决实际感兴趣的反问题。该研究将解决算法开发，分析和软件实现中的问题。这些项目非常适合培养学生在数值优化，这是一个高需求的信息技术产业领域。