Harmonic analysis, non-convex optimization, and large data sets

调和分析、非凸优化和大数据集

• 批准号: 1620455
• 负责人: Thomas Strohmer
• 金额: $ 18万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-10-01 至 2019-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620455&HistoricalAwards=false
• 关键词: Harmonic; analysis; non; convex; optimization

Future scientific and technological progress will depend heavily on the generation of new information technology capabilities and novel methods from signal and image processing to deal with today's massive volumes of data. A research effort is proposed to create mathematical concepts and computational methods to address some of the key challenges in this important area. In particular, the PI will focus on the areas of imaging, high-dimensional data analysis, machine learning, and information theory. The project uses tools from computational harmonic analysis, operator theory, random matrix theory, and optimization yielding efficient numerical algorithms with rigorously-established properties under carefully stated conditions. The payoffs for society at large are many, including new information technology capabilities, improved methods for signal- and image processing, as well as better understanding of data mining tools for Big Data.Two concrete topics of this research effort are:(i) Fast and reliable algorithms of non-convex problems: When dealing with massive data sets, many tasks involve the use of a heuristic algorithm to solve a non-convex optimization problem. Often these heuristic algorithms get stuck in local minima, that are far away from the global minimum. We will develop fast numerical algorithms that come with theoretical performance guarantees for a range of important data analysis tasks; (ii) Efficient algorithms for heterogenous and high-dimensional data: Existing methods for high-dimensional data are often computationally rather expensive and rely on stationarity and homogeneity of the data, thus limiting their use for massive, heterogenous data sets. The PI will derive a framework of computationally efficient methods for properly fusing and efficiently processing heterogeneous, high-dimensional data.

未来的科学和技术进步将在很大程度上取决于新信息技术能力以及来自信号和图像处理的新方法，以处理当今的大量数据。提出了一项研究工作，以创建数学概念和计算方法，以应对这一重要领域的一些关键挑战。特别是，PI将专注于成像，高维数据分析，机器学习和信息理论的领域。该项目使用计算谐波分析，运算符理论，随机矩阵理论和优化的工具，在仔细陈述的条件下，具有严格建立的特性，得出有效的数值算法。 The payoffs for society at large are many, including new information technology capabilities, improved methods for signal- and image processing, as well as better understanding of data mining tools for Big Data.Two concrete topics of this research effort are:(i) Fast and reliable algorithms of non-convex problems: When dealing with massive data sets, many tasks involve the use of a heuristic algorithm to solve a non-convex optimization 问题。这些启发式算法通常会陷入与全球最低限度相距甚远的本地最小值。我们将开发具有一系列重要数据分析任务的理论性能保证的快速数值算法； （ii）异源和高维数据的有效算法：现有的高维数据方法通常在计算上相当昂贵，并且依赖数据的平稳性和同质性，从而限制了它们用于大规模，异源数据设置的使用。 PI将得出一个计算有效方法的框架，以正确融合和有效地处理异质，高维数据。