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

未来的科学和技术进步将在很大程度上取决于新的信息技术能力的产生和信号和图像处理的新方法，以处理今天的海量数据。提出了一项研究努力，以创造数学概念和计算方法，以解决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、机器学习和信息论等领域。该项目使用来自计算调和分析、算子理论、随机矩阵理论和优化的工具，在仔细说明的条件下产生具有严格建立的性质的高效数值算法。这项研究的两个具体主题是：(I)快速可靠的非凸问题算法：在处理海量数据时，许多任务涉及使用启发式算法来解决非凸优化问题。这些启发式算法经常陷入局部最小点，远离全局最小值。我们将开发快速的数值算法，为一系列重要的数据分析任务提供理论上的性能保证；(Ii)针对异质和高维数据的高效算法：现有的高维数据方法通常计算成本较高，并且依赖于数据的平稳性和同质性，因此限制了它们对海量、异质数据集的使用。PI将推导出一个计算高效的方法框架，用于适当融合和有效处理不同种类的高维数据。

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.