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以及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，机器学习和信息论领域。该项目使用了计算谐波分析、算子理论、随机矩阵理论和优化等工具，在精心设定的条件下，生成了具有严格建立的性质的高效数值算法。对整个社会的回报是多方面的，包括新的信息技术能力，改进的信号和图像处理方法，以及更好地理解用于大数据的数据挖掘工具。本研究工作的两个具体主题是：(i)快速可靠的非凸问题算法：当处理大量数据集时，许多任务涉及使用启发式算法来解决非凸优化问题。通常这些启发式算法会陷入局部最小值，离全局最小值很远。我们将为一系列重要的数据分析任务开发具有理论性能保证的快速数值算法；（ii）异构和高维数据的高效算法：现有的高维数据处理方法在计算上往往相当昂贵，而且依赖于数据的平稳性和同质性，因此限制了它们对大量异构数据集的使用。PI将推导出一个计算有效方法的框架，用于适当融合和有效处理异构的高维数据。

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.