CIF: Medium: Collaborative Research: Advances in the Theory and Practice of Low-Rank Matrix Recovery and Modeling

该项目涉及当今科学和工程面临的基本挑战之一，即有效处理和分析大量高维数据，如图像、视频、网页和生物信息学数据。简而言之，现在的数据通常有数千甚至数十亿个维度。一方面，大量数据收集的动机是：1)科学发现；2)需要更好的工程系统。另一方面，现在的困难任务是在如此高的维度上进行有意义的推断，并从有限数量的样本数据和有限的计算资源中得出正确的结论。幸运的是，科学或工程数据通常具有非常低的内在复杂性和维度。该项目解决了这种常见情况所提供的机会，建立了实际可能进行可靠推断的条件，并开发了用于从庞大数据集中提取关键信息的计算工具。该跨学科项目预计将有三个成果：1)研究从部分和损坏信息中恢复数据矩阵所需的创新数学的发展；2)开发用于恢复低秩矩阵和使用损坏数据执行精确降维的有效算法；3)开发新的应用程序，这些技术有望大大推进最新技术的发展。有了这些新工具，科学家和工程师将能够有效地从数据中提取正确的信息，这是以前传统技术无法获得或难以处理的。这将有助于开发更好的计算机视觉系统，用于面部识别，更好的视频序列压缩方案，更好地理解基因表达数据，或者更好地搜索网络文档和图像引擎。

This project concerns one of the fundamental challenges facingcontemporary science and engineering today, namely, the efficientprocessing and analysis of massive amounts of high-dimensional data,such as images, videos, web pages, and bioinformatics data. In short,data now routinely lie in thousands or even billions of dimensions. Onthe one hand, massive data collection is motivated by 1) scientificdiscovery and 2) the need for better engineering systems. On the otherhand, the difficult task now is to conduct meaningful inference insuch high dimensions, and draw correct conclusions from limitedamounts of sample data and with limited computationalresources. Fortunately, scientific or engineering data often have verylow intrinsic complexity and dimensionality. This project addressesthe opportunities offered by this common situation, establishesconditions under which reliable inference is actually possible, anddevelops computational tools for extracting key information from hugedata sets.This interdisciplinary project is expected to have three outcomes: 1)the development of innovative mathematics needed to study the recoveryof data matrices from partial and corrupted information 2) thedevelopment of effective algorithms for recovering low-rank matricesand performing accurate dimensionality reduction with corrupted dataand 3) the development of novel applications in which these techniquesare expected to considerably advance the state-of-the-art. With thesenew tools, scientists and engineers will be able to efficientlyextract correct information from data, which was previouslyinaccessible or intractable by conventional techniques. This willenable the development of far better computer vision systems for facerecognition, better compression schemes of video sequences, a betterunderstanding of gene expression data, or better search engines forweb documents and images.