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，即高效地处理和分析海量高维数据，如图像、视频、网页和生物信息学数据。简而言之，数据现在通常存在于数千甚至数十亿的维度中。一方面，大规模数据收集的动机是1)科学发现和2)对更好的工程系统的需求。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理，并从有限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。这个跨学科的项目预计将有三个结果：1)研究从部分和损坏的信息中恢复数据矩阵所需的创新数学；2)开发有效的算法来恢复低阶矩阵并对损坏的数据执行精确的降维；3)开发新的应用程序，其中这些技术有望极大地推进最先进的技术。有了这些新的工具，科学家和工程师将能够有效地从数据中提取正确的信息，而这些信息以前是传统技术无法访问或难以处理的。这将使更好的人脸识别计算机视觉系统的开发，视频序列的更好的压缩方案，对基因表达数据的更好的理解，或者更好的网络文档和图像的搜索引擎。

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.