Numerical Multilinear Algebra in Signal Processing and Environmetrics

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI和她的合作者将开发基于张量的数值方法，主要应用于信号处理和测量，其次是图像处理，数据挖掘和科学计算。在过去，这些学科受益于数值线性代数的进步。类似地，数值多线性代数的目标是找到一个给定张量到简单张量的和的有用分解，例如简单的张量积。这导致有效的数据压缩。通过使用一种新的张量范数，称为张量的迹类范数，可以大大改善信号的去噪、压缩和重建。由于压缩感知的应用，该范数在矩阵秩最小化中引起了广泛的关注。此外，PI将开发新的正则化方法用于张量分解。张量应用中的许多问题都是不适定的，例如在存在噪声的情况下重建多通道信号。 新的数值稳定的方法将在逆问题和不适定问题的框架内进行研究。数学可应用于计量学。 环保署和其他机构使用空气质量模型来决定如何监管排放。这些模型包括受体模型、源解析和污染识别。PI和她的合作者将致力于新的计算方法，以提供更准确的环境数据估计和分析。基于张量的信号处理在生物医学成像中也有许多应用，例如磁共振成像诊断和肿瘤检测中的核磁共振光谱。新的高阶张量分解将促进更好的数据分析工具。该研究项目的一个关键组成部分是与其他科学家以及学生的合作。通过本科生的暑期研究计划，以及研究生的列入，他们将培养新一代的应用数学家在张量为基础的分析方法。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI and her collaborators will develop tensor-based numerical methods for applications primarily to signal processing and environmetrics, and secondarily to image processing, data mining and scientific computing. In the past, these subjects benefited from advances in numerical linear algebra. In analogy, numerical multilinear algebra aims to find useful decompositions of a given tensor into sums of simpler tensors, such as simple tensor products. This leads to efficient data compression. By using a new tensor norm called the trace-class norm for tensors, one can substantially improve de-noising, compression, and reconstruction of signals. This norm has attracted much attention in matrix rank minimization because of compressed sensing applications. Also, the PI will develop new regularization methods for tensor decomposition. Many of the problems in tensor applications are ill-posed, such as reconstruction of multi-channel signals in the presence of noise. New numerical stable methods will be studied in the framework of inverse and ill-posed problems. Mathematics has applications to environmetrics. The EPA and other agencies use air quality models, to decide how to regulate emissions. These models include receptor modeling, source apportionment and pollution identification. The PI and her collaborators will work on new computations methods to provide more accurate estimation and analysis of environmental data. Tensor-based signal processing also has many applications in biomedical imaging, such as magnetic resonance imaging diagnosis and nuclear magnetic resonance spectroscopy in tumor detection. Better data analysis tools will be facilitated by new higher-order tensor decompositions. A key component of the research project is collaboration with other scientists as well as students. Through summer research programs for undergraduates, as well as inclusion of graduate students, they will train a new generation of applied mathematicians in the methods of tensor based analysis.