CAREER: Numerical Multilinear Algebra and Its Applications - From Matrices to Tensors

PI将扩展数值线性代数的广泛研究到数值多线性代数设置，补充和丰富了底层的线性框架。 PI之前的工作通过以下方式为这个新主题奠定了基础：（1）映射可能与不可能，可计算与不可计算之间的边界;（2）将几个矩阵概念扩展到张量（例如特征值，奇异值，Schatten，Ky Fan范数，Perron-Frobeniuis定理）。这个项目采取了下一步--为这些边界内的问题开发必要的算法。虽然目前用于张量问题的几乎所有“算法”都缺乏正确性和收敛性保证，但为这个项目开发的算法努力成为真正意义上的算法，即满足收敛到真正解的基本要求，而不仅仅是静止或不动点。这通常是不可能的，但PI将（i）识别存在有效的、可证明收敛的算法的大量有趣案例;以及（ii）利用多重线性并挖掘现有丰富的线性技术。这些设计原则和目标将适用于以下所有问题的算法开发：（a）低秩张量近似;（B）张量的特征值和奇异值;（c）精确和最小二乘意义上的多线性方程组。PI还将通过提出电信，生物信息学和神经成像中的几种新工具来展示数值多线性代数的实用性。几乎所有的工程，科学和统计计算问题最终都可能被简化为线性代数中的几个标准问题。因此，有人可能会认为，计算线性代数是计算在科学和工程的主力。这个项目是关于通过从“线性”到“多线性”来扩展这个基本的工具库。要做到这一点，人们必须首先认识到，虽然有上述少数线性问题的自然延伸，也有微妙的困难，甚至不可能与这些多线性类似物。该项目的关键是要开辟出一个实质性的易于处理的问题，但仍然在应用程序中有用的子类。该项目还将研究三个具体的应用，大脑成像，手机通信和分析化学。

The PI will extend extensive studies in numerical linear algebra to the numerical multilinear algebra setting, which complements and enriches the underlying linear framework. The PI's prior work has laid foundations for this new subject via (1) mapping the boundary between the possible and impossible, the computable and non-computable; and (2) extending several matrix notions to tensors (e.g. eigenvalues, singular values, Schatten, Ky Fan norms, Perron-Frobeniuis theorem). This project takes the next step -- developing the requisite algorithms for problems that sit within these boundaries. While almost all 'algorithms' currently in use for tensor problems lack correctness and convergence guarantees, the ones developed for this project strive to be algorithms in the true sense of the word, i.e. meeting the basic requirement of convergence to a true solution and not just a stationary or fixed point. This is in general not possible but the PI will (i) identify large classes of interesting cases for which efficient, provably convergent algorithms exist; and (ii) exploit multilinearity and tap the existing rich collection of linear techniques. These design principles and goals will be applied to algorithmic development of all following problems: (a) Low rank tensor approximations; (b) eigenvalues and singular values for tensors; (c) systems of multilinear equations in the exact and least-squares sense. The PI will also demonstrate the utility of numerical multilinear algebra by proposing several new tools in telecommunications, bioinformatics, and neuroimaging.Almost all engineering, scientific, and statistical computing problems may ultimately be reduced to a handful of standard problems in linear algebra. Therefore one may argue that computational linear algebra is the workhorse of computations in science and engineering. This project is about expanding this fundamental arsenal of tools by going from "linear" to "multilinear". To achieve this, one must first realize that while there are natural extensions of the aforementioned handful of linear problems, there are also subtle difficulties and even impossibilities associated with these multilinear analogues. The crux of the project is to carve out a substantial tractable subclass of problems that are nevertheless still useful in applications. The project would also examine three concrete applications to brain imaging, cellular phone communication, and analytical chemistry.