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Research Spotlights
SIAM Review ( IF 10.2 ) Pub Date : 2020-02-14 , DOI: 10.1137/20n974951
Misha E. Kilmer

SIAM Review, Volume 62, Issue 1, Page 131-131, January 2020.
Tensors, or multiway arrays, are often used for storage of high-dimensional data. In order to have compression, completion, or interpretation of such data, the data tensor is factored according to a proposed model consistent with some belief about the data. The first Research Spotlights article, “Generalized Canonical Polyadic Tensor Decomposition," by David Hong, Tamara G. Kolda, and Jed A. Duersch, treats the problem of finding a low-rank tensor decomposition that minimizes a cost functional defined from a componentwise loss function. The difference between the proposed generalized canonical polyadic (GCP) decomposition and the well-known CP decomposition is the flexibility in the choice of loss function, which can be tailored to reflect the appropriate statistical likelihood of a model given the data: low rankness of the tensor is enforced as a constraint. The main result of the paper is in Theorem 3, which addresses the computation of the gradients of a cost function defined in terms of any loss function that is continuously differentiable with respect to its second argument. Their result reveals an elegant expression for the gradients that enables the use of existing kernels for finding CP decompositions to be employed in their algorithm to compute the GCP. The authors use extensive numerical testing, highlighting the flexibility of their algorithm and its ability to efficiently handle missing data while simultaneously illustrating the potential gain to the data analyst in choosing loss functions more consistent with the data. A variety of applications are considered, including analysis of a chat network and explaining factors for mouse neural activity. A “sticky” diffusion process, as defined in the second Research Spotlights article, “Sticky Brownian Motion and Its Numerical Solution," refers to solutions to stochastic differential equations (SDEs) that can spend finite time on a lower-dimensional boundary. Materials like concrete, paint, and toothpaste have something in common: in the limit, the dynamics of the collection of mesoscale particles of which each is comprised approaches a sticky diffusion process. Sticky Brownian motion (SBM), the simplest example of a sticky diffusion process, is the primary focus of the present work. The goals of authors Nawaf Bou-Rabee and Miranda C. Holmes-Cerfon are to call the attention of the greater applied mathematics community to the topic of sticky diffusions as well as to offer a numerical method to allow for simulation of a sticky diffusion with a relatively large time step. These goals are evident in the presentation of material. The mathematical setup of an SBM is given in section 2. Beginning with the Brownian dynamics equations and assumptions on the potential energy function, the dynamics of the probability density are discussed, including asymptotic behavior in the “sticky limit." The authors discuss how to find the generator of the SBM and give some examples. The generator is the basis for the numerical method they develop in section 3, which the authors claim is orders of magnitude faster than alternative methods to simulate a sticky Brownian motion. The authors complement their presentation with a mathematical literature review of some additional methods for characterizing SBM. The final section serves as a well-considered road map for the reader interested in extending this work to a higher-dimensional setting.


中文翻译:

研究热点

SIAM评论,第62卷,第1期,第131-131页,2020年1月。
张量或多路阵列通常用于存储高维数据。为了对这些数据进行压缩,完成或解释,根据与有关数据的某些信念一致的建议模型对数据张量进行分解。David Hong,Tamara G.Kolda和Jed A.Duersch撰写的第一篇Research Spotlights文章“广义规范的多态张量分解”解决了寻找一种低阶张量分解的问题,该张量分解可最大程度地减少由分量损失定义的成本函数建议的广义规范多adadic(GCP)分解与众所周知的CP分解之间的区别是损失函数选择的灵活性,可以对它进行定制,以反映给定数据的模型的适当统计可能性:张量的低秩被强制作为约束。本文的主要结果是在定理3中,该定理解决了根据任何损失函数定义的成本函数梯度的计算,该损失函数相对于其第二个参数可以连续微分。他们的结果揭示了梯度的一种优雅表达,它使得能够使用现有内核来查找将要在其算法中用于计算GCP的CP分解。作者使用了广泛的数值测试,强调了算法的灵活性及其有效处理丢失数据的能力,同时向数据分析师说明了在选择与数据更一致的损失函数时的潜在收益。考虑了多种应用,包括对聊天网络的分析和解释小鼠神经活动的因素。在第二篇“研究热点”文章“粘滞布朗运动及其数值解”中定义的“粘性”扩散过程是指随机微分方程(SDE)的解决方案,该方程可能会在低维边界上花费有限的时间。混凝土,油漆和牙膏有一些共同点:在极限情况下,由中尺度颗粒组成的集合的动力学接近粘性扩散过程,粘性布朗运动(SBM)是粘性扩散过程的最简单例子,是当前工作的重点,作者Nawaf Bou-Rabee和Miranda C的目标是 Holmes-Cerfon将引起更多的应用数学界的注意,以粘性扩散为主题,并提供一种数值方法,以相对较大的时间步长模拟粘性扩散。这些目标在材料介绍中显而易见。SBM的数学设置在第2节中给出。从布朗动力学方程和势能函数的假设开始,讨论了概率密度的动力学,包括“粘性极限”中的渐近行为。找到了SBM的生成器并给出示例,该生成器是他们在第3节中开发的数值方法的基础,作者声称该方法比模拟粘性布朗运动的替代方法要快几个数量级。作者通过对表征SBM的其他一些方法的数学文献综述来补充他们的演讲。最后一节是为有兴趣将这项工作扩展到更高维度的环境的读者精心设计的路线图。
更新日期:2020-02-14
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