Collaborative Research: Numerical algebra and statistical inference

研究人员在这项提议中有两个目标，分别是数值代数和统计推理的界面。第一个目标是扩展随机逼近在各种依赖于数值线性代数的降维方法中的使用，这些方法依赖于有监督和无监督以及线性和非线性的降维方法，并且除了适用于海量数据的计算动机之外，还为这些方法发展统计基础。另一个动机是使用数值多线性代数将这些用于降维的统计方法扩展到多路数据，这是数值分析中的新发展。这些项目将增加统计推断和数字分析之间的互动，并使这两个领域受益，为我们如何看待和执行数据分析提供新的视角。这些技术包括谷歌的PageRank算法、用于发现与疾病相关的遗传变异的遗传方法、用于存储和治疗的医学图像压缩，以及在地质统计学中的应用。在前面的所有案例中，基本思想都是将海量数据浓缩成关于所需目标的有用摘要。这项建议中的两个想法是(1)研究在现代科学、工程和社会应用中产生的海量数据的数值方法如何对数据施加统计假设或模型，(2)研究数据的更复杂的相互作用或属性，而不是用当前方法检查。第一个目标背后的动机是理解随着我们收集更多的数据，计算缩放所需的数值近似如何影响可以从这些数据中提取的信息--对于哪些类型的数据和应用程序，某些数值近似工作得很好。第二个目标背后的动机是超越标准统计方法的广泛类别，考虑对象对之间的关系--为谷歌的PageRank链接的两个网页，两个基因或两个基因座之间的相关性在遗传学应用中。这一目标背后的问题是，是否可以通过检查三个网页或三个地点之间的链接来提取更丰富的信息来源。这一目标涉及的研究包括开发计算效率高的代数方法来提取这些信息，并理解这些方法实现的统计模型。

The investigators have two aims in this proposal that fall at the interface of numerical algebra and statistical inference. The first aim is to extend the use of randomized approximation in a variety of dimension reduction methods that rely on numerical linear algebra both supervised and unsupervised as well as linear and nonlinear and develop a statistical bases for these methods in addition to the computational motivation of being applicable to massive data. The other motivation is to extend these statistical methods for dimension reduction to multiway data using numerical multilinear algebra, a recent new development in numerical analysis. These projects will increase interaction between statistical inference and numerical analysis and benefit both fields, providing new perspectives to how we view and perform data analysis.Numerical methods with statistical implications are central to a variety of technologies used by the general population. These technologies include Google's pagerank algorithm, genetic methods used to find genetic variation related to disease, compressing of medical images for storage and treatment, as well as applications in geostatistics. In all the previous cases the fundamental idea is to condense massive data in a useful summary with respect to a desired goal. The two ideas in this proposal are (1) to study how numerical methods that scale to the massive data generated in modern scientific, engineering, and social applications impose statistical assumptions or models on the data, (2) to study more complex interactions or properties of the data than examined in current methods. The motivation behind the first aim is to understand how numerical approximations required for computational scaling as we collect more data impact the information that can be extracted from these data -- for what type of data and applications do certain numerical approximations work well. The motivation behind the second aim is to go beyond the broad category of standard statistical methods take into account the relation between pairs of objects -- two web pages that are linked for Google's pagerank, the correlation between two genes or two loci in genetics applications. The question behind this aim is whether richer sources of information can be extracted by examining the links between three web pages or three loci. The research involved in this aim consists of the development of computationally efficient algebraic methods to extract this information and understanding the statistical models implemented by these methods.