Collaborative Research: Numerical algebra and statistical inference

研究人员在这个提议中有两个目标，落在数值代数和统计推理的界面上。第一个目标是扩展随机逼近在各种依赖于有监督和无监督以及线性和非线性数值线性代数的降维方法中的使用，并为这些方法开发一个统计基础，以及适用于大量数据的计算动机。另一个动机是利用数值多元线性代数将这些降维的统计方法扩展到多路数据，这是数值分析中的一个新发展。这些项目将增加统计推断和数值分析之间的互动，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。具有统计含义的数值方法是一般人群使用的各种技术的核心。这些技术包括谷歌的网页排名算法、用于发现与疾病相关的遗传变异的遗传方法、压缩医学图像以进行存储和治疗，以及在地质统计学中的应用。在前面的所有案例中，基本思想是将大量数据压缩成一个有用的摘要，以实现期望的目标。本提案中的两个想法是：(1)研究如何将数值方法扩展到现代科学，工程和社会应用中产生的大量数据中，并在数据上施加统计假设或模型；(2)研究比当前方法更复杂的相互作用或数据属性。第一个目标背后的动机是了解当我们收集更多数据时，计算缩放所需的数值近似值如何影响可以从这些数据中提取的信息——对于哪种类型的数据和应用程序，某些数值近似值可以很好地工作。第二个目标背后的动机是超越标准统计方法的广泛范畴，考虑到对象对之间的关系——为b谷歌的网页排名而链接的两个网页，遗传学应用中两个基因或两个位点之间的相关性。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位点之间的链接来提取更丰富的信息来源。这一目标所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.