CAREER: Learning from Coarse, Nonmetric, and Incomplete Data

近年来，我们目睹了在各种背景下获取和分析的数据量的爆炸性增长。数据驱动技术的应用越来越多，不仅在传统的量化科学中，而且在整个社会科学和各种其他非传统情景中，这些非传统情景挑战了我们的许多常见假设。例如，在协作过滤、个性化和预测性医疗以及个性化学习系统等环境中，我们面临着各种各样的挑战，这主要是因为一个重要的--通常是唯一的--数据来源是人。在这些和许多其他现代应用程序中，我们希望了解人们使用人们提供的数据的情况。这带来了几个困难，包括这样的数据通常非常“粗糙”或严重“量化”。它甚至可能是由类别或比较组成的二进制或完全非度量数据。此外，在其中许多情况下，不可能对数据进行完全抽样，而且感兴趣的基础数据可能不断变化，这就需要能够处理不完整观测和动态数据模型的方法。这项研究面临这些困难，因为它建立在利用低维结构执行推理的高效算法设计的最新进展的基础上，通常使用高度不完整和粗糙的观测。这项研究在低阶矩阵恢复、非度量多维缩放、展开和低维动态模型的背景下解决了一些基本的理论和算法问题。它在协作过滤、个性化和预测性医学以及个性化学习系统等领域都有应用。

In recent years, we have witnessed an explosion in the amounts of data being acquired and analyzed in a wide variety of contexts. Data-driven techniques are increasingly applied, not only in the traditional quantitative sciences, but also throughout the social sciences and in a variety of other non-traditional scenarios that challenge many of our common assumptions. For example, in contexts such as collaborative filtering, personalized and predictive medicine, and personalized learning systems, we face a variety of challenges due largely to the fact that an important -- often the only -- source of data is people. In these and many other modern applications, we want to learn about people using the data that people supply.This presents several difficulties, including the fact that such data is often very "coarse" or heavily "quantized". It might even be binary or entirely nonmetric data consisting of categories or comparisons. Moreover, in many of these cases it is impossible to fully sample the data, and the underlying data of interest may be constantly changing, necessitating approaches that can handle incomplete observations and dynamic data models. This research confronts these difficulties by building on recent progress in the design of efficient algorithms for exploiting low-dimensional structure to perform inference, often using highly incomplete and coarse observations. This research addresses a number of fundamental theoretical and algorithmic questions in the context of low-rank matrix recovery, nonmetric multidimensional scaling, unfolding, and low-dimensional dynamic models. It has applications in contexts such as collaborative filtering, personalized and predictive medicine, and personalized learning systems.