CAREER: Learning from Coarse, Nonmetric, and Incomplete Data

近年来，我们目睹了在各种各样的环境中获取和分析的数据量的爆炸式增长。数据驱动技术不仅在传统的定量科学中得到越来越多的应用，而且在整个社会科学和各种其他非传统场景中也得到越来越多的应用，这些场景挑战了我们的许多常见假设。例如，在协作过滤、个性化和预测医学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中，我们希望了解人们如何使用人们提供的数据。这带来了一些困难，包括这样的数据通常非常“粗糙”或严重“量化”的事实。它甚至可能是由类别或比较组成的二进制或完全非度量的数据。此外，在许多情况下，不可能对数据进行完全采样，并且感兴趣的底层数据可能不断变化，因此需要能够处理不完整观察和动态数据模型的方法。本研究通过建立在利用低维结构进行推理的有效算法设计的最新进展来面对这些困难，通常使用高度不完整和粗糙的观测。本研究在低秩矩阵恢复、非度量多维尺度、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。它在协同过滤、个性化和预测医学以及个性化学习系统等环境中都有应用。

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.