AF: Medium: Dropping Convexity: New Algorithms, Statistical Guarantees and Scalable Software for Non-convex Matrix Estimation

来自相机的图像是一个数字矩阵，但大多数数字矩阵看起来并不像图像--图像中的数字矩阵反映了场景的结构。数据分析在科学、工程和商业领域的许多应用可以看作是获取观测矩阵，并拟合低等级或其他结构的矩阵来解释它们之间的关系。图像和视频分析并不是唯一的例子；这个问题出现在社交网络的结构分析、预测用户对新产品和服务的偏好以及许多其他分析任务中。随着这些问题的规模和维度的增加，数据分析师面临着严格和规模之间的差距：从理论上讲，健全的算法通常具有只有在中等规模的数据集上才可行的要求(例如，重复/随机访问数据)，即使这样，也可能无法在“交互时间”(即人类交互分析数据所需的较小时间尺度)内提供答案。因此，实践转向了缺乏严格保证的方法，但这些方法是可扩展的，并已被观察到提供了像样的近似值。该项目旨在通过两项技术意见缩小这一差距：(A)认识到快速矩阵推理需要非凸算法，它侧重于开发对非凸算法的严格分析，以及(B)在算法设计和统计分析阶段本身明确纳入大数据体系结构(核外和分布式多核)。它集中于几个具体的任务，包括通过有效的低阶逼近，最小化低阶矩阵的非凸集上的一般凸函数，稳健的矩阵估计，以及非线性和核矩阵的设置。该项目对研究生进行数学和计算开发方面的培训，这对数据分析非常重要。大数据的承诺只能通过使用数据扩展基础设施以继续提供统计上有意义的见解来实现；该项目旨在为一大套矩阵估计问题实现这一承诺。

An image from your camera is a matrix of numbers, but most matrices of numbers would not look like an image -- the matrix of numbers in an image reflect structure from the scene. Many applications of data analysis across science, engineering, and business can be viewed as taking a matrix of observations and fitting low-rank or otherwise structured matrices to explain their relationships. Image and video analysis is not the only example; the problem arises in structural analysis of social networks, divining user preferences for new products and services, and many other analysis tasks. As the scale and dimensionality of these problems increases, the data analyst is faced with a gap between rigor and scale: theoretically sound algorithms often have requirements (e.g. repeated/random access to data) that are feasible only on medium-scale datasets, and even then may not provide answers in "interactive time" (i.e. smallish time scales required for a human interactively analyzing data). Thus practice has turned towards methods that lack rigorous guarantees, but that are scalable and have been observed to provide decent approximation. This project aims to narrow this gap by two technical observations: (a) Recognizing that fast matrix inference necessitates non-convex algorithms, it focuses on developing a rigorous analysis of the same, and (b) by explicitly incorporating big-data architectures (out of core, and distributed multicore) in the algorithm design and statistical analysis stage itself. it focuses on several specific tasks, including pass-efficient low-rank approximation, minimizing general convex functions over the non-convex set of low-rank matrices, robust matrix estimation, and non-linear and kernel matrix settings. The project trains graduate students in the mathematical and computational development important for data analysis. The promise of big data can only be realized by scaling infrastructure with data to continue to provide statistically meaningful insights; this project aims to realize this promise for a large suite of matrix estimation problems.