通过保证低秩矩阵估计来利用大数据中的结构
Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation
摘要：
低秩建模在信号处理和机器学习中起着举足轻重的作用，其应用范围从协同滤波、视频监控、医学成像到降维和自适应滤波。许多现代高维数据及其相互作用可以被建模为近似位于低维子空间或流形中，可能具有附加结构，并且其适当的开发导致传感，计算和存储成本的显着降低。近年来，在理解如何以可证明的方式使用计算有效的过程来利用低秩结构方面取得了大量进展，包括凸和非凸方法。一方面，凸松弛，如核范数最小化往往导致统计最优的程序估计低秩矩阵，其中一阶方法的开发，以解决计算的挑战;另一方面，有新的证据表明，适当设计的非凸程序，如投影梯度下降，往往提供全局最优的解决方案，在许多问题的计算成本要低得多。这篇调查文章将提供一个统一的概述，这些最新进展低秩矩阵估计不完全测量。注意这些算法的性能的严格表征，并在低秩矩阵有额外的结构特性，需要新的算法设计和理论分析的问题。
英文摘要：
Low-rank modeling plays a pivotal role in signal processing and machine learning, with applications ranging from collaborative filtering, video surveillance, medical imaging, to dimensionality reduction and adaptive filtering. Many modern high-dimensional data and interactions thereof can be modeled as lying approximately in a low-dimensional subspace or manifold, possibly with additional structures, and its proper exploitations lead to significant reduction of costs in sensing, computation and storage. In recent years, there is a plethora of progress in understanding how to exploit low-rank structures using computationally efficient procedures in a provable manner, including both convex and nonconvex approaches. On one side, convex relaxations such as nuclear norm minimization often lead to statistically optimal procedures for estimating low-rank matrices, where first-order methods are developed to address the computational challenges; on the other side, there is emerging evidence that properly designed nonconvex procedures, such as projected gradient descent, often provide globally optimal solutions with a much lower computational cost in many problems. This survey article will provide a unified overview of these recent advances on low-rank matrix estimation from incomplete measurements. Attention is paid to rigorous characterization of the performance of these algorithms, and to problems where the low-rank matrix have additional structural properties that require new algorithmic designs and theoretical analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
