CIF: Medium: Collaborative Research: New Approaches to Robustness in High-Dimensions

大规模数据收集技术的快速发展引发了对高维机器学习的研究。例如，设计推荐系统的问题，比如亚马逊、Netflix和其他在线公司使用的推荐系统，涉及分析描述用户在过去情况下行为的大型矩阵。在社会学中，研究人员感兴趣的是将网络拟合到涉及数百或数千个人的大规模数据集上。在医学成像中，目标是基于最少数量的不完整和可能损坏的测量来重建复杂的现象（例如，大脑图像；心脏跳动的视频）。在这些应用的激励下，本研究的目标是开发和分析模型和算法，以鲁棒和可扩展的方式从这些高维数据集中提取相关结构。该研究利用了凸优化，信号处理和鲁棒统计的工具。它包括三个主要的推力：(1)模型限制性：高维数据的成功方法利用低维结构；然而，许多现实世界的问题超出了现有模型的范围。该提议通过允许多个结构显著扩展了基本设置，导致计算效率高的算法，同时消除了模型不匹配的负面影响。(2)非理想数据：缺失数据在现实问题中普遍存在，并且可能导致高维数据的标准算法出现重大故障。第二个推力设计了非凸问题的松弛和贪心方法。(3)任意异常值：由于各种原因，包括容易出错的传感器和操纵性代理，可能会产生严重误差。第三个要点提出了有效的随机算法来处理任意异常值。

Rapid development of large-scale data collection technology hasignited research into high-dimensional machine learning. Forinstance, the problem of designing recommender systems, such as thoseused by Amazon, Netflix and other on-line companies, involvesanalyzing large matrices that describe users' behavior in pastsituations. In sociology, researchers are interested in fittingnetworks to large-scale data sets, involving hundreds or thousands ofindividuals. In medical imaging, the goal is to reconstructcomplicated phenomena (e.g., brain images; videos of a beating heart)based on on a minimal number of incomplete and possibly corruptedmeasurements. Motivated by such applications, the goal of thisresearch is to develop and analyze models and algorithms forextracting relevant structure from such high-dimensional data sets ina robust and scalable fashion.The research leverages tools from convex optimization, signalprocessing, and robust statistics. It consists of three main thrusts:(1) Model restrictiveness: Successful methods for high-dimensionaldata exploit low-dimensional structure; however, many real-worldproblems fall outside the scope of existing models. This proposalsignificantly extends the basic set-up by allowing for multiplestructures, leading to computationally efficient algorithms whileeliminating negative effects of model mismatch. (2) Non-ideal data:Missing data are prevalent in real-world problems, and can cause majorbreakdowns in standard algorithms for high-dimensional data. Thesecond thrust devises relaxations and greedy approaches for thesenon-convex problems. (3) Arbitrary Outliers: Gross errors can arisefor various reasons, including fault-prone sensors and manipulativeagents. The third thrust proposes efficient and randomized algorithmsto address arbitrary outliers.