CIF: Small: Structured Signal Recovery from Noisy Measurements via Convex Programming: A Framework for Analyzing Performance

随着无处不在的传感（多模态传感器、成像系统和摄像头等）、各种复杂的社交网络和海量的医疗数据（DNA序列、微阵列等）的出现，社会现在正式进入了大数据时代。在这样的环境下，系统有效地推导结构化模型，并从大量高维数据中恢复可靠和可操作的信息的能力将对工程挑战和日常生活产生深远的影响。不幸的是，这些数据通常是嘈杂的、不准确的或部分缺失的。这项研究将发展一个全面的理论，以评估基于凸规划技术的为此目的而设计的非常广泛的算法的性能。这种性能保证将在信号处理、机器学习、统计和数据分析等广泛应用中帮助从业者。近年来，在凸优化和压缩感知方面取得了一些惊人的理论和算法进展，这些进展改变了处理大型噪声数据集的方式。尽管取得了这些成功，但关键的挑战仍然存在，包括需要一个全面的理论来准确预测算法的性能，并超越传统的“顺序”。性能保证。研究人员将进行一项雄心勃勃的研究计划，为各种基于凸优化的信号恢复方法（包括经典LASSO及其变体）提供准确的性能评估。该框架可以处理各种各样的信噪比、不同的测量矩阵集成以及各种成本函数和信号结构。这些技术借鉴了高维几何、统计学和信号处理方面的大量思想，是几个不同研究团体一系列活动的成果。

With the advent of ubiquitous sensing (multi-modal sensors, imaging systems and cameras, etc.), various complex social networks, and the deluge of health-care data (DNA sequences, micro-arrays, etc.), society is now officially in the era of Big Data. In such a setting, the ability to systematically and efficiently derive structured models, and recover reliable and actionable information, from barrages of high dimensional data will have far-reaching impact on engineering challenges and on everyday life. Unfortunately, the data is often noisy, inaccurate, or partially missing. This research will develop a comprehensive theory to assess the performance of a very wide class of algorithms designed for this purpose which are based on convex programming techniques. Such performance guarantees will assist practitioners in a wide array of applications in signal processing, machine learning, statistics and data analysis.Recent years has witnessed some spectacular theoretical and algorithmic advances in convex optimization and compressed sensing that have changed how large noisy data sets are handled. Despite these successes, key challenges remain, including the need for a comprehensive theory that accurately predicts the performance of the algorithms and goes beyond the customary ?order-wise? performance guarantees. The investigators will pursue an ambitious research program to give exact performance evaluations for a wide variety of convex-optimization-based signal recovery methods, including the classical LASSO and its variants. The framework can deal with a wide array of signal-to-noise ratios, different measurement matrix ensembles, and a variety of cost functions and signal structures. The techniques draw upon a host of ideas in high-dimensional geometry, statistics, and signal processing and are the culmination of a flurry of activity by several different research communities.