CIF: Small: Collaborative Research: Compressed Sensing for Coherent Designs under Gaussian/Non-Gaussian Noise

近年来，科学、工程和社会中大量高维数据的爆炸式增长需要新的技术来从高维噪声观测中恢复稀疏信号。为了解决这个问题，本项目旨在开发高效且稳健的高维数据分析方法，这些方法在信号处理、通信、计算生物学、机器学习、图像/视频编码、传感器网络、社会科学等领域有广泛的应用。最近，压缩感知（CS）引起了计算机科学、工程和统计界的广泛关注。然而，只有当特征弱相关时，CS恢复才有意义。一个单独的异常值可能会彻底破坏重建。计算程序在统计精度、算法稳定性和计算方便性等方面无法满足超高维问题的挑战。为了解决这些挑战，研究人员开发了新的非凸正则化技术，以获得预测精度和模型简约性的相干统计模型，远远超出高斯性。对其在估计、预测和稀疏度恢复方面的性能进行了理论分析。提出了一种简单的算法，可用于求解任何非凸惩罚广义线性模型，以及一种超高维数据的非边缘特征筛选的随机化技术。此外，研究人员明确研究了异常值的关键影响，并开发了一个稳健的CS来处理高杠杆点和总异常值。为同时进行高斯/非高斯噪声下的变量选择和离群值识别提供了一个适用于小样本高维问题的统一框架。最后，该项目涉及丰富的激励示例和广泛的应用在各个领域，包括频谱分析，网络拓扑和动力学建模，图形模型，计算生物学，机器学习和图像压缩，作为一个重要组成部分。

The recent explosion of large amounts of high dimensional data in science, engineering, and society demands new technologies to recover sparse signals from high dimensional noisy observations. To address it, this project aims to develop efficient and robust methods for analyzing high-dimensional data, which have wide applications in signal processing, communication, computational biology, machine learning, image/video coding, sensor networks, social science, etc.Recently, compressed sensing (CS) has attracted a good deal of attention from computer science, engineering, and statistics communities. However, the CS recovery only makes sense when the features are weakly correlated. A single rogue outlier may break down the reconstruction completely. The computational procedures cannot meet the challenge of ultrahigh dimensional problems in terms of statistical accuracy, algorithmic stability and computation expediency. To address these challenges, the investigators develop novel nonconvex regularization techniques to attain prediction accuracy and model parsimony for coherent statistical models that go much beyond Gaussianity. Theoretical analysis of its performance in estimation, prediction, and sparsity recovery is conducted. A class of simple algorithms feasible for solving essentially any nonconvex penalized generalized linear models is developed, together with a randomization technique of nonmarginal feature screening for ultra-high dimensional data. Furthermore, the investigators explicitly study the critical effects of outliers and develop a robust CS for handling high leverage points and gross outliers. A unified framework that applies to small-sample-size-high-dimension problems is provided for simultaneous variable selection and outlier identification under Gaussian/non-Gaussian noise. Finally, this project involves rich motivating examples and widespread applications in various areas including spectral analysis, network topology and dynamics modeling, graphical models, computational biology, machine learning, and image compression, as an essential component.