CAREER: Optimizations for Sparse Solutions and Applications

在许多领域，如信号处理、控制、统计、学习、反问题和管理，经常处理“大”数据集以找到“小”解决方案，这些解决方案最终取决于少数因素。由于这些解决方案在某种程度上往往是稀疏的，与通常认为必要的方法相比，挑选稀疏解决方案的方法可以从数量较少的间接测量中找到它们。这是一种新兴的压缩传感技术。在这项研究中，PI建议研究一系列广泛的问题和技术来推进CS。他提出的研究包括引入新的方法来利用解的稀疏性来加速CS计算，开发使用低存储需求的运算并保持对数据中的噪声和错误的鲁棒性的算法，以及发现有效的方法来最小化大类函数的L1范数，例如一阶和高阶差分。该项目将包括一项综合教育计划，包括一门新课程，一名博士生，以及参与莱斯-休斯顿AGEP计划，培养有竞争力的女性和少数族裔研究生。新兴的“压缩传感”技术是对传统数据压缩的补充。虽然传统技术使用更少的比特对数字数据进行编码，以节省存储和传输时间，但新技术可以显著减少与获取数字数据相关的时间、能量和成本。这是通过从比通常需要的次数更少的观测中获取感兴趣对象的数字信息来实现的。例如，由于较低的采样率(因此也就是较低的功率需求)，太空望远镜等飞行器的寿命可以大大延长。高光谱和红外成像设备可以用更小的传感器产生相同的图像，如果使用相同的传感器，则可以产生更高分辨率的图像。因此，这项新技术可以为那些瓶颈在于数据采集成本较高的应用带来突破。

In many areas such as signal processing, control, statistics, learning, inverse problems, and management, "large" data sets are often processed to find "small" solutions, those depending ultimately upon a small number of factors. Since these solutions tend to be sparse in a way, it is possible for methods that pick out the sparse solutions to find them from a reduced number of indirect measurements compared to what are usually considered necessary. This is the emerging technology of compressed sensing (CS). In this research, the PI proposes to study a broad range of issues and techniques to advance CS. His proposed reseach includes the introduction of new methodolgies for exploiting solution sparsity to accelerate CS computation, the development of algorithms that utilize operations requiring low storage and maintain robustness to noise and errors in data, and the discovery of efficient methods for minimizing the l1-norms of wide classes of functions such as first and higher-order differences. This project will include an integrated educational program involving a new course, one Ph.D. student, and the participation in the Rice-Houston AGEP program in producing competitive women and minority graduate students.The new emerging technology of "compressed sensing" is a complement to traditional data compression. While the traditional technology encodes digital data using fewer bits in order to save storage and transmission time, the new technology can significantly reduce the time, energy, and cost associated with the acquisition of digital data. This is achieved by acquiring digit information of an object of interest from a reduced number of obervations than what is usually necessary. For example, the life of an aeriel such as a space telescope can be greatly extended due to a lower sampling rate (and thus a lower power demand). Hyperspectral and infrared imaging devices can produce the same images with smaller sensors, or if with the same sensors, images at higher resolution. As such, the new technology can lead to breakthroughs for applications where the bottleneck lies in the high cost of data acquisition.