FRG: Collaborative Research: Overcomplete Representations with Incomplete Data: Theory, Algorithms, and Signal Processing Applications

在积累的科学结果和最近在稀疏表示方面的突破的推动下，近年来人们对数据不完整的过度扩展越来越感兴趣-这是一个关键的问题，需要统计学家、数学家和工程师之间的密切合作和思想交流。一些指标表明，由这三个社区组成的重点研究小组(FRG)的适当性和及时性是处理这一高潜力但具有挑战性的研究领域的最佳手段。特别是，该项目遵循一个全面和垂直综合的研究计划，用于(1)在过完备的Gabor时频表示和多分辨率小波字典的背景下得出新的统计估计理论结果；(2)利用这些结果来开发针对信号和图像处理中的典型问题的算法，在信号和图像处理中，从业人员经常面临丢失数据或更普遍地不完整的测量；以及(3)解决普遍和重要的交叉应用，包括曲线拟合以及音频和彩色图像增强。为了应对这些迫切的科学需求，并为数学科学的重大发展奠定基础，FRG团队正在利用调和分析和框架理论的最新结果，为过度扩张情况下的统计建模开发一个连贯的框架，包括审查关键的未决问题，例如在贝叶斯框架中选择先验系数分布的影响，以及当潜在预测因素集过度完成时回归的渐近风险界限。作为应对这些重大挑战的决定性的第一步，该团队提出并研究了一种创新的框架系数公共分量模型，该模型将当前使用的方法恢复为特例，但开辟了重要的新发展途径。FRG团队在缺失数据的多分辨率分析、计算贝叶斯推理和自洽方法方面拥有丰富的经验，因此也在开发和应用最先进的程序来实施产生的新算法。专注研究小组(FRG)项目团队结合了来自老牌机构(哈佛大学)和年轻、快速增长的机构(中佛罗里达大学)的科学家。该项目的研究议程将极大地促进对过完整表示法在统计和工程实践中的适用性的理论知识和理解(这是一个新的和重要的数学交叉领域，最近在《纽约时报》、《经济学人》和主流媒体的其他地方报道了许多与“压缩传感”领域有关的重大未决问题)。这最终将导致在必须以非常低的速率(如上文所述的压缩传感系统)收集数据的情况下，或在部分可用数据丢失或严重污染的情况下，开发更有效的信号处理和数据分析算法。后一种情况对于商业应用(例如，蜂窝通信情况下的语音数据)以及军事和国土安全问题(例如，从相关来源恢复未观察到的数据)都特别突出。该项目的另一个好处是它强调数学家、统计学家和工程师通过一个团队进行密切合作，这不仅将导致正在研究的具体问题的解决，而且还将导致新的重要研究领域的形成及其在现实世界中的应用。在国家科学基金会的支持下，该团队培训了一批准备开展数学、统计和工程前沿研究的学生，并定期举办研讨会，以增加新研究人员的参与，并向更广泛的科学界传播成果。

Driven by accumulated scientific results and recent breakthroughs in sparse representations, recent years have seen an ever-increasing interest in overcomplete expansions with incomplete data---a critical subject requiring close cooperation and exchange of ideas amongst statisticians, mathematicians, and engineers. A number of indicators suggest the appropriateness and timeliness of a Focused Research Group (FRG) involving these three communities as the best means to approach to this high-potential yet challenging research area. In particular, this project follows a comprehensive and vertically integrated research plan for (1) deriving new theoretical results for statistical estimation in the context of overcomplete Gabor time-frequency representations and multiresolution wavelet dictionaries; (2) leveraging these results to develop algorithms tailored for canonical problems in signal and image processing, where practitioners are often faced with missing data or more generally incomplete measurements; and (3) addressing ubiquitous and important cross-cutting applications, including curve fitting as well as audio and color image enhancement. To respond to these pressing scientific needs and prepare the ground for significant developments in the mathematical sciences, the FRG team is exploiting recent results from harmonic analysis and the theory of frames to develop a coherent framework for statistical modeling in the case of overcomplete expansions, including an examination of key open questions such as the impact of the choice of prior coefficient distributions in a Bayesian framework and asymptotic risk bounds for regression when the set of potential predictors is overcomplete. As a definitive first step toward these grand challenges, the team proposes and investigates an innovative common-component model for frame coefficients that recovers currently used methods as special cases but opens up important new avenues for advancement. The FRG team has significant prior experience in multiresolution analysis, computational Bayesian inference, and self-consistency methods for missing data, and hence is also developing and applying state-of-the-art procedures to implement the resulting new algorithms.The Focused Research Group (FRG) project team combines scientists from an established institution (Harvard University) and a young, rapidly growing one (University of Central Florida). The project's research agenda is set to substantially advance the theoretical knowledge and understanding of the applicability of overcomplete representations (a new and important cross-cutting area of mathematics, with many major open questions relating to the area of "compressed sensing" recently featured in the New York Times, The Economist, and elsewhere in the mainstream media) in both statistical and engineering practice. This will ultimately lead to development of more efficient algorithms for signal processing and data analysis in situations where data must be collected at a very low rate (as in the compressed sensing regime described above), or when a portion of available data has been lost or highly contaminated. The latter scenario is particularly salient both for commercial applications (e.g., voice data in the case of cellular communications) as well as military and homeland security concerns (for instance, to recover unobserved data from related sources). Another benefit of the project it its emphasis on close collaboration amongst mathematicians, statisticians, and engineers through a single team, which will lead not only to solution of the specific problems under study, but also to formulations of new important areas of research and their application to the real world. Using support from NSF, the team trains a number of students who are ready to carry out research on the cutting edge of mathematics, statistics and engineering, and holds regular workshops to increase the involvement of new researchers and disseminate results to the wider scientific community.