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

在不断积累的科学成果和最近在稀疏表示方面的突破的推动下，近年来人们对不完整数据的过完备扩展越来越感兴趣--这是一个需要统计学家，数学家和工程师之间密切合作和交流思想的关键主题。 一些指标表明，适当和及时的重点研究小组（FRG）涉及这三个社区的最佳手段，以接近这一高潜力，但具有挑战性的研究领域。 特别是，这个项目遵循一个全面的和纵向一体化的研究计划：（1）在过完备的Gabor时频表示和多分辨率小波字典的背景下，导出统计估计的新理论结果;（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定制的算法，其中从业者经常面临缺失的数据或更一般地不完整的测量;以及（3）解决普遍存在的和重要的交叉应用，包括曲线拟合以及音频和彩色图像增强。为了满足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，为过完备扩展情况下的统计建模开发一个连贯的框架，包括对关键开放问题的审查，例如在贝叶斯框架中选择先验系数分布的影响，以及渐近当潜在的预测因子集是过完备的时，回归的风险界限。 作为迈向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殊情况，但开辟了重要的新途径。 FRG团队在多分辨率分析、计算贝叶斯推理和缺失数据的自洽方法方面有着丰富的经验，因此也在开发和应用最先进的程序来实现由此产生的新算法。FRG项目团队由一个成熟的机构（哈佛大学）和一个年轻的、快速发展的机构（中央佛罗里达大学）的科学家组成。 该项目的研究议程将大大推进理论知识和理解的适用性过完备表示（数学的一个新的和重要的交叉领域，与许多重大的开放问题有关的“压缩传感”领域最近在纽约时报，经济学人，和其他地方的主流媒体）在统计和工程实践。这将最终导致在必须以非常低的速率收集数据的情况下（如在上述压缩感测机制中），或者当一部分可用数据已经丢失或高度污染时，开发用于信号处理和数据分析的更有效的算法。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考虑（例如，从相关源恢复未观察到的数据）。该项目的另一个好处是强调数学家、统计学家和工程师通过一个团队进行密切合作，这不仅将导致研究中的具体问题的解决，而且还将制定新的重要研究领域及其在真实的世界中的应用。利用NSF的支持，该团队培训了一些准备在数学，统计和工程前沿进行研究的学生，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.