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

近年来，在积累的科学成果和最近在稀疏表示方面的突破的推动下，人们对不完整数据的过完全展开越来越感兴趣——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，将这三个社区纳入重点研究小组（FRG）是处理这一高潜力但具有挑战性的研究领域的最佳手段，这是适当和及时的。特别是，本项目遵循一个全面的垂直整合的研究计划：(1)在过完备Gabor时频表示和多分辨率小波字典的背景下，为统计估计提供新的理论结果；(2)利用这些结果开发针对信号和图像处理中的典型问题的算法，在这些问题中，从业者经常面临数据缺失或更普遍的不完整测量；(3)解决无处不在和重要的交叉应用，包括曲线拟合以及音频和彩色图像增强。为了应对这些紧迫的科学需求，并为数学科学的重大发展做好准备，FRG团队正在利用谐波分析和框架理论的最新结果，为过完全展开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检查，例如在贝叶斯框架中选择先验系数分布的影响，以及当潜在预测因子集过于完整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组件模型，该模型恢复了当前使用的方法作为特殊情况，但为发展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一致性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现由此产生的新算法。重点研究小组（FRG）项目团队由来自一个成熟机构（哈佛大学）和一个年轻的、快速发展的机构（中佛罗里达大学）的科学家组成。该项目的研究议程旨在实质性地推进理论知识和对过完备表示在统计和工程实践中的适用性的理解（一个新的和重要的数学交叉领域，与“压缩感知”领域有关的许多主要开放问题最近在纽约时报，经济学人和其他主流媒体上出现）。这最终将导致在必须以非常低的速率收集数据（如上文所述的压缩传感系统）或部分可用数据丢失或高度污染的情况下，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，蜂窝通信中的语音数据）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、统计学家和工程师之间通过一个团队的密切合作，这不仅会解决所研究的具体问题，而且还会形成新的重要研究领域的公式，并将其应用于现实世界。在美国国家科学基金会的支持下，该团队培养了一批准备在数学、统计学和工程学的前沿领域进行研究的学生，并定期举办研讨会，以增加新研究人员的参与，并将结果传播到更广泛的科学界。

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.