Optimal sampling and recovery for multilinear signals and systems

多线性信号和系统的最佳采样和恢复

• 批准号: 1319653
• 负责人: Shuchin Aeron
• 金额: $ 48.34万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1319653&HistoricalAwards=false
• 关键词: Optimal; sampling; recovery; multilinear; signals

This research involves the development and validation of a novel framework for signals and systems defined in dimensions greater than two. Such multilinear signals can be viewed as high dimensional data tensors, and the associated systems as multidimensional linear operators acting on this class of signals. These mathematical objects are ubiquitous in science and engineering and of growing practical importance. They arise in time varying tomographic problems such as medical imaging, hyperspectral remote sensing, and seismic data acquisition and processing for rapid exploration for oil and gas, to name but a few applications. More recently multilinear signals have become prominent in the context of data analytics for social networks, online big-data management, data visualization, predictive modeling for disease propagation and forecasting of complex events.Effective multilinear computation and processing requires generalization of the traditional 2-D factorization methods to higher dimensions. To date, tensor factorization techniques are either optimal in terms of compactness of data or operator representation but NP-hard to evaluate, or on the other extreme computationally feasible but can be severely non-compact thereby loosing structural information. This adversely affects the efficiency in sampling (tensor data compression) and computational costs in recovery of multilinear data using regularized inversion algorithms. In this context, the investigators develop fundamentally new approaches for tensor decompositions, which, while being computationally feasible, are also provably compact. Specifically, the investigators will develop: Algorithms for handling large scale processing for multilinear signals and systems; Optimal sampling and recovery of multilinear data as non-trivial extension of the theory of compressive sensing; and Tensor representation based, regularized inversion models and algorithms for multilinear inverse problems.

这项研究涉及到一个新的框架的发展和验证的信号和系统定义的尺寸大于2。这样的多线性信号可以被看作是高维数据张量，并且相关联的系统可以被看作是作用于这类信号的多维线性算子。这些数学对象在科学和工程中无处不在，并且具有越来越重要的实际意义。它们出现在时变层析成像问题中，例如医学成像、高光谱遥感以及用于快速勘探石油和天然气的地震数据采集和处理，仅举几个应用。近年来，多线性信号在社交网络的数据分析、在线大数据管理、数据可视化、疾病传播预测建模和复杂事件预测等方面变得越来越重要。有效的多线性计算和处理需要将传统的二维因式分解方法推广到更高的维度。到目前为止，张量因式分解技术在数据或算子表示的紧凑性方面是最佳的，但NP-难以评估，或者在另一个极端计算上可行，但可能严重非紧凑，从而丢失结构信息。这不利地影响采样（张量数据压缩）的效率和使用正则化反演算法恢复多线性数据的计算成本。在这种情况下，研究人员开发了张量分解的新方法，这些方法在计算上可行的同时，也是可证明的紧凑的。 具体而言，研究人员将开发：用于处理多线性信号和系统的大规模处理的算法;作为压缩感知理论的非平凡扩展的多线性数据的最佳采样和恢复;以及基于张量表示的正则化反演模型和算法。