Optimal sampling and recovery for multilinear signals and systems

本研究涉及开发和验证一个新的框架的信号和系统定义在大于两个维度。这样的多线性信号可以被看作是高维数据张量，而相关的系统则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，如医学成像、高光谱遥感、用于石油和天然气快速勘探的地震数据采集和处理，仅举几例。近年来，多元线性信号在社交网络数据分析、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领域变得越来越突出。有效的多线性计算和处理需要将传统的二维分解方法推广到更高的维度。迄今为止，张量分解技术要么在数据紧凑性或算子表示方面是最优的，但np难以评估，要么在另一个极端计算上可行，但可能严重不紧凑，从而丢失结构信息。这对采样（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产生不利影响。在这种情况下，研究人员开发了张量分解的基本新方法，这些方法虽然在计算上可行，但也可以证明是紧凑的。具体来说，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；多线性数据的最优采样与恢复——压缩感知理论的非平凡扩展以及基于张量表示的正则化多线性逆问题反演模型和算法。

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.