CIF: Medium: Collaborative Research: Subspace Matching and Approximation on the Continuum

在各种各样的应用领域——包括天文成像、显微镜、无线通信、雷达、声纳和许多其他领域——人们必须面对从一个由少量未知（连续值）参数控制的信号中估计或提取信息的问题。针对这些应用中的每一个，过去已经提出了处理这些未知参数的独特方法。本研究涉及开发一个统一的框架，系统、有效地解决所有这些领域的问题，意识到基本限制，并具有可证明的准确性。这个框架有两个主要组成部分。第一个组件涉及模型的开发，其中感兴趣的信号位于或靠近子空间或子空间的并集。在信号的基本参数是连续值的情况下，这是一项具有挑战性的任务。与其试图简单地否定这些问题（一个可能充满困难的程序），本研究涉及到使用有效的局部子空间拟合来发展局部无限范围的子空间的有限逼近，其维度与局部自由度的有效数量相匹配。该框架的第二个组成部分包括用于识别负责从一组离散观测产生信号的子空间范围的技术，这些技术面临着由相干性、参数空间的连续性以及某些信号可能从狭窄（但连续索引，因此是无限的）子空间范围合成的事实所带来的困难。这些组件结合起来，使有效的采集，处理和估计信号产生的参数化子空间模型。

In a diverse range of application areas--including astronomicalimaging, microscopy, wireless communications, radar, sonar, andmany others--one must confront the problem of estimating orextracting information from a signal that is governed by a smallnumber of unknown (continuous-valued) parameters. Customized toeach of these applications, unique methods for handling theseunknown parameters have been proposed in the past. This researchinvolves the development of a unified framework for addressingproblems in all of these areas systematically, efficiently, withawareness of the fundamental limits, and with provable accuracy.There are two main components of this framework. The firstcomponent involves the development of models where the signals ofinterest lie within or close to a subspace or a union of subspaces.In the case where the underlying parameters for the signal arecontinuous-valued, this is a challenging task. Rather than attemptto simply discredit these problems (a program potentially fraughtwith difficulty), this research involves the development of finiteapproximations for a locally infinite range of subspaces usingefficient local subspace fits whose dimensions match the effectivenumber of local degrees of freedom. The second component of thisframework consists of techniques for identifying the range ofsubspaces responsible for generating a signal from a set ofdiscrete observations, confronting difficulties posed by thecoherence, by the continuous nature of the parameter space, and bythe fact that some signals may be synthesized from a narrow (butcontinuously-indexed and therefore infinite) range of subspaces.These components combine to enable the efficient acquisition,processing, and estimation of signals arising from theparameterized subspaces model.