Robust Identification of a Class of Structured Systems with High Dimensional Outputs and Applications

这项研究计划的目标是开发一个全面的框架，用于稳健地识别一类出现在从图像处理到纳米系统的不同领域的多维系统。它的变革性影响和新颖性在于重塑问题，这些问题需要将高维数据流中稀疏编码的信息提取为多维系统识别问题，在动态系统理论、图像处理和机器学习之间建立了新的联系。智能优点：最近传感能力的指数增长对识别理论提出了严重挑战。简而言之，现有技术不足以处理海量数据。本提案力求制定一个专门为应对这一挑战而量身定做的全面、稳健的建模(识别、减少、验证)框架。相对于现有技术的优势包括能够直接适应结构约束(例如周期性)，利用数据中的相关性来实现实质性降维，以及利用优化中的最新结果来为由于较差的缩放特性而挑战当前技术的问题提供易于处理的解决方案。例子(通常是NP难的)是(I)分段仿射混合系统的稳健识别，(Ii)Hammerstein/Wiener系统的稳健识别和(Iii)半盲(In)验证。广泛的影响：增强的数据收集和分析能力可以深刻地影响社会，其好处包括从更安全的自我意识环境到增强的基于图像的治疗。实现这一愿景的一个主要障碍来自维度的诅咒。这项拟议的研究利用了一种隐藏的共性--底层动力学模型的表示比数据的维度简单得多--将关键问题，如数据分割、重建和分类，重新塑造成一种易于处理的形式，极大地促进了几个领域的最新水平。例子包括(但不限于)生物医学图像处理、建筑安全、纳米系统和老化的民用基础设施监测。将通过积极让我们的合作伙伴参与生物医学图像处理和建设安全来将这些成果转化为社会和经济。这项拟议的研究还有可能与工程学和应用数学的其他分支进行重大的交流。一个例子是非线性降维方法和流形发现(机器学习的两个标志)和非线性识别之间的联系。

The objective of this research program is to develop a comprehensive framework for robust identification of a class of multidimensional systems arising in diverse domains ranging from image processing to nano-systems. Its transformative impact and novelty reside in recasting problems that require extracting information sparsely encoded in high dimensional data streams as multidimensional systems identification problems, establishing a new connection between dynamical systems theory, image processing and machine learning.Intellectual Merit: Recent exponential growth in sensing capabilities poses a serious challenge to identification theory. Simply put, existing techniques are ill-equipped to deal with the overwhelming volume of data. The present proposal seeks to develop a comprehensive robust modeling (identification, reduction, validation) framework specifically tailored to address this challenge. Advantages over existing techniques include the abilities to directly accommodate structural constraints (such as periodicity), exploit correlations in the data to accomplish substantial dimensionality reduction and exploit recent results in optimization to furnish tractable solutions to problems that challenge current techniques, due to poor scaling properties. Examples (known to be generically NP-hard) are (i) robust identification of piecewise affine hybrid systems, (ii) robust identification of Hammerstein/Wiener systems and (iii) semi-blind (in)validation.Broader Impact: Enhanced data collection and analysis capabilities can profoundly impact society, with benefits ranging from safer, self aware environments, to enhanced image-based therapies. A major impediment to realizing this vision stems from the curse of dimensionality. The proposed research exploits a hidden commonality --underlying dynamical models having a far simpler representation than the dimension of the data-- to recast key problems, e.g. data segmentation, reconstruction and classification, into a tractable form, significantly advancing the state of the art in several domains. Examples include (but are not limited to) biomedical image processing, building safety, nano-systems and aging civil-infrastructure monitoring. Translation of these results to society and the economy will proceed by actively engaging our partners in bio-medical image processing and building security. The proposed research also has the potential for significant cross--fertilization with other branches of engineering and applied mathematics. An example is the connection between nonlinear dimensionality reduction methods and manifold discovery (both hallmarks of machine learning) and nonlinear identification.