Convex optimization methods for system identification and graphical modeling of time series

该项目旨在开发新的动态系统建模方法，基于凸优化公式和最新的大规模非光滑优化算法。该项目的第一个组成部分涉及时间序列图形模型的估计和拓扑选择。图形模型提供了随机变量之间关系的图形表示，例如，条件依赖。这些关系可以转化为对模型参数的稀疏约束。在图形模型估计中的一个基本挑战是从观测数据中选择稀疏图形拓扑。该项目旨在开发基于非光滑凸正则的稀疏拓扑选择方法。推动这项工作的主要应用是从功能磁共振成像时间序列进行连通性分析。第二部分是基于结构矩阵低阶逼近凸算法的系统辨识新方法。这项工作需要将系统辨识问题描述为约束秩优化问题，并开发用于秩优化问题凸松弛的大规模算法。智能价值该项目结合了最优化、系统论和机器学习的技术来解决动态系统建模中的基本问题。图形模型是机器学习中的一个重要主题，在系统辨识中没有得到广泛的研究。相反，系统识别可以为机器学习问题中的动态方面建模提供工具。凸公式和快速一阶算法的使用将使实际应用中的大型实例的有效解决方案成为可能。广泛影响项目中开发的算法的软件实现将免费提供。这些成果将被整合到加州大学洛杉矶分校电气工程系的研究生优化序列中，特别是关于大规模优化的高级课程。将通过个人学习课程和暑期实习为学生提供研究机会。

ObjectivesThe project aims at developing new methods for dynamical system modeling, based on convex optimization formulations and recent algorithms for large-scale non-smooth optimization. A first component of the project addresses the estimation and topology selection of graphical models of time series. A graphical model provides a graph representation of relations between random variables, for example, conditional dependence. These relations can be translated into sparsity constraints on the parameters of the model. A fundamental challenge in the estimation of a graphical model is the selection of a sparse graph topology from observed data. The project aims at developing methods for sparse topology selection via non-smooth convex regularizations. The main application that motivates this work is connectivity analysis from functional magnetic resonance imaging time series. A second part is concerned with new methods for system identification based on convex algorithms for low-rank approximation of structured matrices. This work requires the formulation of system identification problems as constrained rank optimization problems and the development of large-scale algorithms for convex relaxations of the rank optimization problems.Intellectual MeritThe project combines techniques from optimization, system theory, and machine learning to address fundamental problems in the modeling of dynamical systems.Graphical models, an important topic in machine learning, are not widely studied in system identification. Conversely, system identification can provide tools for modeling dynamical aspects in machine learning problems. The use of convex formulations and fast first-order algorithms will enable an efficient solution of large instances in practical applications.Broader ImpactsSoftware implementations of the algorithms developed in the project will be made freely available. The outcomes will be integrated in the graduate optimization sequence in the Electrical Engineering Department at UCLA, in particular an advanced course on large-scale optimization. Research opportunities will be offered to students via individual study courses and summer internships.