CAREER: Parsimonious Modeling via Matrix Minimization

智力优点：在许多工程应用中，模型的复杂性、阶数或维数等概念可以用矩阵的秩来表示，而先验信息和模型精度往往对应于该矩阵上的凸约束。简约建模涉及在凸约束下最小化矩阵秩的计算问题。例子包括系统识别，模型简化和欧几里得嵌入，在控制，信号处理和机器学习中出现的问题。已知秩最小化问题一般在计算上是难以处理的。 这项工作的灵感来自于最近开发的稀疏信号压缩感知框架。PI和她的合作者的初步工作，指出了这个理论从稀疏向量到低秩矩阵的丰富推广，表明这个难题的某些类别可以有效地解决。拟议的研究建立在压缩感知及其基础数学以及凸优化的基础上。该方案有三个重点：（1）可以有效解决的秩最小化问题的类别的理论表征，（2）针对该问题的有效半定编程算法的开发，（3）关注秩最小化的应用（例如，系统识别）。该计划结合了概念深度的实际影响，统一了现有的简约概念（如向量稀疏性和矩阵秩）以及解决这些问题的计算方法。更广泛的影响：该计划将利用纯数学家和工程师之间的广泛合作，并应激励传统上不考虑的数学领域的研究？适用”。结果和计算工具的开发，可用于研究人员在各个应用领域。该项目的研究将与PI过去的工作整合到UW的新项目驱动的研究生课程中。各级学生都将参与。一个研讨会和一个数学问题解决比赛计划作为华盛顿大学的桥梁计划的一部分，为新来的妇女和少数民族新生。

Intellectual merit: In many engineering applications, notions of complexity, order or dimension of a model can be expressed by the rank of a matrix, while prior information and model accuracy often correspond to convex constraints on this matrix. Parsimonious Modeling involves the computational problem of minimizing matrix rank subject to convex constraints. Examples include problems in system identification, model reduction, and Euclidean embedding, arising in control, signal processing, and machine learning. The rank minimization problem is known to be computationally intractable in general. The work is inspired by the recently developed framework of compressed sensing for sparse signals. A preliminary work by the PI and her collaborators, points to a rich generalization of this theory from sparse vectors to low-rank matrices, showing that some classes of this hard problem can be solved efficiently. The proposed research builds on advances in compressed sensing and its underlying math, as well as convex optimization. The program has three thrusts: (1) theoretical characterization of classes of rank minimization problems that can be solved efficiently, (2) development of efficient semidefinite programming algorithms for this problem, (3) a focus on applications of rank minimization (e.g., in system identification). This program combines practical impact with conceptual depth, unifying existing notions of parsimony (such as vector sparsity and matrix rank) as well as the computational methods to address them.Broader impact: This program will leverage extensive collaborations between pure mathematicians and engineers, and should motivate research in mathematics in areas not traditionally considered ?applied". Results and computational tools developed can be used by researchers in various application fields. Research from this project will be integrated with the PI's past work into a new project-driven graduate course at UW. Students at all levels will be engaged. A workshop and a mathematical problem solving competition is planned as part of UW's BRIDGE program for incoming women and minority freshman.