CAREER: Parsimonious Modeling via Matrix Minimization

智能优点：在许多工程应用中，模型的复杂性、阶数或维度的概念可以用矩阵的秩来表示，而先验信息和模型精度通常对应于对该矩阵的凸约束。简约建模涉及在凸约束条件下最小化矩阵秩值的计算问题。例如，在控制、信号处理和机器学习中出现的系统辨识、模型降阶和欧几里得嵌入问题。通常情况下，秩最小化问题在计算上是困难的。这项工作的灵感来自于最近开发的稀疏信号压缩传感框架。PI和她的合作者的初步工作指出，该理论从稀疏向量到低阶矩阵的丰富推广，表明这类困难问题可以有效地解决。这项拟议的研究建立在压缩传感及其基本数学以及凸优化方面的进展的基础上。该程序有三个主要内容：(1)可以有效解决的一类秩极小问题的理论刻画，(2)该问题的有效半定规划算法的发展，(3)秩极小问题的应用(例如，在系统辨识中)。该计划将实际影响与概念深度相结合，统一了现有的简约概念(如向量稀疏性和矩阵排名)以及解决这些问题的计算方法。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，并应促进传统上未被认为是应用的领域的数学研究。开发的结果和计算工具可供研究人员在各种应用领域使用。该项目的研究将与PI过去的工作结合在一起，成为威斯康星大学一门新的以项目为导向的研究生课程。所有级别的学生都将参与进来。计划举办研讨会和数学问题解决比赛，作为威斯康星大学为新入学女性和少数族裔新生提供的桥牌计划的一部分。

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.