Low-Complexity Modeling of Partially Available Second-Order Statistics: Theory and an Efficient Matrix Completion Algorithm
Low-Complexity Modeling of Partially Available Second-Order Statistics: Theory and an Efficient Matrix Completion Algorithm
复制标题
部分可用的二阶统计量的低复杂度建模：理论和高效的矩阵完成算法
DOI：
10.1109/tac.2016.2595761
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发表时间：
2014
影响因子：
6.8
通讯作者：
T. Georgiou
中科院分区：
文献类型：
--
作者：
A. Zare;Yongxin Chen;M. Jovanović;T. Georgiou
线性系统的状态统计满足某些结构约束，这些约束来自于潜在的动力学和输入扰动的方向性。本文研究部分已知状态统计量的完备化问题。我们的目标是开发工具，可用于大规模动态系统的控制导向建模的背景下。对于我们所考虑的应用类型，状态变量之间的动态相互作用是已知的，而输入激励的方向性和动态通常是不确定的。因此，我们制定的数学问题的目标是识别输入激励的动态和方向性，以解释和完成观察到的样本统计。更具体地说，我们试图解释相关数据与最少数量的可能的输入干扰通道。我们制定这个反问题的秩最小化，并为它的解决方案，我们采用了基于核范数的凸松弛。由此产生的优化问题是铸造作为一个半定规划，可以使用通用求解器来解决。对于这些求解器无法处理的问题大小，我们开发了一个定制的交替最小化算法（AMA）。我们解释AMA作为一个邻近梯度的对偶问题，并证明了次线性收敛的算法与固定的步长。最后，我们用一个例子来说明我们的建模和优化框架的效用，并绘制AMA和常用的交替方向乘法器（ADMM）算法之间的对比。
State statistics of linear systems satisfy certain structural constraints that arise from the underlying dynamics and the directionality of input disturbances. In the present paper, we study the problem of completing partially known state statistics. Our aim is to develop tools that can be used in the context of control-oriented modeling of large-scale dynamical systems. For the type of applications we have in mind, the dynamical interaction between state variables is known while the directionality and dynamics of input excitation is often uncertain. Thus, the goal of the mathematical problem that we formulate is to identify the dynamics and directionality of input excitation in order to explain and complete observed sample statistics. More specifically, we seek to explain correlation data with the least number of possible input disturbance channels. We formulate this inverse problem as rank minimization, and for its solution, we employ a convex relaxation based on the nuclear norm. The resulting optimization problem is cast as a semidefinite program and can be solved using general-purpose solvers. For problem sizes that these solvers cannot handle, we develop a customized alternating minimization algorithm (AMA). We interpret AMA as a proximal gradient for the dual problem and prove sublinear convergence for the algorithm with fixed step-size. We conclude with an example that illustrates the utility of our modeling and optimization framework and draw contrast between AMA and the commonly used alternating direction method of multipliers (ADMM) algorithm.
影响因子：
6.8
作者：
Chen, Yongxin;Georgiou, Tryphon T.;Pavon, Michele
通讯作者：
Pavon, Michele