Parsimonious model identification via atomic norm minimization
Parsimonious model identification via atomic norm minimization
复制标题
通过原子范数最小化进行简约模型识别
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
M. Sznaier
中科院分区：
文献类型：
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作者：
K. Bekiroglu;Burak Yılmaz;C. Lagoa;M. Sznaier
在过去的几年中，相当多的研究工作一直致力于从实验数据中识别简约模型的问题。由于这个问题一般是非凸的，这些方法通常依赖于松弛，如组套索或核范数最小化。然而，虽然这些方法通常在实践中工作得很好，但不能保证使用这些替代物将导致解释实验数据的最简单模型。此外，将稳定性约束的形式化需要大量增加的计算复杂性。或者，稳定性和模型阶数约束可以直接使用基于矩的方法来处理。然而，目前这种方法仅限于相对较小的问题，由于其计算复杂性。受这些困难的启发，最近提出了一种新的方法，该方法基于将LTI系统的响应表示为适当选择的对象（原子）的线性组合的想法，并且观察到最小化原子范数导致稀疏表示。在本文中，我们涵盖了这种新方法的基本原理，并表明它导致了一个非常有效的算法，避免了使用正则化步骤的需要，并自动纳入稳定性约束。此外，这种方法可以扩展到适应非均匀采样和（未知的）初始条件。这些结果说明了几个例子，包括识别一个非常轻阻尼结构的时域和频域测量。
During the past few years a considerably research effort has been devoted to the problem of identifying parsimonious models from experimental data. Since this problem is generically non-convex, these approaches typically rely on relaxations such as Group Lasso or nuclear norm minimization. However, while these approaches usually work well in practice, there is no guarantee that using these surrogates will lead to the simplest model explaining the experimental data. In addition, incorporating stability constraints into the formalism entails a substantial increase in the computational complexity. Alternatively stability and model order constraints can be handled directly using a moments based approach. However, presently this approach is limited to relatively small sized problems, due to its computational complexity. Motivated by these difficulties, recently a new approach has been proposed based on the idea of representing the response of an LTI system as a linear combination of suitably chosen objects (atoms) and the observation that minimizing the atomic norm leads to sparse representations. In this paper we cover the fundamentals of this new approach and show that it leads to a very efficient algorithm, that avoids the need for using regularization steps and automatically incorporates stability constraints. In addition, this approach can be extended to accommodate non-uniform sampling and (unknown) initial conditions. These results are illustrated with several examples, including identification of a very lightly damped structure from time and frequency domain measurements.