This article considers the problem of selecting sensors in a large-scale system to minimize the error in estimating its states, more specifically, the state estimation mean-square error (MSE) and worst-case error for Kalman filtering and smoothing. Such selection problems are in general NP-hard, i.e., their solution can only be approximated in practice even for moderately large problems. Due to its low complexity and iterative nature, greedy algorithms are often used to obtain these approximations by selecting one sensor at a time choosing at each step the one that minimizes the estimation performance metric. When this metric is supermodular, this solution is guaranteed to be (1 - 1/e)-optimal. This is, however, not the case for the MSE or the worst-case error. This issue is often circumvented by using supermodular surrogates, such as the log det, despite the fact that minimizing the log det is not equivalent to minimizing the MSE. Here, this issue is addressed by leveraging the concept of approximate supermodularity to derive near-optimality certificates for greedily minimizing the estimation mean-square and worst-case error. In typical application scenarios, these certificates approach the (1 - 1/e) guarantee obtained for supermodular functions, thus demonstrating that no change to the original problem is needed to obtain guaranteed good performance.

中文翻译：

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卡尔曼滤波器传感器选择的近似超模性


本文考虑在大规模系统中选择传感器以最小化估计其状态的误差的问题，更具体地说，是卡尔曼滤波和平滑的状态估计均方误差 (MSE) 和最坏情况误差。此类选择问题通常是 NP 困难的，即，即使对于中等规模的问题，它们的解决方案也只能在实践中近似。由于其低复杂性和迭代性质，贪婪算法通常用于通过一次选择一个传感器来获得这些近似值，并在每一步选择使估计性能指标最小化的传感器。当该度量是超模时，该解决方案保证是 (1 - 1/e) 最优的。然而，对于 MSE 或最坏情况误差来说，情况并非如此。尽管最小化 log det 并不等于最小化 MSE，但通常可以通过使用超模代理（例如 log det）来规避此问题。在这里，这个问题是通过利用近似超模性的概念来得到近最优证书，以贪婪地最小化估计均方和最坏情况误差来解决的。在典型的应用场景中，这些证书接近为超模函数获得的（1 - 1/e）保证，从而表明不需要改变原始问题即可获得有保证的良好性能。