The selection problem of an optimal set of sensors estimating the snapshot of high-dimensional data is considered. The objective functions based on various criteria of optimal design are adopted to the greedy method: D-optimality, A-optimality, and E-optimality, which maximizes the determinant, minimize the trace of inverse, and maximize the minimum eigenvalue of the Fisher information matrix, respectively. First, the Fisher information matrix is derived depending on the numbers of latent state variables and sensors. Then, the unified formulation of the objective function based on A-optimality is introduced and proved to be submodular, which provides the lower bound on the performance of the greedy method. Next, the greedy methods based on D-, A-, and E-optimality are applied to randomly generated systems and a practical data set of global climates. The sensors selected by the D-optimality objective function works better than those by A- and E-optimality with regard to the determinant, trace of the inverse, and reconstruction error, while those by A-optimality works the best with regard to the minimum eigenvalue. On the other hand, the performance of sensors selected by the E-optimality objective function is worse for all indices and reconstruction error. This might be because of the lack of submodularity as proved in the paper. The results indicate that the greedy method based on D-optimality is the most suitable for high accurate reconstruction with low computational cost.
Randomized Greedy Methods for Weak Submodular Sensor Selection with Robustness Considerations
Elmore Family School of Electrical and Computer Engineering, Purdue University（普渡大学埃尔莫尔电气与计算机工程学院）；Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin（德克萨斯大学奥斯汀分校航空航天工程与应用力学系）
