Sensor Placement by Maximal Projection on Minimum Eigenspace for Linear Inverse Problems

Abstract

Abstract-This paper presents two new greedy sensor placement algorithms, named minimum nonzero eigenvalue pursuit (MNEP) and maximal projection on minimum eigenspace (MPME), for linear inverse problems, with greater emphasis on the MPME algorithm for performance comparison with existing approaches. In both MNEP and MPME, we select the sensing locations one-by-one. In this way, the least number of required sensor nodes can be determined by checking whether the estimation accuracy is satisfied after each sensing location is determined. For the MPME algorithm, the minimum eigenspace is defined as the eigenspace associated with the minimum eigenvalue of the dual observation matrix. For each sensing location, the projection of its observation vector onto the minimum eigenspace is shown to be monotonically decreasing w.r.t. the worst case error variance (WCEV) of the estimated parameters. We select the sensing location whose observation vector has the maximum projection onto the minimum eigenspace of the current dual observation matrix. The proposed MPME is shown to be one of the most computationally efficient algorithms. Our MonteCarlo simulations showed that MPME outperforms the convex relaxation method [1], the SparSenSe method [2], and the FrameSense method [3] in terms of WCEV and the mean square error (MSE) of the estimated parameters, especially when the number of available sensor nodes is very limited.

Sensor Scheduling With Time, Energy, and Communication Constraints

“…These bounds are also reflected in the results from Fig. 2 where we can observe that for k ≈ n the decrease in MSE achieved by the proposed method, with the increased number of selected sensors k, is larger that of a random sensor selection algorithm when k n. These insights confirm previous experimental results from the literature, like [4], where the methods proposed for sensors selection differ mostly when k ≈ n and are similar when k n (or k ≈ m) where even random selections provide good estimation accuracy (low MSE and WCE).…”

“…2: the estimated curve is above the one empirically observed. It is clear from the figure that the largest differences are for low k (on the same order with n) while the gap closes for k approaching m. As we will also see from the results section, the largest differences between the performance of the methods we analyze are for low values of k. Indeed, past research [4] has shown by numerical experimentation that most of the sensor selection methods proposed in the literature perform similarly in the regime k n. Also, Result 4 shows that the MSE(A 1 ) and WCE(A 1 ) decrease on average linearly with the number of selected sensors. Dependencies with the other dimensions are also linear and intuitive: increasing the number of parameters to estimate (n) and the total number of available sensors (m) leads to worse performance; increasing the energy, essentially the signal to noise ratio, of the measurement matrix (α) improves performance.…”

“…A proof is given in [39,Chapter 4]. This is one of the reasons given in [4] that the goal to increase all the eigenvalues λ i with each new measurement. Consider also the following two results.…”

“…Fig. 4 shows the simulation results where we compare with FrameSense [6], convex relaxation ( 1 followed by rounding, using the log determinant approach to minimize VCE) [8], SparSenSe [13] and MPME [4]. All measurement matrices used here are α−tight with α = 100.…”

“…The sensor placement (and in general the sensor management) problems have been extensively studied in the past. A general approach is to use greedy methods based on a minimum eigenspace approach [4] or with submodularity based performance guarantees [5] that provide results within (1 − e −1 ) of the optimal solution. Another popular greedy sensor selection method, called FrameSense [6], initially activates all the sensors and then removes one at each step based on a "worst-out" principle to optimize its submodular objective function.…”