Near-Optimal Sensor Placement for Linear Inverse Problems (1305.6292v4)

Abstract: A classic problem is the estimation of a set of parameters from measurements collected by only a few sensors. The number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates. Unfortunately, the selection of the optimal sensor locations is intrinsically combinatorial and the available approximation algorithms are not guaranteed to generate good solutions in all cases of interest. We propose FrameSense, a greedy algorithm for the selection of optimal sensor locations. The core cost function of the algorithm is the frame potential, a scalar property of matrices that measures the orthogonality of its rows. Notably, FrameSense is the first algorithm that is near-optimal in terms of mean square error, meaning that its solution is always guaranteed to be close to the optimal one. Moreover, we show with an extensive set of numerical experiments that FrameSense achieves state-of-the-art performance while having the lowest computational cost, when compared to other greedy methods.

Near-Optimal Sensor Placement for Linear Inverse Problems

The paper "Near-Optimal Sensor Placement for Linear Inverse Problems," by Juri Ranieri, Amina Chebira, and Martin Vetterli, presents a novel approach to sensor placement, crucial for effective parameter estimation in scenarios where only limited sensors are feasible due to various constraints. The proposed solution, FrameSense, leverages the concept of frame potential (FP) to achieve efficient sensor placement, thereby minimizing the mean square error (MSE) of parameter estimations.

Core Contribution

The central contribution of this work is the FrameSense algorithm, a greedy method for selecting optimal sensor locations in the context of linear inverse problems. It optimizes the frame potential, a scalar metric measuring matrix row orthogonality. FrameSense is distinguished as the first algorithm with near-optimal performance concerning MSE, consistently delivering solutions close to the optimal.

Methodological Insights

In linear inverse problems, where the goal is to infer unknown parameters from limited sensor data, sensor placement directly impacts the accuracy of parameter recovery. FrameSense operates by a “worst-out” procedure, incrementally improving sensor placement by removing rows that overly contribute to the FP, thus promoting row orthogonality in the selected submatrix. The objective function — minimizing the FP — aligns with lowering the MSE, where the latter is notorious for being fraught with local minima making direct minimization approaches suboptimal or infeasible.

Theoretical Considerations

The paper establishes the theoretical underpinnings of FrameSense by displaying its submodular cost function traits, which enable guarantees on solution quality, a rarity for combinatorial optimization problems. Furthermore, under certain conditions (akin to the Restricted Isometry Property), FrameSense also ensures near-optimal MSE performance.

Comparative Analysis

The paper contrasts FrameSense against traditional methods such as convex optimization and other greedy algorithms. Numerically, FrameSense showcases superior MSE performance and lower computational demand. It efficiently handles matrix sizes and configurations where traditional strategies falter due to computational expense or suboptimal outcomes.

Practical Implications and Future Work

FrameSense offers practical advancements in scenarios like Wireless Sensor Networks, where strategic sensor deployment can lead to significant improvements in field reconstruction accuracy. In practical settings, as evidenced by the thermal map reconstruction experiments on multi-core processors, FrameSense accurately predicts temperature distributions with efficiently sparse sensor arrangements, optimizing resource use while maintaining data fidelity.

The paper suggests future avenues, including adaptations of FrameSense within convex optimization frameworks to exploit both methodologies' strengths and refine sensing energy exploitation strategies, which could further enhance practical sensor network implementations.

Conclusion

Overall, this research delineates a methodologically rigorous approach to sensor placement for inverse problem applications, emphasizing computational efficiency and MSE minimization. FrameSense emerges as an advantageous tool for engineering applications requiring precise data inference from constrained sensor data on physical phenomena, marking a substantive contribution to sensor network design and optimization.