The mean square error in Kalman filtering sensor selection is approximately supermodular

Abstract

This work considers the problem of selecting sensors in large scale system to minimize the state estimation mean-square error (MSE). More specifically, it leverages the concept of approximate supermodularity to derive near-optimality certificates for greedy solutions of this problem in the context of Kalman filtering. It also shows that in typical application scenarios, these certificates approach the typical 1/e guarantee. These performance bounds are important because sensor selection problems are in general NP-hard. Hence, their solution can only be approximated in practice even for moderately large problems. A common way of deriving these approximations is by means of convex relaxations. These, however, come with no performance guarantee. Another approach uses greedy search, although also in this case typical guarantees do not hold since the MSE is neither submodular nor supermodular. This issue is commonly addressed by using a surrogate supermodular figure of merit, such as the log det. Unfortunately, this is not equivalent to minimizing the MSE. This work demonstrates that no change to the original problem is needed to obtain performance guarantees.