  This work 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 logdet, despite the
fact that minimizing the logdet 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.
