Lower Bounds on the Minimax Risk for the Source Localization Problem

The "source localization" problem is one in which we estimate the location of a point source observed through a diffusive medium using an array of sensors. We obtain lower bounds on the minimax risk (mean squared-error in location) in estimating the location of the source, which apply to all estimators, for certain classes of diffusive media, when using a uniformly distributed sensor array. We show that for sensors of a fixed size, the lower bound decays to zero with increasing numbers of sensors. We also analyze a more physical sensor model to understand the effect of shrinking the size of sensors as their number increases to infinity, wherein the bound saturates for large sensor numbers. In this scenario, it is seen that there is greater benefit to increasing the number of sensors as the signal-to-noise ratio increases. Our bounds are the first to give a scaling for the minimax risk in terms of the number of sensors used.