Fisher Information and the Minimum Mean-Square Error for Locating Closely Spaced Point Sources

Abstract

We determine the optimal average error of locating closely spaced sources from noisy superpositions of a known smooth response with unknown complex coefficients. Relative phases control how changes in one location can be offset by the other unknowns, so local Fisher information alone does not determine the attainable error. For every fixed number $K\ge2$ of sources with independent uniform phases, we prove matching Bayes risk bounds jointly in separation and noise, for all sufficiently small separations and all noise levels. Odd source counts have two risk regimes; even counts have an intermediate regime in which mean squared error is linear in the noise standard deviation. A projected least-squares estimator attains these orders without knowing the phases or the prior density. Bounded nonuniform phase laws preserve the orders under stated support conditions, whereas priors concentrated near poorly identified configurations can change the exponents. A common finite comparison yields both a global inverse bound and paths controlled by the original prior. These give sufficient conditions under which the risk has the order of the pointwise capped Cramér--Rao bound, and a lower-tail calculation explains the power and logarithmic regimes. We also give conditional guarantees when the cluster center, separation, and source count are unknown, with an additional low-noise requirement for count selection. A calibrated covariance factor model illustrates the same criterion beyond source localization.