  Classical results in asymptotic statistics show that the Fisher information
matrix controls the difficulty of estimating a statistical model from observed
data. In this work, we introduce a companion measure of robustness of an
estimation problem: the radius of statistical efficiency (RSE) is the size of
the smallest perturbation to the problem data that renders the Fisher
information matrix singular. We compute RSE up to numerical constants for a
variety of test bed problems, including principal component analysis,
generalized linear models, phase retrieval, bilinear sensing, and matrix
completion. In all cases, the RSE quantifies the compatibility between the
covariance of the population data and the latent model parameter.
Interestingly, we observe a precise reciprocal relationship between RSE and the
intrinsic complexity/sensitivity of the problem instance, paralleling the
classical Eckart-Young theorem in numerical analysis.
