  In this paper, we study a nonlinear spiked random matrix model where a
nonlinear function is applied element-wise to a noise matrix perturbed by a
rank-one signal. We establish a signal-plus-noise decomposition for this model
and identify precise phase transitions in the structure of the signal
components at critical thresholds of signal strength. To demonstrate the
applicability of this decomposition, we then utilize it to study new phenomena
in the problems of signed signal recovery in nonlinear models and community
detection in transformed stochastic block models. Finally, we validate our
results through a series of numerical simulations.
