Imagine passing a signal $v$ through a noisy channel, where $v \in \mathbb{C}^N$ is a deterministic unit vector. We assume that the recipient observes a corrupted version of the signal in the form of $\theta vv^* + M$, where $\theta \in \mathbb{R}$ is the strength of the signal and $M$ is a random Hermitian $N \times N$ matrix representing the noise. We consider two questions:

1. (Detection) Is it possible for the recipient to conclude that a signal has been passed based on the observation?
2. (Recovery) If so, is it possible for the recipient to recover the signal

For rotationally invariant noise, Benaych-Georges and Nadakuditi answered the detection question in terms of the outlying eigenvalues and the recovery question in terms of the corresponding eigenvectors. Their proof crucially relies on the fact that the eigenvectors of a rotationally invariant ensemble are Haar distributed (in particular, delocalized). 

We consider a general class of noise that includes non mean-field models such as random band matrices in regimes where the eigenvectors are known to be localized. In contrast to the usual approach to outliers via the resolvent, our analysis relies on moment method calculations for general vector states and a seemingly innocuous isotropic global law.