Outlier theory beyond non-degeneracy/invertibility assumptions (ReLU and zero diagonal entries)

In the high-dimensional regime $d,n\to\infty$ with $n/d\to\alpha\in(0,\infty)$, for weighted sample-covariance/Hessian-type matrices of the form $H_n=\frac1n\sum_{\ell=1}^n D_{\ell\ell}y_\ell y_\ell^\top$ arising from Gaussian-mixture features and data-dependent diagonal gates $D_{\ell\ell}=w(s_\ell)$, develop an outlier theory when $\mathbb P(D_{\ell\ell}=0)>0$: characterize limiting outlier locations and eigenvector alignments, and identify outlier phase transitions, without assuming diagonal weights are bounded away from zero.