BBP Phase Transition for an Extensive Number of Outliers

Random-matrix theory helps disentangle signal from noise in large data sets. We analyze rectangular $p\times q$ matrices $W=W_0+M$ in which the noise $M$ generates a Marchenko-Pastur bulk, whereas the signal $W_0$ injects an extensive set of degenerate singular values. Keeping $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}$ $W_0\mathrm{/}q$ finite as $p,q\to \mathrm{\infty }$, we show that the singular value density obeys a quartic equation and derive explicit asymptotics in the strong-signal regime. The resulting generalized Baik-Ben Arous-Péché phase diagram yields a scaling law for the critical signal strength and clarifies how a finite density of spikes reshapes the bulk edges. Numerical simulations validate the theory and illustrate its relevance for high-dimensional inference tasks.