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 $\mathrm{rank}$ $W_0/q$ finite as $p,q \to \infty$, we show that the trace of the resolvent of $W^{\top} W$ obeys a quartic equation for one degenerate signal, yielding an exact spectral density, and derive explicit asymptotics in the strong-signal regime. We map out a detailed generalized Baik-Ben Arous-P\'ech\'e (BBP) phase diagram and clarify how a finite density of spikes reshapes the bulk edges. We further derive a $1/3$-scaling law for the critical signal strength in terms of the rank ratio for rectangular matrices in the finite-to-extensive-rank crossover. Numerical simulations validate the theory and illustrate its relevance for high-dimensional inference tasks with multiple degenerate signals and more general signal distributions.