BBP Phase Transition for an Extensive Number of Outliers

TL;DR

This paper studies the phase transition in the singular value spectrum of large noisy matrices with extensive signals, providing a detailed phase diagram and asymptotic analysis relevant for high-dimensional data inference.

Contribution

It introduces a new phase diagram for extensive outliers in large random matrices, deriving explicit asymptotics and a scaling law for critical signal strength.

Findings

Singular value density follows a quartic equation in the strong-signal regime

The phase diagram generalizes Baik-Ben Arous-Péché transition for extensive outliers

Numerical simulations confirm theoretical predictions

Abstract

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 singular value density obeys a quartic equation and derive explicit asymptotics in the strong-signal regime. The resulting generalized Baik-Ben Arous-P\'ech\'e 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.