BBP Phase Transition for an Extensive Number of Outliers

Niklas Forner, Alexander Maloney, Bernd Rosenow

arXiv:2511.18501

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-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.

Description

This repository includes links, code, scripts, and data to generate the figures in the paper.

Requirements

The data in this project was generated via Python with the following packages:

• numpy
• matplotlib.pyplot
• time
• scipy.stats

Figures

Figure 01: Figure Name

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