Speaker
Description
Detecting signals within nearly continuous spectra remains a fundamental challenge in statistical inference, particularly in high-noise regimes where standard random matrix theory approaches, such as the Baik-Ben Arous-Péché (BBP) transition, frequently fail. We introduce a methodology that bridges statistical mechanics and statistical inference. By treating the empirical spectrum as an effective equilibrium field theory, we establish a formal mapping between anomaly detection and the Functional Renormalisation Group (FRG) flow of non-equilibrium systems using a stochastic field framework, wherein the noise-to-signal ratio functions mathematically as a physical temperature. Instead of treating signals as isolated statistical outliers, the FRG approach models them as ordered domains emerging within a thermalised background of fluctuations. This dynamic induces a structural deformation of the spectral geometry. By using the scale-dependent canonical dimension of this geometry as a highly sensitive order parameter, we demonstrate how the system undergoes a sharp dimensional phase transition. Expanding upon fundamental formulations, our results show that this transition directly correlates with stochastic ergodicity breaking, a spontaneous $\mathbb{Z}_2$ symmetry breaking in the effective potential, and a measurable deviation of eigenvector statistics from the universal Porter-Thomas distribution. Consequently, this enables signal resolution at signal-to-noise thresholds significantly below standard limits, even when the underlying data distribution closely follows the Marchenko-Pastur law. Furthermore, we validate the framework on realistic datasets, including those typical in computer vision, and critical phenomena, such as the determination of the phase transition temperature in physical systems. The FRG thus provides a rigorous, physics-grounded technique for extracting extensive-rank signals in complex datasets.
| Affiliation | CEA Paris-Saclay |
|---|---|
| Career status | Senior |