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Signal detection is one of the main challenges of data science. As it oftenhappens in data analysis, the signal in the data may be corrupted by noise.There is a wide range of techniques aimed at extracting the relevant degrees offreedom from data. However, some problems remain difficult. It is notably thecase of signal detection in almost continuous spectra when the signal-to-noiseratio is small enough. This paper follows a recent bibliographic line whichtackles this issue with field-theoretical methods. Previous analysis focused onequilibrium Boltzmann distributions for some effective field representing thedegrees of freedom of data. It was possible to establish a relation betweensignal detection and $\mathbb{Z}_2$-symmetry breaking. In this paper, weconsider a stochastic field framework inspiring by the so-called "Model A", andshow that the ability to reach or not an equilibrium state is correlated withthe shape of the dataset. In particular, studying the renormalization group ofthe model, we show that the weak ergodicity prescription is always broken forsignals small enough, when the data distribution is close to theMarchenko-Pastur (MP) law. This, in particular, enables the definition of adetection threshold in the regime where the signal-to-noise ratio is smallenough.

Functional renormalization group for signal detection and stochastic ergodicity breaking