Functional renormalization group for signal detection and stochastic ergodicity breaking

TL;DR

The paper tackles detecting weak signals embedded in almost-continuous spectra common in high-dimensional covariance data, where standard methods falter near the MP universality class. It builds a non-equilibrium stochastic field theory, cast as Model A Langevin dynamics, and analyzes its Wilsonian FRG flow under the Local Potential Approximation using the Martin-Siggia-Rose formalism to connect dynamics with equilibrium properties. The key result is the identification of two dynamical regimes: in pure MP/noise, ergodicity breaks and the RG flow diverges, while a sufficiently strong fixed-rank signal (characterized by a parameter $\beta$) can stabilize a finite flow and enable a detectable transition, with a threshold $\beta_c \approx 0.2$. This provides a principled, field-theoretic criterion for signal detectability in low-SNR, high-dimensional data and suggests avenues beyond the Local Potential Approximation to sharpen the threshold and boundary between signal and noise.

Abstract

Signal detection is one of the main challenges of data science. As it often happens 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 of freedom from data. However, some problems remain difficult. It is notably the case of signal detection in almost continuous spectra when the signal-to-noise ratio is small enough. This paper follows a recent bibliographic line which tackles this issue with field-theoretical methods. Previous analysis focused on equilibrium Boltzmann distributions for some effective field representing the degrees of freedom of data. It was possible to establish a relation between signal detection and $\mathbb{Z}_2$-symmetry breaking. In this paper, we consider a stochastic field framework inspiring by the so-called "Model A", and show that the ability to reach or not an equilibrium state is correlated with the shape of the dataset. In particular, studying the renormalization group of the model, we show that the weak ergodicity prescription is always broken for signals small enough, when the data distribution is close to the Marchenko-Pastur (MP) law. This, in particular, enables the definition of a detection threshold in the regime where the signal-to-noise ratio is small enough.

Figures (11)

• Figure 1: Depending on the nature of the underlying data, an empirical spectrum can exhibit some localized spikes (left) out of a bulk (i.e. noise, in red) made of delocalized eigenvectors (i.e. relevant information, in blue), in which case the cut-off $\Lambda$ provides a clean separation between delocalized eigenvectors and localized ones. In the case of nearly continuous spectra (right), the position of the cut-off $\Lambda$ is arbitrary, and the separation of the signal can become impossible.
• Figure 2: Convergence toward Marchenko-Pastur law for large size $N \times P$ Wishart matrices ($P / N = 0.4$).
• Figure 3: Behaviour of the canonical dimensions (left) of local couplings ($u_{2n}$) of $\phi^{2n}$ interaction for Marchenko-Pastur law and shape of the effective potential in the deep IR (right), depending on the strength of the signal embedded in a random Wishart matrix. Pictures taken from LahocheSignal2022.
• Figure 4: Behaviour of the threshold function for a typical regulator $R_k$ (the dashed blue curve).
• Figure 5: Typical shape of $\rho(p^2)$ for MP law. Note that we labelled the abscissa variables with $k^2$, the renormalization group scale.

Theorems & Definitions (1)

• Remark 1