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Functional Renormalization Group Approach for Signal Detection

Vincent Lahoche, Dine Ousmane Samary, Mohamed Tamaazousti

TL;DR

The paper reframes data analysis as an analogue Euclidean field theory and uses nonperturbative functional renormalization group methods to study signal detection in nearly-continuous spectra. By constructing an analogue field theory via maximum entropy and applying the effective average action framework, it identifies two regimes (nonlocal near Gaussian and local near the tail) and shows that signals shift the RG flow away from non-Gaussian fixed points, effectively restoring symmetry or driving symmetry breaking depending on the spectrum. Across Marchenko-Pastur, Wigner, and tensor universality classes, the work demonstrates that only a small set of local couplings governs infrared behavior for purely noisy data, while a sufficiently strong signal can render these couplings irrelevant and reveal a detectable phase transition. The approach provides a universal diagnostic: the change in the number and relevance of couplings near the tail of the spectrum and the associated shift in RG trajectories, offering a principled, theory-driven criterion for signal detection in high-dimensional datasets with nearly continuous spectra. The framework is poised to inform practical algorithms for spectral data analysis and tensor PCA by connecting covariance structure, universality, and RG flow to detection thresholds and phase-transition-type behavior.

Abstract

This review paper uses renormalization group techniques for signal detection in nearly-continuous positive spectra. We highlight universal aspects of the analogue field-theory approach. The first aim is to present an extended self-consistent construction of the analogue effective field-theory framework for data, which can be viewed as a maximum entropy model. In particular and exploiting universality arguments, we justify the $\mathbb{Z}_2$-symmetry of the classical action and we stress the existence of a large-scale (local) regime and of a small-scale (nonlocal) regime. Secondly and related to noise models, we observe the universal relation between phase transition and symmetry breaking in the vicinity of the detection threshold. Finally, we discuss the issue of defining the covariance matrix for tensorial-like data. Based on the cutting graph prescription, we note the superiority of definitions based on complete graphs of large size for data analysis.

Functional Renormalization Group Approach for Signal Detection

TL;DR

The paper reframes data analysis as an analogue Euclidean field theory and uses nonperturbative functional renormalization group methods to study signal detection in nearly-continuous spectra. By constructing an analogue field theory via maximum entropy and applying the effective average action framework, it identifies two regimes (nonlocal near Gaussian and local near the tail) and shows that signals shift the RG flow away from non-Gaussian fixed points, effectively restoring symmetry or driving symmetry breaking depending on the spectrum. Across Marchenko-Pastur, Wigner, and tensor universality classes, the work demonstrates that only a small set of local couplings governs infrared behavior for purely noisy data, while a sufficiently strong signal can render these couplings irrelevant and reveal a detectable phase transition. The approach provides a universal diagnostic: the change in the number and relevance of couplings near the tail of the spectrum and the associated shift in RG trajectories, offering a principled, theory-driven criterion for signal detection in high-dimensional datasets with nearly continuous spectra. The framework is poised to inform practical algorithms for spectral data analysis and tensor PCA by connecting covariance structure, universality, and RG flow to detection thresholds and phase-transition-type behavior.

Abstract

This review paper uses renormalization group techniques for signal detection in nearly-continuous positive spectra. We highlight universal aspects of the analogue field-theory approach. The first aim is to present an extended self-consistent construction of the analogue effective field-theory framework for data, which can be viewed as a maximum entropy model. In particular and exploiting universality arguments, we justify the -symmetry of the classical action and we stress the existence of a large-scale (local) regime and of a small-scale (nonlocal) regime. Secondly and related to noise models, we observe the universal relation between phase transition and symmetry breaking in the vicinity of the detection threshold. Finally, we discuss the issue of defining the covariance matrix for tensorial-like data. Based on the cutting graph prescription, we note the superiority of definitions based on complete graphs of large size for data analysis.
Paper Structure (41 sections, 6 theorems, 220 equations, 41 figures, 2 tables)

This paper contains 41 sections, 6 theorems, 220 equations, 41 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Renormalization group and AI.
  3. State of the art.
  4. Purpose of this paper.
  5. Remark for the physicist reader:
  6. Motivations and outline
  7. Short review on the spiked matrix models
  8. The nearly continuous spectra issue
  9. Working methodology
  10. Analogue field theory for spectral analysis
  11. Maximum entropy estimator for spins
  12. Analogue effective field theory candidates
  13. Gaussian model
  14. Nearly Gaussian distribution for noisy data
  15. Effective model in the large-N limit.
  16. ...and 26 more sections

Key Result

Theorem 1

Let $Q=\beta uu^T+M/\sqrt{N}$ a spiked Wigner matrix with $u^2=1$ and $M\in GOE$. We have:

Figures (41)

  • Figure 1: Illustration of Kadanoff's block-spin RG transformation $\mathcal{T}$: Spins in the initial lattice, with spacing a, are averaged into blocks of four spins. The interactions between individual spins are replaced by interactions between blocks with a spacing of $2a$.
  • Figure 2: Illustration of the convergence toward universal laws. On both sides we show eigenvalues histograms for Wigner (on the left) and white Wishart (on the right) matrices of size $10^4$. The blue lines materialize the limit Wigner semi-circle ($\mu_W$) and MP ($\mu_{MP}$) laws.
  • Figure 3: On the left: Typical empirical spectrum exhibiting some localised spikes out of a bulk (in red) made of delocalized eigenvectors. The cut-off $K=\Lambda$ provides a clean separation between delocalized eigenvectors (noise) and localized ones (information). On the right: Arbitrariness in the choice of the cut-off $\Lambda$ in nearly continous spectra.
  • Figure 4: Covariance matrix for nearly continuous datasets. On the top Qualitative illustration of the deviations from the universal MP law (in blue), obtained by completely randomize the data matrix. On the bottom: Illustration for a spectrum obtained by adding large rank deterministic matrix to a purely Gaussian Wishart noise.
  • Figure 5: Eigenvalue density spectrum for a scalar field theory. On right: Energy density spectrum $\tilde{\rho}(E^2)$ for free particles with mass $m^2=1$ in dimension $d=5$ (in blue) and $d=3$ (in orange). On left: Eigenvalue density spectrum $\tilde{\mu}(1/E^2)$ of the free propagator $\hat{H}_0^{-1}$ for $d=5$ (in blue) and $d=3$ (in orange).
  • ...and 36 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Definition 2
  • Theorem 3
  • Definition 3
  • Definition 4
  • Remark 1
  • Definition 5
  • Proposition 1
  • ...and 7 more