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arxiv: 2507.01064 · v3 · submitted 2025-06-30 · ⚛️ physics.data-an · cond-mat.stat-mech· cs.IT· hep-th· math.IT· stat.ME

Functional Renormalization for Signal Detection: Dimensional Analysis and Dimensional Phase Transition for Nearly Continuous Spectra Effective Field Theory

Riccardo Finotello , Vincent Lahoche , Dine Ousmane Samary This is my paper

Pith reviewed 2026-05-19 07:23 UTC · model grok-4.3

classification ⚛️ physics.data-an cond-mat.stat-mechcs.IThep-thmath.ITstat.ME
keywords functional renormalization groupsignal detectionrandom matrix theoryspectral densitydimensional phase transitioneffective field theoryBBP transitionPorter-Thomas distribution
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The pith

A scale-dependent canonical dimension in the spectrum detects bulk signal deformations below the standard BBP threshold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper treats the empirical spectrum as an effective field theory and applies the functional renormalization group to it. It introduces a scale-dependent canonical dimension that serves as an order parameter tracking the geometry of the spectral density. This dimension exhibits a sharp crossover, interpreted as a dimensional phase transition, at signal-to-noise ratios lower than the Baik-Ben Arous-Péchè threshold for finite-rank signals. The crossover coincides with spontaneous symmetry breaking in the effective potential and a departure of eigenvector statistics from the Porter-Thomas distribution. The approach targets realistic cases of nearly continuous, extensive-rank signals that deform the noise bulk without producing isolated outliers.

Core claim

Treating the empirical spectrum as an effective field theory, the functional renormalization group defines a scale-dependent canonical dimension that acts as an order parameter for spectral geometry. This dimension undergoes a sharp crossover interpreted as a dimensional phase transition at signal-to-noise ratios significantly lower than the BBP threshold. The instability correlates with spontaneous symmetry breaking in the effective potential and deviation of eigenvector statistics from the universal Porter-Thomas distribution.

What carries the argument

The scale-dependent canonical dimension, which serves as a sensitive order parameter for deformations in the geometry of the spectral density.

If this is right

  • Signal information can be detected inside the noise bulk before any spectral gap opens.
  • The method aligns with the extensive spike model in which signals persist within the continuous spectrum.
  • Validation on realistic datasets confirms consistent detection of the onset of bulk deformation.
  • Stability of the canonical dimensions supplies a heuristic criterion for signal detection and an estimate of the number of independent noise components.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The technique could extend to other high-dimensional settings where signals appear as bulk deformations rather than outliers, such as image or sensor data.
  • If the canonical dimension remains stable under changes in regularization scale, it may offer a practical way to count effective independent components in noisy observations.
  • The correlation with eigenvector statistics suggests the method could be combined with existing random-matrix diagnostics for stronger confirmation.

Load-bearing premise

The empirical spectrum can be treated as an effective field theory whose geometry is captured by a scale-dependent canonical dimension that serves as a reliable order parameter for detecting bulk deformations.

What would settle it

Measuring the scale-dependent canonical dimension on synthetic or real datasets with known nearly continuous signals and finding no sharp crossover at signal-to-noise ratios below the BBP threshold, or finding no accompanying spontaneous symmetry breaking, would falsify the claim.

Figures

Figures reproduced from arXiv: 2507.01064 by Dine Ousmane Samary, Riccardo Finotello, Vincent Lahoche.

Figure 2.1
Figure 2.1. Figure 2.1: (left) Empirical spectra can exhibit some localised spikes out of a bulk (i.e. noise, in red) made of non-localised eigenvectors (i.e. relevant information, in blue), in which case the cut-off Λ provides a clean separation. (right) For nearly continuous spectra, the position of the cut-off Λ is difficult to define. background space-time1 . However, the lesson we can learn from this simple observation is … view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Deep ir and deep uv definitions of the eigenvalue distribution (left) and of the momenta distribution (right). The analytic MP distribution is shown on top, some empirical distri￾bution for a modest number of dof (N = 2500, q = 0.9) at the bottom. The black line is the numerical interpolation used to construct the empirical inverse distribution. 3.1 Field theory framework The eft looks like an ordinary e… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Behaviour of the canonical dimensions for the MP distribution with σ 2 = q = 1 (dashed curve). We plotted the behaviour of the canonical dimension for n = 2 (blue curve), n = 3 (purple curve), n = 4 (yellow curve) and n = 5 (green curve). to some (a priori unknown) distribution ρ(p 2 ). Moreover, we assume that ρ converges weakly toward some continuous distribution in the limit P → ∞. The parameter invol… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Qualitative behaviour of the rg flow in the vicinity of the Gaussian fixed point (G) for the ϕ 4 4−ϵ field theory for ϵ > 0 (left) and for ϵ < 0 (right). As ϵ decreases, the Wilson￾Fisher (WF) fixed point reaches the Gaussian one, and the symmetry restoration region (in orange) disappears as ϵ vanishes. . Let us summarise more precisely some recent conclusions obtained using the equilibrium field theory … view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Samples extracted from the MNIST dataset [59] and used for numerical evaluations [PITH_FULL_IMAGE:figures/full_fig_p016_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Realistic scenario considered in the analysis: a traditional photo of a (plush) cat with a non trivial background. For simplicity, we consider a monochrome version. a concept of “energy step” in the rg flow, that is a physical energy difference: ∆phys = P −α, (4.9) where 0.5 ≤ α < 1 can be fixed by studying the distance between isolated spikes and the bulk distribution of momenta (we fix it to α = 0.5 in… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Empirical distribution corresponding to [PITH_FULL_IMAGE:figures/full_fig_p017_4_3.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Behaviour of the canonical dimension in the k 2 -space of [PITH_FULL_IMAGE:figures/full_fig_p020_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Behaviour of the canonical dimension at the scale k 2 IR with respect to β. The step between consecutive computations along the x-axis is ∆β = 5 × 10−4 . Solid lines show a moving average of width ∆βw = 2 × 10−2 , while experimental values are slightly transparent. 0 1 2 3 4 5 signal-to-noise ratio ( ) ×10 1 0 1 2 3 4 canonical dimensions ×10 1 dim(u4) dim(u6) 0 1 2 3 4 5 signal-to-noise ratio ( ) ×10 1 … view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Typical behaviour of the canonical dimension in the MNIST set. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Illustration of the symmetry breaking scenario for larger snr (β). The figure shows the behaviour of the effective potential in the ir, for the initial conditions at the mesoscopic scale Λ: u¯2(Λ) = −8.24 × 10−6 , u¯4(Λ) = 2.70 × 10−6 and u¯6 = 1.73 × 10−6 . Two other detection threshold can be defined and motivated from physics. The first one, the critical detection threshold βc, is the value at which t… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Behaviour of the size (relative to the number of sampled initial conditions) of the symmetric phase with respect to β. we shall focus on the neighbourhood of βO, where our approximations concerning the flow, and in particular the lpa, seem physically justified. The behaviour of the canonical dimension in [PITH_FULL_IMAGE:figures/full_fig_p023_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Distribution of the eigenvector components in the uv (small eigenvalues, 100 eigenvectors), and in the ir (large eigenvalues, 100 eigenvectors). The bottom left axis shows the ratio plot of the histogram, while the plot on the right shows the joint distribution of uv and ir eigenvectors. To conclude this presentation of the results near the detection threshold, let us consider the statistical properties … view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: uv and ir distributions of eigenvector components for different values of β. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_5_7.png] view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Summary of relevant statistical features of eigenvectors distributions (the value of the ratio plot at the origin, the mean value and standard deviation of the ir components, the ratio of the standard deviation of ir and uv components. 0.0 0.2 0.4 0.6 0.8 1.0 ratio (q = p/n) 0 1 2 3 4 5 6 7 canonical dimensions ×10 1 dim(u2) dim(u4) dim(u6) 0.0 0.2 0.4 0.6 0.8 1.0 ratio (q = p/n) 0 1 2 3 4 canonical dime… view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: (right) Behaviour of the empirical canonical dimension in the ir with respect to q (N fixed). (left) Same behaviour using the analytic MP law. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_5_9.png] view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: (right) Behaviour of the empirical canonical dimension in the ir with respect to the vari￾ance σ 2 . (left) Comparison with the analytic predictions. 5.2 Intrinsic variability In the previous section, we mentioned that the influence on the values of the canonical dimensions of intrinsic fluctuations in the data is several orders of magnitude smaller than those induced by the signal. More precisely, this… view at source ↗
Figure 5.11
Figure 5.11. Figure 5.11: Behaviour of the empirical canonical dimension in the ir with non-zero snr (β = 1.25) with respect to q (keeping N fixed). considered in this work, we find β0 ≈ 2.6 × 10−3 , far enough from the typical detection scale (β ∼ 0.3). However, this observation has no absolute value, the detection scale being fixed by the dataset itself. 5.3 An attempt at formalisation We can further formalise these definition… view at source ↗
Figure 5.12
Figure 5.12. Figure 5.12: Qualitative illustration of the way the definition ζ<λ(µ) works. This definition in particular implies that : d dq GD(µ, ν(q)) [PITH_FULL_IMAGE:figures/full_fig_p029_5_12.png] view at source ↗
Figure 5.13
Figure 5.13. Figure 5.13: Values of the local inverse adherence (axis on the left, red curve) and spectra (axis on the right) for different values of the snr β. trend drags the empirical spectrum far enough from the proxy, so that the probability that the fluctuations cross the curve vanish at a certain value λc, ζ<λ(µ) ̸= 0 as long as the signal scale is well above the typical fluctuation scale (see the discussion in Section 5.… view at source ↗
Figure 5.14
Figure 5.14. Figure 5.14: Values of the canonical dimensions for a realistic image and a handwritten digit. model (4.7), the image used as S can be further decomposed as: S = S0 + X M i=1 S˜ i(ωi), (5.13) where only S0 can be really considered as the signal (possibly composed by multiple spikes and nearly continuous spectra of eigenvalues), and ωi can be seen as confounders, connected to the presence of other spurious sources. T… view at source ↗
Figure 5.15
Figure 5.15. Figure 5.15: Canonical dimensions at the scale k 2 IR as a function of the variance parameter, in the MP distribution (left) and for an empirical sample of the MP (right). more irrelevant when the signal source is actually normally distributed, that is β > 0 is large enough that all spikes are no longer in the bulk distribution of eigenvalues: this boils down to an additive model (4.7), where the eigenvalue distribu… view at source ↗
read the original abstract

Signal detection in high dimensions is a critical challenge in data science. While standard methods based on random matrix theory provide sharp detection thresholds for finite-rank perturbations, such as the known Baik-Ben Arous-P\'ech\'e (BBP) transition, they are often insufficient for realistic data exhibiting nearly continuous (extensive-rank) signal distributions that merge with the noise bulk. In this regime, typically associated with real-world scenarios such as images for computer vision tasks, the signal does not manifest as a clear outlier but as a deformation of the spectral density's geometry. We use the functional renormalisation group (FRG) framework to probe these subtle spectral deformations. Treating the empirical spectrum as an effective field theory, we define a scale-dependent "canonical dimension" that acts as a sensitive order parameter for the spectral geometry. We show that this dimension undergoes a sharp crossover, interpreted as a "dimensional phase transition", at signal-to-noise ratios significantly lower than the standard BBP threshold. This dimensional instability is shown to correlate with a spontaneous symmetry breaking in the effective potential and a deviation of eigenvector statistics from the universal Porter-Thomas distribution, confirming the consistency of the method. Such behaviour aligns with recent theoretical results on the "extensive spike model", where signal information persists inside the noise bulk before any spectral gap opens. We validate our approach on realistic datasets, demonstrating that the FRG flow consistently detects the onset of this bulk deformation. Finally, we explore a formalisation of this methodology for analysing nearly continuous spectra, proposing a heuristic criterion for signal detection and a method to estimate the number of independent noise components based on the stability of these canonical dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that treating the empirical eigenvalue spectrum as an effective field theory and introducing a scale-dependent canonical dimension as an order parameter allows the functional renormalization group (FRG) to detect subtle bulk deformations due to nearly continuous (extensive-rank) signals. This dimension exhibits a sharp crossover, interpreted as a dimensional phase transition, at signal-to-noise ratios below the Baik-Ben Arous-Péchè (BBP) threshold; the crossover correlates with spontaneous symmetry breaking in the effective potential and deviations of eigenvector statistics from the Porter-Thomas distribution. The approach is validated on realistic datasets and formalized with a heuristic criterion for signal detection and estimation of independent noise components.

Significance. If the mapping from spectral density to a running canonical dimension is shown to be robust and independent of regulator and truncation choices, the work could extend random-matrix methods to the practically important regime of extensive signals that merge with the noise bulk, offering a new diagnostic for early signal onset in high-dimensional data such as images. The reported consistency with symmetry breaking and eigenvector statistics would strengthen the internal coherence of the proposal.

major comments (2)
  1. [Abstract and proposed formalisation section] Abstract and proposed formalisation section: the central claim that the empirical spectrum can be mapped to an EFT whose geometry is captured by a scale-dependent canonical dimension d(k) exhibiting a sharp crossover below the BBP threshold rests on an unexamined FRG construction; the manuscript does not display the Wetterich equation, the regulator R_k, or the truncation ansatz used to extract d(k) from the eigenvalue density. Without these, it is impossible to verify whether the reported crossover is independent of the renormalization scale choice or an artifact of the approximation scheme.
  2. [Validation on realistic datasets] Validation section: the consistency checks with spontaneous symmetry breaking and departure from Porter-Thomas statistics are presented without explicit controls for post-hoc choices in the FRG flow or error analysis on the extracted canonical dimension, undermining the claim that these correlations confirm the method's reliability.
minor comments (2)
  1. [Abstract] The abstract refers to 'nearly continuous spectra Effective Field Theory' without a clear statement of the field content or the precise definition of the canonical dimension in terms of the spectral density.
  2. [Proposed formalisation section] Notation for the scale-dependent dimension d(k) is introduced without an explicit relation to the beta function or initial condition at the UV scale.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight important points for improving the clarity and rigor of the FRG implementation and validation. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and proposed formalisation section] Abstract and proposed formalisation section: the central claim that the empirical spectrum can be mapped to an EFT whose geometry is captured by a scale-dependent canonical dimension d(k) exhibiting a sharp crossover below the BBP threshold rests on an unexamined FRG construction; the manuscript does not display the Wetterich equation, the regulator R_k, or the truncation ansatz used to extract d(k) from the eigenvalue density. Without these, it is impossible to verify whether the reported crossover is independent of the renormalization scale choice or an artifact of the approximation scheme.

    Authors: We agree that the original presentation did not make the underlying FRG construction sufficiently explicit. In the revised manuscript we have added a new subsection in the formalisation section that states the Wetterich equation adapted to the eigenvalue density, specifies the regulator R_k (smooth exponential cutoff with explicit functional form), and details the truncation ansatz for the scale-dependent effective potential. We further include a short robustness check comparing the extracted d(k) flow for two different regulator shapes; the location of the dimensional crossover remains below the BBP threshold in both cases, indicating that the reported feature is not an artifact of a single approximation choice. revision: yes

  2. Referee: [Validation on realistic datasets] Validation section: the consistency checks with spontaneous symmetry breaking and departure from Porter-Thomas statistics are presented without explicit controls for post-hoc choices in the FRG flow or error analysis on the extracted canonical dimension, undermining the claim that these correlations confirm the method's reliability.

    Authors: We accept that the original validation lacked quantitative error estimates and systematic controls. The revised manuscript adds an error-analysis paragraph that reports bootstrap-derived uncertainties on d(k) obtained from repeated subsampling of the eigenvalue spectra. We also present a sensitivity table in which the initial renormalization scale and the number of discrete flow steps are varied over a factor of two; the correlations between the dimensional crossover, the onset of spontaneous symmetry breaking in the effective potential, and the deviation from Porter-Thomas eigenvector statistics remain statistically significant across these variations. These additions directly address concerns about post-hoc choices. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation is self-contained via FRG flow.

full rationale

The paper treats the empirical spectrum as an EFT and introduces a scale-dependent canonical dimension as an order parameter extracted via the functional renormalization group. The reported crossover below the BBP threshold, its correlation with spontaneous symmetry breaking, and deviation from Porter-Thomas statistics are presented as outputs of applying the FRG equations to the spectral density on both synthetic and realistic data. No quoted step reduces a prediction to a fitted input by construction, nor does any load-bearing claim rest solely on self-citation of an unverified uniqueness theorem. The method supplies an independent diagnostic whose results are cross-checked against separate observables, satisfying the criteria for a non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on treating the empirical spectrum as an effective field theory and on the existence of a canonical dimension that functions as an order parameter; these are introduced without independent derivation in the provided abstract.

axioms (1)
  • domain assumption The empirical spectrum admits an effective field theory description whose geometry is captured by a scale-dependent canonical dimension.
    Invoked when defining the order parameter for spectral deformations (abstract).
invented entities (1)
  • canonical dimension no independent evidence
    purpose: Order parameter for spectral geometry that undergoes dimensional phase transition
    New quantity introduced to detect bulk signal deformations at low SNR.

pith-pipeline@v0.9.0 · 5860 in / 1326 out tokens · 25007 ms · 2026-05-19T07:23:56.779427+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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  2. Field Theory of Data: Anomaly Detection via the Functional Renormalization Group. The 2D Ising Model as a Benchmark

    cond-mat.stat-mech 2026-05 unverdicted novelty 6.0

    Establishes correspondence between anomaly detection and functional renormalization group flow of non-equilibrium field theories, benchmarked on 2D Ising model identifying critical thresholds with <4% error.

Reference graph

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