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arxiv: 2504.03631 · v4 · submitted 2025-04-04 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn

Diagrammatics of free energies with fixed variance for high-dimensional data

Tobias K\"uhn This is my paper

Pith reviewed 2026-05-22 21:09 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nn
keywords Feynman diagramsfree energyfixed varianceperturbative expansionspin systemsIsing modelhigh-dimensional statisticsmessage-passing algorithms
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The pith

Feynman diagrams organize perturbative free energies when variance is held fixed, without assuming a Gaussian expansion point.

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

High-dimensional systems with many interacting parts are fully described by their free energy, yet standard perturbative methods often become unwieldy because their terms lack systematic organization. This work extends an earlier diagrammatic framework to the case of free energies computed at fixed variance. The diagrams remain valid even when the underlying theory is non-Gaussian, which permits direct application to the free energy of a spin system previously studied by Maillard et al. The approach completes the thermodynamic-limit perturbative derivation for that system, supplies resummed expressions for entropies from limited samples, and yields new results for the Ising model. These advances matter for high-dimensional statistics and complex networks, where variance constraints are common but Gaussian assumptions are not.

Core claim

The central claim is that a set of Feynman diagrams can be defined for the perturbative expansion of the free energy when the variance of the fluctuating variables is held fixed. This organization does not require the expansion to begin from a Gaussian theory and therefore applies directly to non-Gaussian models such as the spin system of Maillard et al. 2019, allowing the authors to finish the derivation of its free energy in the thermodynamic limit.

What carries the argument

Feynman diagrams for free-energy perturbations at fixed variance, which systematically collect and organize all diagrammatic contributions without reference to a Gaussian reference state.

If this is right

  • The completed free-energy expression supplies the basis for message-passing algorithms on the spin system in the thermodynamic limit.
  • Resummed series give practical estimates of entropy for poorly sampled high-dimensional data using only limited statistics.
  • The same diagrams produce revised expressions for the Ising-model free energy that were unavailable under the earlier Gaussian-centered framework.
  • The method extends naturally to other high-dimensional statistics problems such as matrix factorization.
  • The approach applies to the statistical mechanics of complex networks where variance is fixed by data constraints.

Where Pith is reading between the lines

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

  • The fixed-variance condition may allow systematic improvement of variational approximations in data-limited regimes beyond the examples treated here.
  • Because the diagrams do not presuppose Gaussianity, the same machinery could be tested on other non-Gaussian spin-glass or Potts models.
  • The resummed entropy estimators could be validated by subsampling from large synthetic datasets whose exact entropies are known.

Load-bearing premise

The diagrammatic rules remain complete and closed when the theory is non-Gaussian yet the variance is constrained.

What would settle it

Compute the next few orders of the diagrammatic series for the Ising free energy and check whether they reproduce the known high-temperature or low-temperature expansions term by term.

Figures

Figures reproduced from arXiv: 2504.03631 by Tobias K\"uhn.

Figure 1
Figure 1. Figure 1: Sketch of the cancellation mechanism of one-line reducible and cactus diagrams. 2.1.2. Diagrammatics of Γ K As for the first Legendre transform (with respect to j only) the first two orders of Γ K are given as Γ K [m, v] = ΓK 0 [m, v] − − + . . . (12) The contribution (7) generates all connected diagrams, according to the well￾established linked-cluster theorem [31][13, appendix A.3]. Then, as shown in [13… view at source ↗
Figure 2
Figure 2. Figure 2: Sketch of (a) a cactus diagram and (b) an improper pseudo-cactus diagram. 2.1.4. Cancellations in higher orders We now proceed to higher orders, leveraging the ideas employed for the second-order terms. To do this, we introduce the notion of a counter diagram: it has the same form as a diagram in the diagrammatic expansion of W, but it contributes with the opposite sign. We will see that there are counter … view at source ↗
Figure 3
Figure 3. Figure 3: Ring diagrams with possible pairs of nodes with identical indices indicated by dashed lines. same one we obtain from assembling the sub-diagrams and adding them afterwards. The only aspect that is different than in [13] is that we stick together the subdiagrams at two legs, instead of one. The resulting differences in the symmetry factors are the same, no matter if we construct the diagrams with ∂Γ K V ∂v … view at source ↗
read the original abstract

Systems with many interacting stochastic constituents are fully characterized by their free energy. Computing this quantity is therefore the objective of various approaches, notably perturbative expansions, which are applied in problems ranging from high-dimensional statistics to complex systems. However, a lot of these techniques are complicated to apply in practice because they lack a sufficient organization of the terms of the perturbative series. In this manuscript, we tackle this problem by using Feynman diagrams, extending a framework introduced earlier to the case of free energies at fixed variances. This diagrammatics do not require the theory to expand around to be Gaussian, which allows its application to the free energy of a spin system studied to derive message-passing algorithms by Maillard et al. 2019. We complete their perturbative derivation of the free energy in the thermodynamic limit. Furthermore, we derive resummations to estimate the entropies of poorly sampled systems requiring only limited statistics and we revisit earlier approaches to compute the free energy of the Ising model, revealing new insights due to the extension of our framework to the free energy at fixed variances. We expect our approach to be particularly useful for problems of high-dimensional statistics, like matrix factorization, and the study of complex networks.

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

1 major / 2 minor

Summary. The manuscript introduces a Feynman diagrammatic framework for computing free energies with fixed variance in high-dimensional interacting systems. It extends an earlier framework to non-Gaussian cases, allowing application to the spin system studied by Maillard et al. 2019. The paper completes the perturbative derivation of the free energy in the thermodynamic limit, derives resummations for estimating entropies from limited statistics, and revisits the Ising model free energy with new insights.

Significance. If the diagrammatic rules correctly handle non-Gaussian contributions by reorganizing higher cumulants under fixed variance, this work could offer a practical tool for free energy calculations in complex systems and high-dimensional statistics, such as matrix factorization and network studies. The completion of the Maillard et al. derivation and the resummation techniques are notable strengths if verified.

major comments (1)
  1. The central extension to non-Gaussian theories with only fixed variance relies on the assumption that the diagrammatic organization remains complete without a Gaussian expansion point. However, standard perturbative methods depend on vanishing higher cumulants via Wick's theorem or equivalent. The manuscript needs to provide explicit rules or a derivation showing how third- and higher-order cumulants are accounted for or eliminated in the diagram set, as this is load-bearing for the claim of completing the Maillard et al. 2019 derivation.
minor comments (2)
  1. The abstract mentions 'resummations to estimate the entropies of poorly sampled systems' but does not specify the order of the resummation or the statistics required; clarifying this would improve readability.
  2. Ensure that all references to prior work, including Maillard et al. 2019, are consistently cited with full details.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: The central extension to non-Gaussian theories with only fixed variance relies on the assumption that the diagrammatic organization remains complete without a Gaussian expansion point. However, standard perturbative methods depend on vanishing higher cumulants via Wick's theorem or equivalent. The manuscript needs to provide explicit rules or a derivation showing how third- and higher-order cumulants are accounted for or eliminated in the diagram set, as this is load-bearing for the claim of completing the Maillard et al. 2019 derivation.

    Authors: We thank the referee for identifying this key point. Our diagrammatic rules are obtained by reorganizing the cumulant expansion of the free-energy functional subject to the fixed-variance constraint; the constraint is enforced at the level of the measure, so the diagrams are defined without reference to a Gaussian expansion point and without invoking Wick's theorem. Third- and higher-order cumulants therefore appear as additional vertex types whose contributions are systematically included according to the same combinatorial rules already introduced in our earlier Gaussian-variance framework. While this organization is implicit in the derivation presented in Sections 2 and 3, we agree that an explicit statement of the rules for non-Gaussian cumulants would strengthen the manuscript and make the completion of the Maillard et al. derivation fully transparent. We will therefore add a short dedicated subsection (or appendix) that derives the diagrammatic rules from the constrained cumulant-generating function and illustrates how the higher cumulants enter the diagram set. revision: yes

Circularity Check

0 steps flagged

Extension of prior diagrammatic framework to fixed-variance case relies on one non-load-bearing self-reference but remains independently organized

full rationale

The manuscript extends an earlier framework (likely by the same author) to free energies at fixed variance and completes the Maillard et al. 2019 perturbative derivation for a spin system. No equation or step in the provided abstract or outline reduces a central prediction or completeness claim to a fitted input or self-citation by construction. The key assertion—that diagrammatics apply without a Gaussian expansion point—functions as an independent reorganization rather than a definitional loop or forced resummation. External benchmarks (thermodynamic-limit completion, entropy estimation) are not shown to collapse into the cited prior work. This yields a minor self-citation score without compromising the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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