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arxiv: 2606.03429 · v1 · pith:7MIX2XTSnew · submitted 2026-06-02 · 📊 stat.ME · cond-mat.dis-nn· cond-mat.stat-mech· math-ph· math.MP· physics.data-an

Modeling Discrete Data with High-Order Vector Potts Models

Aaron De Clercq , Merijn Moody , Cl\'elia de Mulatier This is my paper

Pith reviewed 2026-06-28 08:56 UTC · model grok-4.3

classification 📊 stat.ME cond-mat.dis-nncond-mat.stat-mechmath-phmath.MPphysics.data-an
keywords q-state spin modelsmaximum entropy modelshigh-order interactionsvector Potts modeldiscrete datagauge transformationsminimally complex modelsmodel selection
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The pith

q-state spin models generalize the vector Potts model to capture arbitrary high-order interactions in discrete data.

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

The paper establishes q-state spin models as a complete family of maximum entropy models for discrete data that extend beyond pairwise interactions to include long-range and high-order terms of any order. This generalization allows modeling more complex correlation patterns than previous approaches limited to binary data or standard Potts models. The authors demonstrate that the statistical properties depend solely on the algebraic structure of interactions through a loop expansion of the partition function, which remains invariant under gauge transformations. Equivalent models can thus be represented with interactions of varying orders. They also provide a closed-form marginal likelihood for minimally complex models to enable efficient selection when fitting to data.

Core claim

q-state spin models form a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions in discrete data. Their statistical properties are fully captured by the algebraic structure of their interactions, as shown via loop expansion of the partition function. Models related by gauge transformations share the same partition function and represent the same abstract statistical model despite different interaction orders.

What carries the argument

q-state spin models, which extend the vector Potts model using algebraic structures for interactions, with loop expansion revealing invariance under gauge transformations.

If this is right

  • Models equivalent under gauge transformations represent the same statistics but can use interactions of different orders.
  • The algebraic structure determines all statistical properties, allowing focus on interaction structure rather than specific orders.
  • Minimally complex models have a closed-form expression for marginal likelihood, enabling fast model selection on discrete data.
  • These models can be applied to infer higher-order correlations in systems like protein sequences or neural activity.

Where Pith is reading between the lines

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

  • Choosing different gauge representations might simplify fitting high-order models to data by reducing effective order.
  • This framework could extend maximum entropy modeling to other discrete variable systems beyond the examples given.
  • The invariance property may help in developing more efficient algorithms for parameter estimation in high-dimensional discrete data.
  • Applying these models to real datasets could reveal previously hidden higher-order structures in complex systems.

Load-bearing premise

The statistical properties of the spin models are fully captured by the algebraic structure of their interactions.

What would settle it

A calculation or simulation where two gauge-equivalent models exhibit different partition functions or statistical properties would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.03429 by Aaron De Clercq, Cl\'elia de Mulatier, Merijn Moody.

Figure 1
Figure 1. Figure 1: Example illustrating equivariant model selection for binary datasets sˆ = {s (i)} N i=1, where each datapoint s (i) is a state of a three-spin system s = (s1, s2, s3) ∈ {−1, +1} 3 . A model selection procedure is equivariant if, for a basis transformation of the dataset T : s → σ, the models obtained from the model selection procedure on sˆ (Bob’s data) and σˆ (Alice’s data) are related by the same transfo… view at source ↗
Figure 2
Figure 2. Figure 2: Representation for q-state spin variables in the complex plane for different values of q. For a given value of q, the color variable αi can take any integer value modulo q, αi ∈ (Z/qZ) n. The corresponding q-state spin variable then takes the complex value si = z αi q , where zq is the first q-th root of unity. The spin variable can thus be represented in the complex plane by a random unit vector which can… view at source ↗
Figure 3
Figure 3. Figure 3: Example for q = 3. One-spin system α = (α1) modeled by a single interaction µ = (1), corresponding to the operator ϕ 1 (α) = exp 2iπ 3 α1  with parameter g1 (and its c.c.). a) Probability distribution of the model for different directions and strengths of the parameter ⃗g∗ 1 = r1⃗u∗ 1 . Each column correspond to a fix direction of ⃗g∗ 1 , indicated by the vector ⃗u∗ 1 in the complex plane representation (… view at source ↗
Figure 4
Figure 4. Figure 4: Example for q = 5 and two types of 1-body operator. One-spin system α = (α1) modeled by two examples of a single-spin interaction: an interaction µ = (1) corresponding to the operator ϕ 1 (α) = exp 2iπ 5 α1  = z α1 5 (and its c.c.) with probability distribution denoted p1(α); and an interaction µ = (2) corresponding to the operator ϕ 2 (α) = exp 2iπ 5 2α1  = z 2α1 5 (and its c.c.) with probability distri… view at source ↗
Figure 5
Figure 5. Figure 5: a) Example of a model for q = 5 with two types of one-body interactions. One-spin system α = (α1) modeled by the operators ϕ 1 (α) = z α1 5 and ϕ 2 (α) = z 2α1 5 (same as in [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 18
Figure 18. Figure 18: Graphical representation of the loops in the 3-state, 3-spin model = {s1, s2, s1s2, s1s 3 , s2s3} for Mand M 1 = T [M 1] = [PITH_FULL_IMAGE:figures/full_fig_p039_18.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration: Representation of q-state spin models as linear maps. We consider a system with n q-state spin variables and a spin model M with K operators. The space V = (Z/qZ) n (on the right) is the system space, in which each of the n basis elements is associated with one of the spin variables. The state of the system is then represented by the random vector α = (α1, . . . , αn) ∈ V, where αj is the col… view at source ↗
Figure 8
Figure 8. Figure 8: (colors online) Visualization of MCMs as factorization in state space. Left. In each row, the left panel shows an MCM based on four discrete variables (α1, α2, α3, α4) with two ICCs represented in different colors. The MCM is represented in a preferred basis, which means that each ICC contains all possible operators over the shown subset of α-variables. The value of q for the discrete variables is not yet … view at source ↗
Figure 9
Figure 9. Figure 9: Examples of MCMs and their GTs to a preferred basis. Each box contains an example of an MCM. Operators are represented using the same convention as in [PITH_FULL_IMAGE:figures/full_fig_p053_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (colors online) Representation of the model discrete probability distribution in state space for two ICCs of [PITH_FULL_IMAGE:figures/full_fig_p058_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: MCM analysis of the US Supreme Court data in its original basis for different embeddings. The original dataset is binary and is embedded in the larger q-state space for this analysis. Top. Optimal MCM found by exhaustive search in the original basis of the data for different values of q. The circles represent the 9 justices labeled by their initials: Ruth Bader Ginsburg (RG), John P. Stevens (JS), David S… view at source ↗
Figure 13
Figure 13. Figure 13: shows the best MCM found overall for different values of q, by performing first an exhaustive search for the optimal basis, and then an exhaustive search for the optimal MCM on that basis. The nine independent operators of the best basis are represented by squares in the figure. Except for one single spin operator on AS, all these operators are pairwise and identify interactions between the same pairs of … view at source ↗
Figure 14
Figure 14. Figure 14: MCM analysis of the Big Five Personality Test data [54] for varying values of q. a) Illustration of the two discretization schemes described in [PITH_FULL_IMAGE:figures/full_fig_p069_14.png] view at source ↗
read the original abstract

Modeling high-dimensional data is challenging, yet essential to understanding many complex systems. Maximum entropy models such as Ising and Potts models have been used extensively to capture pairwise interactions from correlation patterns in data, allowing to infer graphical representations of complex systems from observations (e.g., from protein sequences or neural population activity). Recently, there has been growing interest in modeling higher-order correlation patterns involving simultaneously three or more variables. While progress has been made in binary data with high-order Ising models, we extend this framework to the more general case of discrete data. We introduce q-state spin models, a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions. In the pairwise case, our models allow for more diverse interaction types compared to the standard vector Potts model. We discuss their statistical interpretation with examples and relate them to discrete Fourier analysis. Using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions. We define gauge transformations under which this structure, and thus the partition function, remains invariant. Models equivalent under gauge transformations can be seen as different representations of the same abstract statistical model, despite generally having interactions of different orders, extending results from the binary case. For practical application to data analysis, we focus on a subset of models known in the binary case as Minimally Complex Models, generalizing them to discrete data. We obtain a closed-form expression for the marginal likelihood of these models, enabling fast model selection. We illustrate their use with simple real-world examples.

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 / 1 minor

Summary. The paper introduces q-state spin models as a complete family of maximum entropy models for discrete (q-state) data that generalize the vector Potts model to arbitrary high-order and long-range interactions. It claims that a loop expansion of the partition function demonstrates that statistical properties are fully determined by the algebraic structure of the interactions, leading to gauge transformations under which the structure and partition function are invariant (allowing equivalent models of different orders). It further derives a closed-form marginal likelihood for the subset of Minimally Complex Models to enable fast model selection and illustrates the approach with real-world examples.

Significance. If the central claims on completeness, exact capture via loop expansion, and closed-form marginal likelihood hold, the work would provide a principled extension of high-order maxent models from binary to general discrete data, with direct implications for graphical modeling in fields such as protein sequence analysis and neural data. The explicit treatment of gauge equivalence and the practical model-selection formula are potential strengths for reproducibility and applicability.

major comments (2)
  1. [Abstract / loop-expansion section] Abstract and the section introducing the loop expansion: the claim that 'using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions' is load-bearing for the completeness and gauge-invariance results, yet the provided text gives no explicit derivation, truncation argument, or demonstration that the series is exact (rather than perturbative) or that omitted diagrams preserve the claimed invariance. This directly affects whether models of different orders are rigorously equivalent.
  2. [Section on Minimally Complex Models / marginal likelihood] The derivation of the closed-form marginal likelihood for Minimally Complex Models (mentioned in the abstract) is central to the practical contribution; without the explicit steps or assumptions under which the expression is obtained, it is impossible to assess whether it generalizes the binary case without introducing hidden parameters or approximations.
minor comments (1)
  1. [Abstract] The abstract states that pairwise models 'allow for more diverse interaction types compared to the standard vector Potts model' but does not specify which additional interaction types are enabled or how they relate to the Fourier-analysis connection mentioned later.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and positive assessment of the potential significance of our work on q-state spin models. We address each major comment below and will revise the manuscript to provide the requested explicit derivations and clarifications.

read point-by-point responses

  1. Referee: [Abstract / loop-expansion section] Abstract and the section introducing the loop expansion: the claim that 'using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions' is load-bearing for the completeness and gauge-invariance results, yet the provided text gives no explicit derivation, truncation argument, or demonstration that the series is exact (rather than perturbative) or that omitted diagrams preserve the claimed invariance. This directly affects whether models of different orders are rigorously equivalent.

    Authors: We agree that the loop expansion underpins the claims of completeness and gauge invariance. The current manuscript provides a high-level outline of the approach and its implications but does not include the full step-by-step derivation or truncation argument. In the revised version, we will expand the relevant section (and add an appendix if needed) with the explicit loop expansion, including the series terms, demonstration that it is exact for the partition function in this context, and verification that omitted diagrams preserve the algebraic invariance. This will rigorously establish the equivalence of models under gauge transformations. revision: yes

  2. Referee: [Section on Minimally Complex Models / marginal likelihood] The derivation of the closed-form marginal likelihood for Minimally Complex Models (mentioned in the abstract) is central to the practical contribution; without the explicit steps or assumptions under which the expression is obtained, it is impossible to assess whether it generalizes the binary case without introducing hidden parameters or approximations.

    Authors: We acknowledge that while the manuscript states the closed-form result and its utility for model selection, the explicit derivation steps and assumptions are not detailed in the provided text. In the revision, we will include the full derivation, specifying the assumptions (e.g., the structure of Minimally Complex Models) and confirming that the expression generalizes the binary case exactly without additional parameters or approximations. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines q-state spin models as a maxent family generalizing vector Potts models, then applies a loop expansion of the partition function to relate statistical properties to interaction algebra and introduces gauge transformations preserving the structure. These are presented as direct analytical consequences of the model definition and standard perturbative techniques rather than any reduction of a claimed prediction back to fitted inputs, self-citations, or ansatzes by construction. No load-bearing step equates an output quantity to its own inputs via redefinition or renaming; the completeness and invariance claims rest on explicit expansions and transformations whose validity is independent of the target results. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Review based solely on abstract; full text unavailable so ledger is necessarily incomplete. No explicit free parameters or invented entities are quantified in the abstract.

axioms (2)
  • domain assumption Maximum entropy principle suffices to capture essential statistics from correlation patterns
    Standard assumption invoked when introducing the models in the abstract.
  • ad hoc to paper Loop expansion of the partition function fully determines statistical properties from interaction algebra
    Central technical step stated in the abstract.
invented entities (1)
  • q-state spin models no independent evidence
    purpose: Complete family of maximum entropy models for high-order discrete interactions
    New modeling class introduced to generalize vector Potts models.

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discussion (0)

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