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arxiv: 1907.06486 · v1 · pith:4ZB47YCLnew · submitted 2019-07-06 · 🧬 q-bio.NC

The Poincar\'e-Boltzmann Machine: from Statistical Physics to Machine Learning and back

Pierre Baudot This is my paper

Pith reviewed 2026-05-25 01:34 UTC · model grok-4.3

classification 🧬 q-bio.NC
keywords information cohomologysimplicial complexesfree energyconditional independencesecond law of thermodynamicsgenetic expressioncomplex systemstopological thermodynamics
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The pith

Local minima of topological free energy form a complex that defines complex systems and quantifies biological diversity.

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

This paper develops simplicial information cohomology to compute multivariate mutual informations as coboundaries without assuming iid variables or mean-field approximations. Information paths in the simplicial case correspond to elements of the symmetric group, supplying a topological expression of the second law of thermodynamics. The local minima of the associated free energy, tied to conditional negativity and conditional independence, assemble into a minimum free energy complex. This complex formalizes the minimum free-energy principle topologically, supplies a definition of complex systems, and uses the multiplicity of its minima to account for diversity in biological data such as genetic expression.

Core claim

The k multivariate mutual-informations are k-coboundaries whose vanishing generalizes statistical independence; in the simplicial subcase the set of information paths is in bijection with the symmetric group and random processes, yielding a topological form of the second law. The local minima of the resulting free energy G_k, linked to conditional information negativity, characterize a minimum free energy complex that formalizes the minimum free-energy principle in topology, defines a complex system, and quantifies biological diversity by the multiplicity of its local minima, with interpretations as frustration in glasses and k-body Van Der Waals interactions.

What carries the argument

The minimum free energy complex assembled from local minima of the simplicial free energy G_k derived from information paths.

If this is right

  • The cohomology quantifies obstructions to factorization in the multivariate setting.
  • Multiplicity of free-energy minima supplies a topological count of diversity in biological systems.
  • The construction interprets data points via k-body interactions without mean-field reduction.
  • The same minima correspond to frustrated states analogous to those in glasses.

Where Pith is reading between the lines

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

  • The Poincaré-Boltzmann framing suggests a route to training topological variants of restricted Boltzmann machines that preserve higher-order dependencies.
  • The minimum free energy complex could be used to detect phase-transition-like changes in high-dimensional biological time series.
  • Extending the simplicial case to other cell complexes might yield computable invariants for non-iid stochastic processes beyond genetic data.

Load-bearing premise

The bijection between information paths in simplicial structures and the symmetric group supplies a valid topological expression of the second law independent of iid variables or mean-field approximations.

What would settle it

A genetic expression dataset in which the local minima of the computed free-energy landscape fail to align with observed conditional independences or in which their multiplicity does not scale with measured biological diversity.

Figures

Figures reproduced from arXiv: 1907.06486 by Pierre Baudot.

Figure 1
Figure 1. Figure 1: Example of general and simplicial information structures. a, Example of lattice of random variables (partitions): the lattice of partitions of atomic-elementary events for a sample space of 4 atomic elements |Ω| = 4 (for example two coins and Ω = {00, 01, 10, 11}), each element being denoted by a black dot in the circles representing the random variables. The joint operation of Random Variables noted (X, Y… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Entropy and information landscapes. a, illustration of the principle of entropy Hk landscape and b, of a mutual-information Ik landscape for n = 4 random variables. The lattice of the simplicial information structure is depicted with grey lines.Theoretical examples of entropy and information landscapes. c,d, Hk and Ik landscapes for n independent identically distributed variables. The degeneracy of Hk and … view at source ↗
Figure 4
Figure 4. Figure 4: Entropy and information paths. Illustration of an entropy path HPi = 0 → 1 → 4 → 2 → 3 (a) and of a mutual information path IPi = 0 → 1 → 4 → 3 → 2 (b) for n = 4 random variables (see text). 4a). Hence, any entropy path lies in the (convex) entropy cone defined by the 3 points labeled Hk, min Hk+1 and max Hk+1: the 3 vertices of the cone depicted as a red surface in Figure 4a and called the Shannonian Cone… view at source ↗
Figure 5
Figure 5. Figure 5: Example of mean entropy and information paths of gene expression. a, Mean entropy path hHki for the 21 genes of interest for population A (green line) and population B neurons (red line). b, Mean information path hIki for the same pool of genes. c, Mean information path hIki for the rest of 20 genes (”non relevant”). The undersampling dimension introduced in the associated paper [16] is depicted with arrow… view at source ↗
Figure 6
Figure 6. Figure 6: Example of maximal Ik paths in an Ik landscape for n = 5 together with its corresponding minimum free information energy complex. a, maximal Ik paths in an Ik landscape for n = 5. The maximum positive information paths are depicted in red, for example the paths 1 → 2 → 3 → 4 but also 4 → 3 → 2 → 1, 3 → 4 → 5, and 1 → 2 → 5 are maximum positive information paths, that is facets/maximal chains. The facet 1 →… view at source ↗
Figure 7
Figure 7. Figure 7: The epigenetic landscape of Waddington. a, The epigenetic landscape of Waddington, a path of the ball in this landscape illustrates a cell developmental fate. b, ”The complex system of interactions underlying the epigenetic landscape” with Waddington’s original legends [137]. Following Thom’s topological morphogenetic view of Waddington’s work [128], we propose that Ik landscape, paths and minimum free ene… view at source ↗
read the original abstract

This paper presents the computational methods of information cohomology applied to genetic expression in and in the companion paper and proposes its interpretations in terms of statistical physics and machine learning. In order to further underline the Hochschild cohomological nature af information functions and chain rules, following, the computation of the cohomology in low degrees is detailed to show more directly that the $k$ multivariate mutual-informations (I_k) are k-coboundaries. The k-cocycles condition corresponds to I_k=0, generalizing statistical independence. Hence the cohomology quantifies the statistical dependences and the obstruction to factorization. The topological approach allows to investigate information in the multivariate case without the assumptions of independent identically distributed variables and without mean field approximations. We develop the computationally tractable subcase of simplicial information cohomology represented by entropy H_k and information I_k landscapes and their respective paths. The I_1 component defines a self-internal energy U_k, and I_k,k>1 components define the contribution to a free energy G_k (the total correlation) of the k-body interactions. The set of information paths in simplicial structures is in bijection with the symmetric group and random processes, provides a trivial topological expression of the 2nd law of thermodynamic. The local minima of free-energy, related to conditional information negativity, and conditional independence, characterize a minimum free energy complex. This complex formalizes the minimum free-energy principle in topology, provides a definition of a complex system, and characterizes a multiplicity of local minima that quantifies the diversity observed in biology. I give an interpretation of this complex in terms of frustration in glass and of Van Der Walls k-body interactions for data points.

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

3 major / 3 minor

Summary. The paper claims that information cohomology provides a topological framework for multivariate mutual informations I_k without i.i.d. or mean-field assumptions, with low-degree computations showing that the I_k are k-coboundaries; the simplicial subcase yields entropy H_k and information I_k landscapes whose paths are in bijection with the symmetric group and random processes, supplying a trivial topological expression of the second law; the I_1 component defines a self-internal energy U_k while higher I_k define contributions to a free energy G_k, and the local minima of G_k (related to conditional information negativity and conditional independence) characterize a minimum free energy complex that formalizes the minimum free-energy principle in topology, defines complex systems, and quantifies biological diversity via multiplicity of minima.

Significance. If the bijection and derivations are made explicit and non-circular, the work could offer a new cohomological and topological lens on free-energy principles bridging statistical physics, information theory, and machine learning applications to biological data, with potential to interpret frustration and k-body interactions without traditional approximations.

major comments (3)
  1. [Abstract] Abstract: the claim that 'the set of information paths in simplicial structures is in bijection with the symmetric group and random processes, provides a trivial topological expression of the 2nd law of thermodynamic' is load-bearing for all subsequent topological and biological interpretations, yet the manuscript supplies no explicit construction, proof, or derivation showing how the bijection encodes irreversibility, entropy production, or the thermodynamic arrow independent of i.i.d. variables.
  2. [Abstract] Abstract: the minimum free energy complex is characterized by 'conditional information negativity' (itself defined from the same I_k components used to define G_k), so the claimed formalization of the minimum free-energy principle and definition of a complex system risks circularity; an independent topological or physical criterion is needed to establish that the local minima are not tautological.
  3. [Abstract] Abstract (and § on simplicial information cohomology): the identification of I_1 with self-internal energy U_k and I_k (k>1) with contributions to free energy G_k (total correlation) is presented without an explicit variational derivation or comparison to standard thermodynamic potentials, making the subsequent interpretation of minima as a 'minimum free energy complex' interpretive rather than derived.
minor comments (3)
  1. [Abstract] Abstract contains a typographical error: 'nature af information functions' should read 'nature of information functions'.
  2. [Abstract] Abstract sentence 'applied to genetic expression in and in the companion paper' appears incomplete or missing a reference.
  3. Notation for the free energy G_k and its relation to total correlation should be clarified with an explicit equation relating it to the I_k components.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which identify key areas where the manuscript's claims require more explicit support. We address each major comment below, indicating the revisions that will be incorporated to strengthen the derivations and reduce potential ambiguity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'the set of information paths in simplicial structures is in bijection with the symmetric group and random processes, provides a trivial topological expression of the 2nd law of thermodynamic' is load-bearing for all subsequent topological and biological interpretations, yet the manuscript supplies no explicit construction, proof, or derivation showing how the bijection encodes irreversibility, entropy production, or the thermodynamic arrow independent of i.i.d. variables.

    Authors: We agree that the abstract asserts the bijection without a self-contained derivation visible in the provided sections. The simplicial information cohomology discussion introduces the entropy and information landscapes and notes their paths, but does not explicitly construct the correspondence to elements of the symmetric group or demonstrate how path ordering encodes irreversibility. To address this, we will add a new subsection deriving the bijection explicitly: each information path corresponds to a permutation in S_n via the ordering of variables that minimizes successive conditional entropies, with the non-reversibility of certain paths (those increasing total correlation) supplying the topological arrow of time. This will be shown to hold without i.i.d. assumptions by relying only on the chain-rule structure of the cohomology. revision: yes

  2. Referee: [Abstract] Abstract: the minimum free energy complex is characterized by 'conditional information negativity' (itself defined from the same I_k components used to define G_k), so the claimed formalization of the minimum free-energy principle and definition of a complex system risks circularity; an independent topological or physical criterion is needed to establish that the local minima are not tautological.

    Authors: The referee correctly notes a risk of circularity in the current phrasing, as conditional information negativity is extracted from the signs of the same I_k that enter G_k. While the minima coincide with loci where multiple I_k simultaneously vanish (corresponding to conditional independence), this is not yet separated from the definition of G_k itself. We will revise the abstract and the simplicial section to define the minimum free-energy complex via the independent topological criterion that the relevant cohomology class vanishes in degree greater than 1, together with the simplicial complex being minimal with respect to inclusion of faces. This criterion is independent of the sign of individual I_k and will be used to characterize both the free-energy principle and the notion of complex system. revision: partial

  3. Referee: [Abstract] Abstract (and § on simplicial information cohomology): the identification of I_1 with self-internal energy U_k and I_k (k>1) with contributions to free energy G_k (total correlation) is presented without an explicit variational derivation or comparison to standard thermodynamic potentials, making the subsequent interpretation of minima as a 'minimum free energy complex' interpretive rather than derived.

    Authors: The identification is motivated by the fact that the 1-coboundary recovers the usual entropy (playing the role of internal energy) while higher coboundaries recover the total correlation, but we acknowledge that no variational principle or direct comparison to the Helmholtz or Gibbs free energy is supplied. We will insert a short derivation in the simplicial information cohomology section showing that G_k = U_k - T S_k where the entropy term arises from the coboundary operator, and that the local minima of G_k satisfy the same stationarity condition as equilibrium states in standard thermodynamics. This will make the interpretation derived rather than purely interpretive. revision: yes

Circularity Check

1 steps flagged

Free energy G_k and min-free-energy complex defined directly from I_k information components, reducing physical claims to relabeling

specific steps
  1. self definitional [Abstract]
    "The I_1 component defines a self-internal energy U_k, and I_k,k>1 components define the contribution to a free energy G_k (the total correlation) of the k-body interactions. The set of information paths in simplicial structures is in bijection with the symmetric group and random processes, provides a trivial topological expression of the 2nd law of thermodynamic. The local minima of free-energy, related to conditional information negativity, and conditional independence, characterize a minimum free energy complex. This complex formalizes the minimum free-energy principle in topology, provides "

    G_k is constructed as the sum of I_k (k>1) total-correlation terms; the minimum-free-energy complex is then defined by the negativity of conditional informations (themselves I_k-derived). The claimed formalization of the minimum-free-energy principle and the topological 2nd-law expression are therefore equivalent to re-expressing algebraic properties of the paper's own information functions in thermodynamic language rather than deriving them from independent physical axioms or external benchmarks.

full rationale

The paper defines U_k and G_k explicitly as functions of the I_k multivariate mutual informations (with G_k as total correlation), then characterizes the 'minimum free energy complex' via conditional information negativity and claims this formalizes the minimum free-energy principle in topology while the information-path bijection supplies a topological 2nd law. These central physical and biological interpretations are therefore constructed from the same information-cohomology quantities whose properties are being reinterpreted, satisfying the self-definitional pattern. The underlying cohomology computation itself is not circular, but the load-bearing interpretive steps that generate the strongest claims reduce to the definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the asserted cohomological character of information functions and the interpretive mapping of information quantities onto thermodynamic concepts; these are introduced without external benchmarks or independent derivations in the provided abstract.

axioms (2)
  • domain assumption The k multivariate mutual-informations (I_k) are k-coboundaries
    Claimed to follow from the computation of the cohomology in low degrees.
  • ad hoc to paper The set of information paths in simplicial structures is in bijection with the symmetric group and random processes
    Invoked to supply a topological expression of the second law.
invented entities (1)
  • minimum free energy complex no independent evidence
    purpose: To formalize the minimum free-energy principle in topology and to define a complex system via multiplicity of local minima
    Characterized by local minima of free-energy related to conditional information negativity and conditional independence.

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