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Shannon invariants: A scalable approach to information decomposition

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Pith's one-line read This paper proves that the average degree of redundancy and the average degree of vulnerability are Shannon invariants, so they can be computed from Shannon entropies without choosing a PID measure.

desk verdict The invariant identities are correct and worth publishing, but the interpretive claims and empirical sections need substantial tightening. read the letter →

arxiv 2504.15779 v1 pith:VOOC4NNU submitted 2025-04-22 cs.IT cs.AIcs.LGmath.ITnlin.AOphysics.data-an

Aaron J. Gutknecht , Fernando E. Rosas , David A. Ehrlich , Abdullah Makkeh , Pedro A. M. Mediano , Michael Wibral This is my paper · ORCID
classification cs.ITcs.AIcs.LGmath.ITnlin.AOphysics.data-an MSC 94A1794A15
keywords partialinformationdecompositionShannoninvariantsredundancy-synergyindexdualmutualhigher-orderdeepneuralnetworks
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Partial information decomposition splits the information a set of sources carries about a target into many fine-grained atoms, but the atoms cannot be uniquely defined and their number explodes with the number of sources. This paper shows that two aggregate quantities of those atoms, the average degree of redundancy and the average degree of vulnerability, are Shannon invariants: their values are fixed by ordinary Shannon entropies alone and are therefore the same under every decomposition. The redundancy average equals the ratio of the sum of individual mutual informations to the joint mutual information, and the vulnerability average equals the corresponding ratio for conditional mutual informations. Because these ratios are easy to compute, the framework brings information-decomposition reasoning to systems with many variables, such as the layers of deep neural networks, and gives the long-used redundancy-synergy index a precise meaning.

What carries the argument

The load-bearing object is the PID atom lattice: each information atom $\Pi(\alpha)$ is indexed by an antichain $\alpha$ of source subsets, and the consistency equation (A7) fixes how atoms compose every mutual information term. On this lattice the paper puts two integer counting functions: $r(\alpha)$, the number of singleton source sets $\{i\}\in\alpha$, and $v(\alpha)$, the number of indices that appear in every subset of $\alpha$. The key identity is a double-counting argument: summing the marginal mutual informations over $i$ counts each atom once per accessible singleton source, giving $\sum_i I(X_i;Y)=\sum_\alpha r(\alpha)\Pi(\alpha)$, and summing the conditional mutual informations counts each atom once per critically necessary source, giving $\sum_j I(X_j;Y|X_{-j})=\sum_\alpha v(\alpha)\Pi(\alpha)$. Normalizing by $I(X;Y)$ converts these identities into the two Shannon-invariant averages.

What would settle it

For a three-source distribution, compute $\bar r$ and $\bar v$ from their entropy formulas, then compute the atom-level sums under any decomposition satisfying Eq. (A7) that permits a negative atom. If the inequality $I_r^{(0)}/I(X;Y) \ge 1-\bar r$ is violated, or if $\bar r<1$ occurs while every source-level-synergy atom $I_r^{(0)}$ is zero, the interpretive reading of the invariants fails even though the entropy formulas still hold.

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Extended reading notes

Core claim

The paper's central claim is that although individual PID atoms are not Shannon quantities, certain averages over them are. For $n$ sources $X_1,\dots,X_n$ and a target $Y$, it defines the average degree of redundancy as $\bar r = \sum_{i=1}^n I(X_i;Y) / I(X;Y)$ and proves in Proposition 1 that this equals the atom-weighted average of $r(\alpha)$, the number of singleton sources through which an atom is accessible. It defines the average degree of vulnerability as $\bar v = \sum_{j=1}^n I(X_j;Y|X_{-j}) / I(X;Y)$ and proves in Proposition 2 that this equals the atom-weighted average of $v(\alpha)$, the number of sources on which an atom critically depends. The paper then reads the redundancy-synergy index through the first average, $\mathrm{RSI} = (\bar r - 1)I(X;Y)$, and introduces a dual index $\mathrm{DRSI} = (1-\bar v)I(X;Y)$, showing both are Shannon invariants. If the claims are right, these averaged quantities are properties of the joint distribution itself, not of any chosen decomposition.

Load-bearing premise

The interpretation that $\bar r<1$ means source-level synergy and $\bar v<1$ means robustness, and the bounds in Appendix B, all assume that no information atom is negative; several PID measures allow negative atoms when there are more than two sources.

Editorial extensions

If this is right

  • The redundancy-synergy index is no longer a heuristic: $\mathrm{RSI}=(\bar r-1)I(X;Y)$, so its sign and magnitude are set by how the average degree of redundancy deviates from 1.
  • The new dual index satisfies $\mathrm{DRSI}=(1-\bar v)I(X;Y)$, giving a second, vulnerability-based balance measure that is also computable from Shannon entropies.
  • Under non-negative atoms, $\bar r<1$ forces some source-level synergy and $\bar v<1$ forces some robust information, with quantitative bounds in Appendix B for $\bar r>1$ and $\bar v>1$.
  • Shannon invariants require only entropies of small subsets, so the computational cost scales linearly with the number of sources, unlike a full PID.
  • In the trained feed-forward classifier studied here, redundancy increases across layers and training while vulnerability decreases; in the autoencoder, larger bottlenecks show higher redundancy and lower vulnerability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If these invariants are as fundamental as claimed, they can serve as consistency checks: any proposed PID measure must reproduce the same $\bar r$ and $\bar v$ from its atoms, so deviations signal a violation of the counting identities.
  • The same double-counting logic likely extends to other coefficient choices: any linear combination of atoms whose coefficients are functions of $r(\alpha)$ and $v(\alpha)$ may be Shannon-invariant whenever the coefficient pattern collapses by the same counting argument, suggesting a larger family of invariants than the two introduced here.
  • The vulnerability average could be deployed directly as a robustness diagnostic for machine-learning representations, for example comparing models of different widths or after pruning, because it needs only conditional mutual informations that can be estimated from finite samples.
  • The paper's network analyses treat the training set as the full population; applying the same invariants on held-out data would test whether the redundancy and vulnerability trends reflect generalization rather than memorization of the training set.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper introduces two aggregate quantities, the average degree of redundancy rbar and the average degree of vulnerability vbar, defined respectively as the ratio of the sum of marginal mutual informations to the joint mutual information (Eq. 6) and the ratio of the sum of conditional mutual informations to the joint mutual information (Eq. 10). The main theoretical results, Propositions 1 and 2, show that these quantities can be rewritten as weighted averages of PID atom degrees (Eqs. 5 and 9), and are therefore independent of the choice of PID measure. The paper further shows that the redundancy-synergy index satisfies RSI = (rbar-1)I(X;Y) (Cor. 1), and introduces a dual index DRSI = (1-vbar)I(X;Y) (Prop. 4, Cor. 2). It then applies these invariants to quantized deep neural networks, reporting layer- and training-dependent trends for an MNIST classifier and a face autoencoder. The core derivations are elementary counting arguments based on the PID consistency equation (A7), and the paper explicitly notes that the interpretational claims require nonnegative PID atoms.

Significance. The theoretical core is sound and potentially useful: if the nonnegativity assumption holds, rbar and vbar provide scalable summaries of high-order information structure that circumvent the super-exponential cost of full PID, and they yield a rigorous interpretation of the widely used RSI as well as a new DRSI. The derivations are parameter-free, transparent, and correct as algebraic identities under the standard PID consistency equation. The paper also frames the empirical analysis carefully as an exact computation on the training-set distribution, avoiding the injectivity pitfalls of continuous neural activations. The main limitations are that the interpretative conclusions depend on an unverified nonnegativity property of PID atoms in the neural data, and the empirical trends are presented without statistical tests or null models. With those caveats addressed, the framework would be a valuable contribution to multivariate information theory and its applications.

major comments (2)
  1. [Section III and Appendix B, Eqs. (21), (B2)–(B9)] The interpretation of rbar and vbar as average degrees of redundancy and vulnerability, and the threshold statements such as rbar<1 implying source-level synergy, presuppose nonnegative PID atoms. The manuscript explicitly assumes nonnegativity in Section III (after Prop. 3) and in Appendix B (Prop. 5, Prop. 6, Cor. 3, Cor. 4), but it never verifies this property for the activation distributions analyzed in Section IV. Since several PID measures admit negative atoms for n>2, the values in Figures 2 and 3 are strictly statements about the Shannon ratios, not about nonnegative PID atoms; consequently, statements such as "redundancy increases with depth" in Section IV.B are not fully supported as claims about actual redundant information atoms. The authors should either verify nonnegativity for the specific PID they intend (or for the data at hand), or explicitly weaken the interpretational language throughout.
  2. [Section IV, Figures 2 and 3] The empirical claims rest on medians and min-max ranges over only 10 runs, with no null models, confidence intervals, or statistical tests. For example, the asserted monotonic increase of redundancy with depth and the decrease of vulnerability in the MNIST classifier (Fig. 2c,d), and the ordering by bottleneck size in the autoencoder (Fig. 3d,e), could plausibly fall within run-to-run variability. The manuscript should either add appropriate statistical analyses (e.g., permutation tests against null models, or at least confidence intervals) or explicitly describe these results as qualitative, exploratory observations rather than established signatures.
minor comments (6)
  1. [Appendix B, Corollary 5, Eqs. (B10)–(B11)] The summation in Corollary 5 runs over k=1 to n for I_r^(k) and I_v^(k), but the derivation and the displayed threshold correspond to proper redundancy/vulnerability, i.e., k>=2. As written, the equivalence is false; for example, with n=2 and λ=0.5, taking I_r^(1)/I=0.6 and I_r^(0)/I=0.4 gives rbar=0.6, so the left-hand side of (B10) holds but the claimed rbar>1.5 does not. The index should be corrected to k=2,...,n.
  2. [Section IV.B and Figure 2 caption] The text says the architecture has three hidden layers, while the Figure 2 caption and the layer labels L3–L5 in the text indicate five hidden layers. Please reconcile this inconsistency.
  3. [Figure 3 caption] The phrase "The inset in E shows" should refer to panel (e) or "the inset in e"; also, Figure 2's caption contains the typo "wile" for "while".
  4. [Definition 1] Definition 1 defines Shannon-invariance for linear combinations of atoms, but the two main quantities rbar and vbar are ratios of such combinations. Consider extending the definition to include normalized invariants, or add a remark that all results apply to ratios of Shannon-invariant linear combinations.
  5. [Section II.B and II.C] The statements that rbar and vbar lie in [0,n] are asserted before the Appendix B proofs; a one-line justification using I(X_i;Y) ≤ I(X;Y) and I(X_j;Y|X_-j) ≤ I(X;Y) would improve readability.
  6. [References] References [42] and [49] are the same Chechik et al. conference paper and should be merged or cross-referenced.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the Shannon-invariant identities are algebraic consequences of the PID consistency equations and are computed directly from Shannon entropies; the acknowledged non-negativity assumption is an interpretation caveat, not a circular step.

full rationale

The derivation is self-contained. Propositions 1 and 2 are obtained by expanding each marginal and conditional mutual information according to the PID consistency equations (A7) and (A8); every atom with r(alpha)=k appears in exactly k terms of sum_i I(X_i;Y), and every atom with v(alpha)=j appears in exactly j terms of sum_j I(X_j;Y|X_-j). These counting identities are the entire content of the proofs, and they involve no fitted parameters, no tuning, and no appeal to the values being predicted. Propositions 3 and 4 and Corollaries 1 and 2 are algebraic rearrangements of the same two invariants together with the definitions of RSI and DRSI; the DRSI is introduced as a definition and then shown to equal an entropy expression. The empirical section computes rbar and vbar directly from empirical distributions of quantized activations, so there is no subset-fit-to-prediction structure. The main caveat, explicitly acknowledged in Section III and Appendix B, is that the redundancy/synergy interpretation of thresholds (for example, rbar<1 implying source-level synergy) assumes non-negative PID atoms; this is an unverified modelling assumption for the neural data and an acknowledged limitation of some PID measures, but it is not circular because the invariant identities themselves do not depend on it. The paper's citations to prior work by the same authors ([27], [37], [39]) supply the atom/antichain formalism and the nninfo implementation; they are not used to force the central result, and no uniqueness theorem is imported. Hence no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted; the central formulas are direct entropy expressions. The interpretational claims rely on the existence of a PID with non-negative atoms and the consistency equation A7. No new physical or mathematical entities are introduced.

assumptions (3)
  • domain assumption A PID exists and satisfies the consistency equations A7 and A8, meaning every mutual information term is the sum of the atoms contained in it.
    All theoretical results in Propositions 1 to 4 count atoms using this minimal consistency structure. It is standard in PID but is an assumption about the decomposition's semantics.
  • domain assumption All PID atoms are non-negative.
    Needed for Propositions 5 and 6, the inequalities in Section IID, and for interpreting RSI and DRSI signs as presence of redundancy or synergy. The paper states this assumption and notes some PID measures violate it.
  • domain assumption Neural activations can be modeled by stochastically quantized discrete variables so that Shannon information quantities are well-defined.
    Section IV.A argues that continuous deterministic activations have infinite channel capacity, and the analysis is performed on activations quantized to eight levels, with the training set treated as a complete population.

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Pith. "Pith review of Shannon invariants: A scalable approach to information decomposition." pith.science (2026). https://pith.science/paper/VOOC4NNU

@misc{pith2026250415779,
  author       = {Pith},
  title        = {Pith review of: Shannon invariants: A scalable approach to information decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VOOC4NNU}},
  note         = {Machine review of arXiv:2504.15779}
}
read the original abstract

Distributed systems, such as biological and artificial neural networks, process information via complex interactions engaging multiple subsystems, resulting in high-order patterns with distinct properties across scales. Investigating how these systems process information remains challenging due to difficulties in defining appropriate multivariate metrics and ensuring their scalability to large systems. To address these challenges, we introduce a novel framework based on what we call "Shannon invariants" -- quantities that capture essential properties of high-order information processing in a way that depends only on the definition of entropy and can be efficiently calculated for large systems. Our theoretical results demonstrate how Shannon invariants can be used to resolve long-standing ambiguities regarding the interpretation of widely used multivariate information-theoretic measures. Moreover, our practical results reveal distinctive information-processing signatures of various deep learning architectures across layers, which lead to new insights into how these systems process information and how this evolves during training. Overall, our framework resolves fundamental limitations in analyzing high-order phenomena and offers broad opportunities for theoretical developments and empirical analyses.

Figures

Figures reproduced from arXiv: 2504.15779 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Lattice of PID atoms for [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

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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. Full citation record

  1. The mathematical landscape of partial information decomposition: A comprehensive review of properties and measures

    cs.IT 2026-03 conditional novelty 6.0 of 10

    A comprehensive survey plus new theorems systematically comparing PID redundancy measures and their axioms, with a Z3-checked map of property implications and incompatibilities.

  2. Redundancy Maximization as a Principle of Associative Memory Learning in Hopfield Networks

    cs.IT 2025-11 conditional novelty 6.0 of 10

    Maximizing per-neuron redundancy between recurrent and target inputs in Hopfield networks yields a memory capacity of 1.59–1.70, versus 0.14 for the Hebbian rule.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.