Information Structure in Mappings: An Approach to Learning, Representation, and Generalisation

Henry Conklin

REVIEW 4 major objections 4 minor 1 cited by

Information Structure in Mappings: An Approach to Learning, Representation, and Generalisation

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This thesis establishes that the way a neural network maps inputs to vector representations can be quantified by three information-theoretic structures—regularity, variation, and disentanglement—and that this 'information structure'…

desk verdict A genuinely useful entropy estimator with an unproven invariance property; the two-phase training story is solid, but the cross-scale headline needs a dimension-sweep before I'd trust it. read the letter →

arxiv 2505.23960 v1 pith:6YKTSE4M submitted 2025-05-29 cs.LG cs.AIcs.CL

Henry Conklin This is my paper

classification cs.LGcs.AIcs.CL

keywords informationstructuresoftentropyrepresentationalestimationneuralnetworkinterpretabilitycompositionalgeneralisationregularityandvariationlargelanguagemodelsemergentcommunication

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

This thesis argues that how a neural network organises information can be read directly from the mapping between inputs and vector representations, using three primitive structures: one-to-one regularity, one-to-many variation, and many-to-one disentanglement. The author quantifies these with information-theoretic measures—entropy, conditional entropy, mutual information, and Jensen-Shannon divergence—and introduces a fast, differentiable 'soft entropy' estimator that makes the analysis tractable for models from 1 million to 12 billion parameters. Across emergent-communication agents, sequence-to-sequence transformers, and large language models, the same picture emerges: training first aligns representations with lexical information, then slowly compresses and contextualises them, and the models that separate different contexts most cleanly generalise best. If right, the framework gives a principled, scalable way to describe what deep networks learn and which design choices shape their ability to generalise.

What carries the argument

The load-bearing object is the mapping f: X -> Y from input labels (tokens, bigrams, parts of speech) to vector representations, analysed through three primitive structures: one-to-one regularity (measured by mutual information), one-to-many variation (conditional entropy), and many-to-one disentanglement (Jensen-Shannon divergence). To make these measurable in continuous high-dimensional space, the paper introduces soft entropy: representations are L2-normalised onto the unit sphere, compared by cosine similarity against uniformly sampled anchor points, and passed through a softmax to form a categorical distribution whose entropy and derived quantities are computed exactly. This estimator is differentiable, parallelisable, and cheap enough to run throughout training, which is what lets the same analysis scale from a 1-million-parameter model to a 12-billion-parameter LLM. The earlier dimension-wise binning version of the estimator is what supports the single-task transformer experiments; soft entropy extends it to large models.

What would settle it

Run the soft-entropy estimator on synthetic point clouds whose true entropy is known—for example, uniform points on the unit sphere versus tightly clustered Gaussians—while sweeping the number of anchor points and the softmax temperature; if the estimated entropy changes substantially with these hyperparameters on data whose true entropy is fixed, the measure is an artifact and the training-phase and generalisation claims built on it do not follow.

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Extended reading notes

Core claim

The central discovery is that the representational spaces of trained neural networks carry a measurable 'information structure' that mirrors the structure of natural language: mappings are neither perfectly regular nor randomly variable, but combine systematic one-to-one alignment with context-dependent variation and controlled ambiguity. Quantifying these properties with entropy-based measures reveals two distinct phases of training. In the first, the model rapidly aligns representations with token and part-of-speech information and disentangles them; in the second, much longer phase, representations compress and become more contextualised, with bigram-level regularity and disentanglement rising while token-level regularity falls. This second phase is where out-of-distribution generalisation emerges, and across runs of the same model, better generalisation is predicted by higher bigram disentanglement (r = 0.65, p = 0.04) and higher bigram regularity (r = 0.61, p = 0.06). The same trajectory is found in large language models up to 12 billion parameters, where representation structure at the end of pre-training correlates with downstream GLUE performance.

Load-bearing premise

The whole argument depends on the assumption that the soft-entropy estimator's categorical distribution over anchor points faithfully represents the information content of the continuous representation space, rather than being an artifact of its own settings such as the number of anchors, the softmax scale, or the model's dimensionality.

Editorial extensions

If this is right

If correct, transformer training can be described as a two-phase process: fast lexical alignment and disentanglement, followed by a slow compression and contextualisation phase in which out-of-distribution generalisation emerges.
Representation structure at the end of pre-training predicts downstream performance after fine-tuning, so information-structure measures could be used to select checkpoints or architectures without running expensive downstream evaluations.
Larger models are less regular at the token level but more regular with respect to context (bigrams and trigrams); this contextual regularity, not raw capacity, is what tracks generalisation.
Capacity constraints regularise emergent languages: smaller agents in communication games develop more regular mappings, consistent with cognitive accounts of language regularisation.
Meta-learning objectives that optimise update steps for similar examples can bias representational structure toward better out-of-distribution generalisation, showing that information structure is not fixed by architecture alone.

Reading between the lines

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

The estimator's differentiability suggests a direct extension the thesis does not pursue: using soft entropy as a training objective or regularizer to steer models toward desirable information structures.
The two-phase trajectory found here may be the same phenomenon as 'grokking' in smaller models, with the slow second phase being where robust generalisation appears; the thesis does not make this identification explicitly.
The framework's reliance on token and bigram labels implies a testable extension to non-linguistic inputs, such as vision or speech, where the analogue of context would need to be defined independently of words.
If the information-structure account is general, one would expect the same end-of-training bigram-style contextual regularity to predict generalisation in multimodal and continual-learning settings, which the thesis leaves untested.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take. The genuinely new thing is the soft entropy estimator in Chapter 5, and it deserves to be taken seriously: L2-normalized representations, uniform anchor points on the sphere, softmax similarities, then standard entropy. It is differentiable, fast, and the validation against existing estimators shows it behaving sensibly. The information-structure framing (regularity/variation/disentanglement as one-to-one/one-to-many/many-to-one) is not mathematically deep, but it is clearly defined and useful for comparing representations across tasks and scales. The two-phase training result—rapid lexical alignment, then slow contextual compression—comes through consistently in seq2seq transformers and LLMs, and the code and data appear to be available. That is a real contribution. So is the capacity story: smaller models regularize, larger models compress more per dimension, and the meta-learning chapter gives a different handle on the same theme.

The soft spots are real but addressable. The biggest one is exactly the stress-test concern: Eq. 5.7 has no proven invariance to hidden dimensionality d or anchor count n. Cosine similarities of random unit vectors concentrate near zero as d grows, so the softmax distribution approaches uniform and soft entropy drifts toward log n. The paper compares models with very different hidden sizes (14M to 12B), so the "scale conditions structure" claim could be partly a geometric artifact. The benchmarking appendix covers heads, mean, and scale, but not d or n. That is a load-bearing gap, not a nit. Chapter 4 has a related issue: dimension-wise binning with an unexamined bin count, and the headline correlations (r = 0.65, p = 0.04) come from 10 runs with no confidence intervals, after selecting the label sets with the clearest effect. That is post hoc selection and should be labeled as pilot evidence.

None of this kills the thesis. The framework is coherent, the estimator is a genuine practical step, and the empirical pattern is consistent across multiple independent setups. But the strongest version of the central claim—the same information-structure picture from 1M to 12B parameters—is not yet established, because the estimator's behavior across dimensions is unverified. The fix is straightforward: synthetic dimension and anchor-count sweeps, and same-dimension comparisons wherever possible. If those check out, the paper becomes quite valuable. The citation pattern is fine; the thesis leans on the author's own prior papers, but those are real published papers and the code is available, so that is not a smell. This is for interpretability researchers and anyone building intrinsic representation metrics. I'd send it to a serious referee, with a clear request to focus on the estimator's invariance and the small-sample correlations.

Referee Report

4 major / 4 minor

Summary. The thesis develops a framework for quantifying 'information structure' in mappings between input spaces and neural representations, using information-theoretic measures of regularity (mutual information), variation (conditional entropy), disentanglement (Jensen-Shannon divergence), and compression (normalised entropy). It applies this framework to emergent communication models, sequence-to-sequence transformers, and large language models up to 12B parameters, claiming two distinct phases of training, systematic effects of model capacity on representational structure, and correlations between structure and generalisation. To scale the analysis to LLMs it introduces 'soft entropy' (Eq. 5.7), an anchor-based estimator of entropy over continuous representation spaces. The manuscript is a PhD thesis drawing on several prior papers by the author and collaborators, with code made available.

Significance. If the proposed measures are faithful and comparable across model sizes, the framework would provide a useful, efficient, and non-parametric complement to probing and behavioural interpretability. The thesis asks an important question --- whether representational structures analogous to those of natural language emerge in neural networks and predict generalisation --- and it makes a credible effort to answer it with first-principles information-theoretic definitions. Credit is due for releasing code, for validating soft entropy against existing estimators, and for explicitly discussing limitations of dimension-wise discretisation. However, the central cross-scale claims depend on the soft entropy estimator being invariant, or at least stable, with respect to hidden dimensionality and anchor count; that invariance is not established, and the current evidence is insufficient to rule out geometric artifacts.

major comments (4)

[§5.4.1, Eq. (5.7)] The soft entropy estimator has no demonstrated invariance or stability with respect to hidden dimensionality d and anchor count n. For two independent random unit vectors in d dimensions the dot product has variance proportional to 1/d, so as d grows all logits concentrate near zero, the softmax approaches a uniform distribution over anchors, and soft entropy approaches log n. Section 5.6.3 compares models that differ in hidden width (14M to 12B parameters), so the reported scale-conditions-structure relationships may be partly a geometric artifact of dimensionality rather than of representational content. Appendix B.1 benchmarks heads, mean, and scale, but not d or n. The manuscript needs either a consistency theorem, or explicit experiments varying n and d (including matching widths across model sizes or normalising for dimensionality) to show that the quantities compared across models are measuring the same thing.
[§4.2.1, Eq. (4.1)] The dimension-wise discretisation used in Chapter 4 depends on the number of bins N, a free hyperparameter, and no sensitivity analysis is reported. The chapter acknowledges that dimension-wise binning limits tracking of cross-dimensional dependencies, but the bin count itself is load-bearing because all Chapter 4 conclusions about training phases, capacity effects, and generalisation correlations are computed through this estimator. Without evidence that the reported trajectories and correlations are robust to N (and to sample size, despite the Miller-Madow correction), the results cannot be distinguished from artifacts of the binning choice.
[§4.3.1.2] The headline correlations between bigram disentanglement/regularity and out-of-distribution generalisation are based on 10 runs with no confidence intervals, and Section 4.3 states that only the label sets with the 'clearest effect on model performance' are reported. This is a post hoc selection, and with four measures, three label levels, and two datasets, the uncorrected p-values (r = 0.65, p = 0.04; r = 0.61, p = 0.06) are weak evidence for the specific claim that bigram-level structure predicts generalisation. The authors should report all comparisons, provide bootstrap or otherwise constructed confidence intervals, and either correct for multiple comparisons or pre-specify the primary analysis.
[§5.3, Eq. (5.6)] Equation (5.6) appears inconsistent with its prose description. The text says residual information is what remains after subtracting the highest regularity from the entropy of the space, but the equation writes residual(Y,set) = regularity(Y,smallest set)/H(Y), which is the explained proportion, not the residual. If residual information is used in any quantitative claim or figure, this sign error would invert the reported quantity. Please correct the equation or clarify what is actually computed.

minor comments (4)

[Abstract] The abstract contains the duplicated phrase 'large large-scale neural networks'; this should be corrected.
[§5.4.1] The description of anchor sampling says points are 'uniformly sampled from the surface of the unit sphere' by normalising standard normal draws, which is correct for the surface measure, but the later phrase 'drawing n samples from a standard normal and taking their euclidean norm' is ambiguous about normalisation; consider rewording to avoid implying the norm itself is the sampled quantity.
[§4.3] Table 4.1 reports summary statistics with 95% CIs for the measures, but the generalisation correlations in Section 4.3.1.2 are reported without any measure of uncertainty; adding the same confidence-interval convention there would help readers calibrate the strength of the claim.
[§5.6.4] The claim that representation structure at the end of pre-training predicts GLUE performance after fine-tuning would be stronger if the number of GLUE tasks and the selection of which structures are reported were stated in the main text, since Appendix B.2 shows per-task correlations that may vary widely.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity found; the thesis's empirical correlations are external to its information-theoretic definitions, and self-citations are transparent rather than load-bearing.

full rationale

The thesis builds its measures from standard information-theoretic quantities defined in Chapter 2 (entropy, conditional entropy, mutual information, Jensen-Shannon divergence) and applies them to vector spaces via explicit estimators in Chapters 4-5. The key empirical claims—e.g., that runs with higher bigram disentanglement generalise better (r=0.65, p=0.04) and higher bigram regularity correlates with generalisation (r=0.61, p=0.06)—are correlations against held-out accuracy, an external performance measure not constructed from the information measures. Similarly, the Chapter 5 claim that representation structure at the end of pre-training predicts GLUE performance after fine-tuning is a predictive correlation against an independent benchmark. The soft entropy estimator (Eq. 5.7) is a proposed estimation procedure, not a derived theorem; the absence of a formal invariance proof for hidden dimensionality or anchor count is a correctness/validity concern, not circularity, because the reported quantities are defined as soft entropy and are benchmarked against existing estimators in Section 5.5. The reliance on the author's own prior papers (Conklin & Smith 2022, 2024) is transparent, and the framework is restated within this thesis rather than being justified solely by self-citation. No step reduces, by the paper's own equations or by a self-citation chain, to its own inputs.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims rest on several measurement assumptions but no invented physical entities. The free parameters are discretisation bins, soft entropy anchor count, softmax scale, and number of heads; all affect the estimated entropy and therefore the reported structure scores. The axioms are the validity of information-theoretic quantities as measures of linguistic structure, the compositionality-as-generalization premise, and the faithfulness of the estimators.

free parameters (4)

number of bins N in dimension-wise discretisation = unspecified in visible text
Chapter 4 cuts each dimension into N equal-width bins; N controls entropy estimates and is not derived from theory.
soft entropy anchor count n = unspecified in visible excerpt
Number of uniform anchor points on the unit sphere in Eq. 5.7; a hyperparameter controlling the categorical distribution's resolution.
soft entropy scale or temperature = benchmarked in Appendix B.1
The dot product in Eq. 5.7 may be scaled; the appendix on 'mean and scale' suggests a scale parameter that affects softmax sharpness.
number of heads for multi-head soft entropy = benchmarked in Appendix B.1
Reshaping hidden dimensions into heads changes entropy estimates and therefore the structure scores.

assumptions (4)

domain assumption Shannon entropy, conditional entropy, mutual information, and Jensen-Shannon divergence are valid quantifications of regularity, variation, and disentanglement in mappings.
Chapter 2 equates the three structural primitives with these quantities; no proof is given that these quantities uniquely capture the intended linguistic structures.
domain assumption In a reconstruction game, successful out-of-distribution generalization implies compositional structure.
Chapter 3, Section 3.2 assumes a language must be compositional to generalize, following Brighton 2002 and Kottur 2017; this premise drives interpretation of the variation measures.
domain assumption Discretising or softmaxing representation space yields a distribution whose entropy reflects information content of the space.
Chapters 4 and 5 rely on binning or soft entropy estimators; acknowledged limitations (loss of cross-dimensional dependencies, dependence on anchors) make this an assumption rather than a theorem.
domain assumption Token, bigram, and trigram labels are sufficient to characterize the information structure relevant to generalization.
Experiments condition on these labels; other structures such as syntax or morphology are not tested, so the generality of the conclusions is limited.

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Cite this review

Pith. "Pith review of Information Structure in Mappings: An Approach to Learning, Representation, and Generalisation." pith.science (2026). https://pith.science/paper/6YKTSE4M

@misc{pith2026250523960,
  author       = {Pith},
  title        = {Pith review of: Information Structure in Mappings: An Approach to Learning, Representation, and Generalisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YKTSE4M}},
  note         = {Machine review of arXiv:2505.23960}
}

read the original abstract

Despite the remarkable success of large large-scale neural networks, we still lack unified notation for thinking about and describing their representational spaces. We lack methods to reliably describe how their representations are structured, how that structure emerges over training, and what kinds of structures are desirable. This thesis introduces quantitative methods for identifying systematic structure in a mapping between spaces, and leverages them to understand how deep-learning models learn to represent information, what representational structures drive generalisation, and how design decisions condition the structures that emerge. To do this I identify structural primitives present in a mapping, along with information theoretic quantifications of each. These allow us to analyse learning, structure, and generalisation across multi-agent reinforcement learning models, sequence-to-sequence models trained on a single task, and Large Language Models. I also introduce a novel, performant, approach to estimating the entropy of vector space, that allows this analysis to be applied to models ranging in size from 1 million to 12 billion parameters. The experiments here work to shed light on how large-scale distributed models of cognition learn, while allowing us to draw parallels between those systems and their human analogs. They show how the structures of language and the constraints that give rise to them in many ways parallel the kinds of structures that drive performance of contemporary neural networks.

Figures

Figures reproduced from arXiv: 2505.23960 by the authors.

**Figure 2.1.** Three basic kinds of mapping structure we consider here, labelled with their [PITH_FULL_IMAGE:figures/full_fig_p034_2_1.png] view at source ↗

**Figure 2.2.** Plots showing suprisal (left) and entropy (right) values for a single event. The [PITH_FULL_IMAGE:figures/full_fig_p038_2_2.png] view at source ↗

**Figure 2.3.** A distribution with 4 events is shown above in 4 different versions. On the [PITH_FULL_IMAGE:figures/full_fig_p040_2_3.png] view at source ↗

Figures from the paper (30 more)

**Figure 2.4.** Figure 2.4: The probabilities of the 10 most frequent words from both documents plot [PITH_FULL_IMAGE:figures/full_fig_p041_2_4.png]

**Figure 2.5.** Figure 2.5: The same distributions shown in figure 2.3 but now labelled with the efficiency of each distribution. Referring back to figure 2.4, note that the document with just the words ‘king lear’ repeated has lower entropy than the full text of the play, but higher efficienc…

**Figure 2.6.** Figure 2.6: Relationships between basic information-theoretic quantities. Given two [PITH_FULL_IMAGE:figures/full_fig_p046_2_6.png]

**Figure 2.7.** Figure 2.7: Jensen Shannon divergences computed for two sets of distributions {a,b} [PITH_FULL_IMAGE:figures/full_fig_p048_2_7.png]

**Figure 3.1.** Figure 3.1: A depiction of the probability tensor built with equation [PITH_FULL_IMAGE:figures/full_fig_p060_3_1.png]

**Figure 3.2.** Figure 3.2: The Freedom measure applied to a regular and variable language. At top [PITH_FULL_IMAGE:figures/full_fig_p063_3_2.png]

**Figure 3.3.** Figure 3.3: Column entropies for two different roles, object and verb. The mean of [PITH_FULL_IMAGE:figures/full_fig_p064_3_3.png]

**Figure 3.4.** Figure 3.4: A model is fit to a sliding window of data from 100 epochs at a time across [PITH_FULL_IMAGE:figures/full_fig_p068_3_4.png]

**Figure 3.5.** Figure 3.5: Caption Opposite [PITH_FULL_IMAGE:figures/full_fig_p070_3_5.png]

**Figure 4.1.** Figure 4.1: a depiction of basic quantities we measure and how they relate to each [PITH_FULL_IMAGE:figures/full_fig_p079_4_1.png]

**Figure 4.2.** Figure 4.2: Examples of Measures defined here using the dimension-wise discretisation [PITH_FULL_IMAGE:figures/full_fig_p086_4_2.png]

**Figure 4.3.** Figure 4.3: Each facet shows a different measure (along the y axis) against training [PITH_FULL_IMAGE:figures/full_fig_p088_4_3.png]

**Figure 4.4.** Figure 4.4: Spearman correlation coefficients between the loss minimised during train [PITH_FULL_IMAGE:figures/full_fig_p091_4_4.png]

**Figure 5.1.** Figure 5.1: Three basic kinds of mapping structure we consider here, labelled with [PITH_FULL_IMAGE:figures/full_fig_p100_5_1.png]

**Figure 5.2.** Figure 5.2: An exemplar showing how different information theoretic quantities relate [PITH_FULL_IMAGE:figures/full_fig_p105_5_2.png]

**Figure 5.3.** Figure 5.3: A visual depiction of the soft entropy estimation process (continued on next [PITH_FULL_IMAGE:figures/full_fig_p108_5_3.png]

**Figure 5.4.** Figure 5.4: The soft entropy estimator applied to exemplar distributions. Examples [PITH_FULL_IMAGE:figures/full_fig_p111_5_4.png]

**Figure 5.5.** Figure 5.5: Comparison of Entropy Estimation Methods | In each facet the x-axis reflects [PITH_FULL_IMAGE:figures/full_fig_p114_5_5.png]

**Figure 5.6.** Figure 5.6: Information Structure with respect to 256,000 sentences from wikipedia over [PITH_FULL_IMAGE:figures/full_fig_p118_5_6.png]

**Figure 5.7.** Figure 5.7: Information Structure with respect to 256,000 sentences from wikipedia over [PITH_FULL_IMAGE:figures/full_fig_p120_5_7.png]

**Figure 5.8.** Figure 5.8: Analyses computed on pythia decoder-only models ranging from 14m pa [PITH_FULL_IMAGE:figures/full_fig_p123_5_8.png]

**Figure 5.9.** Figure 5.9: Analyses computed on pythia decoder-only models ranging from 14m pa [PITH_FULL_IMAGE:figures/full_fig_p124_5_9.png]

**Figure 5.10.** Figure 5.10: Scaling comparison of depth vs. dimensionalities on BERT models. All [PITH_FULL_IMAGE:figures/full_fig_p126_5_10.png]

**Figure 5.11.** Figure 5.11: Scatterplots showing model performance on two GLUE benchmarks, senti [PITH_FULL_IMAGE:figures/full_fig_p128_5_11.png]

**Figure 6.1.** Figure 6.1: Figures reproduced from O’Donnell (2015) show three different approaches to decomposing a word. Colour in each case indicates lexicon entries. A) decomposes words fully into each individual affix, B) stores the entire word in memory without decomposition and C) spli…

**Figure 6.2.** Figure 6.2: The dependency-tree forms for the logical forms of two sentences. Shown [PITH_FULL_IMAGE:figures/full_fig_p141_6_2.png]

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Forward citations

Cited by 1 Pith paper

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

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cs.CL 2025-07 conditional novelty 6.0 of 10

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Works this paper leans on

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Reviewed August 7, 2026 · model on record in the stance chip above.

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