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REVIEW 3 major objections 6 minor 6 references

Logit Distance Bounds Representational Similarity

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proves that small logit distance between softmax-style models forces high linear representational similarity, and that KL divergence cannot supply the same guarantee.

desk verdict Small logit distance provably implies high mCCA (that's the core result); the d_rep bound needs spectral diagnostics, but the paper deserves a serious referee. read the letter →

arxiv 2602.15438 v3 pith:HVNVB6Z4 submitted 2026-02-17 cs.LG cs.AIstat.ML

Beatrix M. G. Nielsen , Emanuele Marconato , Luigi Gresele , Andrea Dittadi , Simon Buchholz This is my paper · ORCID
classification cs.LGcs.AIstat.ML
keywords logitdistancerepresentationalsimilaritylinearidentifiabilityknowledgedistillationcanonicalcorrelationanalysissoftmaxmodelsconcepttransferKLdivergence
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 paper asks whether two models that make almost the same predictions must also think almost alike. For the broad family of discriminative models where the output probability is a softmax over inner products f(x)^T g(y), exact identifiability already says equal distributions imply embeddings equal up to an invertible linear map. The paper shows this conclusion survives approximately when closeness is measured by logit differences: small average squared logit distance yields a quantitative lower bound on mean canonical correlation between representations, and an upper bound on a new dissimilarity measure that vanishes exactly on the linear equivalence class. The same guarantee cannot come from KL divergence: the paper proves KL can bound logit distance only through a factor that blows up when probabilities approach zero. That is why KL-based distillation can match a teacher's probabilities while destroying linearly encoded concepts, whereas logit-distance distillation preserves them.

What carries the argument

The central object is the logit distance d_logit^2 = E_x ||u(x)-u'(x)||^2, the mean squared difference in pre-softmax scores. Around it the paper builds the linear-identifiability machinery: two models (f,g) and (f',g') are considered equivalent when f = A f' and g = A^{-⊤}g', and the matrix A is recovered from shifted unembedding matrices whose columns are g(y)-g(pivot) for m labels. The smallest singular value σ_min of all such matrices converts logit closeness into representation closeness. The new dissimilarity d_rep averages the alignment error f(x) - A_J f'(x) over every pivot and every m-label subset, and it is zero exactly on the equivalence class.

What would settle it

On a dataset with more labels than embedding dimension, train a teacher and a student with logit-distance distillation, then measure d_logit, mCCA, d_rep, and the smallest singular value σ_min of the teacher's shifted unembedding matrices. If the student's mCCA falls below the lower bound in Theorem 3.4, or d_rep exceeds C d_logit/σ_min, the central claim would be false; if σ_min is so small that the bound is vacuous in practice, the practical force of the claim would be unsupported.

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Extended reading notes

Core claim

The paper proves two concrete inequalities. If d_logit^2 = E_x||u(x)-u'(x)||^2 is the mean squared difference in logits, then the mean canonical correlation between the two models' embedding spaces is at least 1 - d_logit^2/(m μ_m), where μ_m is the smallest nonzero eigenvalue of the logit covariance. The paper also defines a representation dissimilarity d_rep that is zero exactly when the models are related by the linear identifiability symmetry f = A f', g = A^{-⊤}g', and proves d_rep ≤ C d_logit/σ_min, where σ_min is the smallest singular value of the shifted unembedding matrices. The proof decomposes logit differences into a regression residual plus an alignment error, then uses von Neum

Load-bearing premise

The d_rep and concept-transfer bounds require general position—every set of m shifted unembedding vectors must be linearly independent—so if real unembeddings are nearly collinear or lie in a lower-dimensional subspace, σ_min approaches zero and those bounds become vacuous.

Editorial extensions

If this is right

  • If two models have small d_logit and their unembedding matrices are well conditioned, their embedding spaces are nearly an invertible linear map apart, and linearly encoded concepts in one will be almost linearly recoverable in the other.
  • Logit-distance distillation should replace KL-based distillation whenever a teacher's linearly encodable attributes matter; the paper's experiments on synthetic and image data confirm this.
  • Matching KL can drive a student's representations arbitrarily far from the teacher's linear geometry while predictions look close; Theorem 3.3 shows why the KL route cannot give a practically useful bound.
  • For next-token predictors with very large vocabularies, the condition k > m+1 is naturally satisfied, so the guarantees extend to autoregressive language models.
  • Minimizing an L1 logit loss also drives d_logit down, so the framework justifies a simple, stable training objective with the same theoretical protection.

Reading between the lines

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

  • Editorial: the same bound suggests logit-distance matching as a generic representation-alignment objective for model stitching or cross-model transfer, not only for distillation; the paper does not discuss this use.
  • Editorial: Theorem 4.3 says concept degradation is controlled by the operator norm of the concept's unembedding coordinates; a testable prediction is that concepts aligned with low-variance unembedding directions will be the first to break under small d_logit.
  • Editorial: the general-position assumption could likely be relaxed to a lower-rank statement: when σ_min ≈ 0, only the span of the unembeddings is identifiable, and d_rep should be measured in that subspace. The authors flag the assumption but do not explore this weakening.
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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

3 major / 6 minor

Summary. The paper studies approximate versions of linear identifiability for softmax-type discriminative models p(y|x) ∝ exp(f(x)^T g(y)). It defines a logit distance d_logit, proves that it is a metric, shows that KL divergence both lower-bounds and (under a τ-lower-boundedness assumption) upper-bounds d_logit^2, proves an mCCA lower bound in terms of d_logit^2/(m μ_m), introduces a "linear identifiability dissimilarity" d_rep, and proves that d_rep is controlled by d_logit up to a 1/σ_min factor under a general-position assumption. It also gives a concept-transfer bound and reports distillation experiments on Synth, CIFAR-100, and SUB, comparing KL students with L1/L2 logit-distance students. The central claim is that small logit distance implies high linear representational similarity, and that this implication is not available with practically useful constants through KL divergence.

Significance. The paper addresses a real and timely question: whether the exact linear identifiability theorem for this model class is robust to distributional closeness. Theorem 3.4, if its hypotheses are met, is a nontrivial quantitative robustness statement under the comparatively mild condition μ_m > 0, and it applies to both embeddings and unembeddings. The appendix proofs are detailed and largely self-contained; I checked the algebra in the central bounds and it is correct under the stated assumptions. The paper also provides code and a clear experimental comparison. However, the headline d_rep guarantee (Theorem 3.9) is conditional on a much stronger general-position assumption and on an unverified smallest-singular-value constant. As written, the experiments do not establish that this particular theoretical bound is operative in the settings reported. The mCCA half of the paper is more robust, but the quantitative link to the experiments is also incomplete because μ_m is not reported.

major comments (3)
  1. [§3.4, Assumption 3.5, Theorem 3.9] The d_rep bound scales as C·d_logit/σ_min, where σ_min is the smallest singular value across every shifted-unembedding matrix ̃L_J for every pivot ̃y and every m-subset J. Assumption 3.5 requires all these matrices to be invertible. For the trained classifiers in the experiments (CIFAR-100 with m=50, k=100; SUB with m=10, k=33), unembedding vectors are learned, not random, and can be nearly collinear or confined to a lower-dimensional subspace. The paper reports d_rep values in Tables 1, 2, and 4 but never reports σ_min or checks Assumption 3.5. If any ̃L_J is singular, d_rep is undefined by Definition 3.7 and Theorem 3.9 does not apply; if σ_min is very small, the bound is vacuous. The random-subsampling procedure in §K.2 cannot certify all subsets. This is a load-bearing gap: the experimental d_rep improvements for L1/L2 students could occur even if the theoretical d_rep bound is vacuo
  2. [§3.2, Theorem 3.4, Appendix F.14] Theorem 3.4 gives mCCA ≥ 1 − d_logit^2/(m μ_m), with μ_m the smallest nonzero eigenvalue of the logit covariance. The proof is correct under the full-rank assumptions in Appendix F. However, the paper never reports μ_m for any of the datasets. Without μ_m, the displayed experimental mCCA values cannot be compared with the theorem: d_logit may be large in absolute terms, and if μ_m is small the lower bound is negative. For the claim that the theory explains the observed mCCA improvements, please report μ_m (or a lower bound) together with d_logit in the extended tables, or state clearly when the bound is nontrivial.
  3. [§5, Proposition 4.1, Theorem 4.3] The concept-preservation result Theorem 4.3 has a constant ∥A∥_op, where A expresses the concept weight matrix in the unembedding basis. The paper does not estimate ∥A∥_op or any related constant on SUB, so the quantitative bound is not connected to the reported Acc(C) improvements. The empirical difference between KL students and logit students on SUB is striking, but as presented it is evidence for the qualitative phenomenon rather than a validation of the specific bound. Please either estimate the relevant constants on the experimental models or explicitly delimit Theorem 4.3 as a qualitative statement.
minor comments (6)
  1. [Throughout] There is an inconsistent use of 'Assumption' and 'Theorem' when referring to Assumptions 2.1, 3.2, and 3.5 (e.g., Corollary 3.6 says 'satisfy Thm. 3.5' for what is Assumption 3.5). Please standardize the cross-references.
  2. [Definition 3.7] The display for d_rep^2 is malformed: the expectation and norm are missing delimiters, and the factor 1/(kJ) is ambiguous because J is used both as a label-set index and as the combinatorial count. Rename the count, e.g., N = binom(k−1,m), and add explicit parentheses.
  3. [Appendix D.3, proof of Theorem D.8] The triangle-inequality proof contains typos: p(y_j|x') should be p_i(y_j|x), and some norm signs are missing in the algebra. The intended argument is clear, but the appendix should be checked carefully for these presentation errors.
  4. [§K.2] For CIFAR-100, d_rep is estimated by sampling 200 label subsets per pivot out of roughly 5×10^28 possible subsets. Since d_rep is an average over all pivots and all subsets, the estimate may have high variance, especially for the KL students with large reported standard deviations. Please report bootstrap confidence intervals or, where feasible, use a stratified sampling scheme.
  5. [§5.2 and Table 2] The statement that KL students score 'slightly above a random classifier (1/33 ≈ 0.03)' is correct (Acc(C) = 0.055), but the phrasing could be sharpened because 0.055 is also far from the teacher's 0.92. Consider reporting the chance level explicitly in the table or caption.
  6. [§6, Limitations] The limitation paragraph for Assumption 3.5 says the assumption holds 'with high probability for random vectors in high dimension.' While true for random vectors, this does not address trained unembeddings. Please replace this argument with a concrete empirical check or a weaker alternative condition.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the central mCCA bound is independent and the d_rep bound is a genuine norm-equivalence theorem, though d_rep is purpose-built.

full rationale

The derivation chain is essentially self-contained. Theorem 3.4 (mCCA) targets an independently defined similarity measure and follows from Theorem F.1 via von Neumann's trace inequality; it does not presuppose d_rep. The d_rep guarantee (Thm 3.9 / Cor 3.10) is admittedly close to a change of norm: Theorem H.1 expresses the squared logit distance as a weighted sum of the same residuals ||f(x)-A_J f'(x)|| that define d_rep, and Lemma H.3 converts the weighting to the smallest singular value sigma_min. But this is a proved mathematical identity plus a norm inequality, not a fitted parameter renamed as a prediction or a definition that assumes the conclusion. The paper is also explicit that d_rep was designed to vanish exactly on the linear identifiability class, so the proximity of the bound to the definitions is transparent rather than hidden. The self-citations (Nielsen et al. 2025, Marconato et al. 2025) are motivational/contextual; the key non-robustness of KL is re-derived in Appendix G, and the linear identifiability theorem is attributed to Roeder et al. 2021. The empirical validation uses mCCA and linear-probe accuracy as external benchmarks. The concern that Assumption 3.5 and sigma_min may be vacuous for trained classifiers is a correctness/robustness risk, not a circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 1 invented entities

The paper honestly states its assumptions. The central theoretical contribution is conditional on the softmax model family and two strong structural assumptions (τ-lower-boundedness and general position). The only hand-chosen quantity is τ; all bounds are parameter-free otherwise. No hidden fitted constants.

free parameters (1)
  • τ (lower bound on class probabilities)
    Introduced in Assumption 3.2. The KL→logit bound in Thm 3.3 grows as 4 log(τ)^2/τ, so the theorem's usefulness depends on τ, which is not measured in experiments.
assumptions (6)
  • domain assumption Softmax linear model family Θ with p(y|x) ∝ exp(f(x)^T g(y)) (Eq. 1)
    Central modeling assumption; restricts the class of discriminative models. For autoregressive LMs this corresponds to next-token prediction with linear readout.
  • domain assumption Diversity: span{u(x)} = R^m (Assumption 2.1)
    Needed for linear identifiability; imported from Roeder et al. 2021.
  • domain assumption τ-lower-bounded probabilities (Assumption 3.2)
    Needed for the KL upper bound; equivalent to bounded embeddings/unembeddings under diversity.
  • domain assumption General position: every m-subset of shifted unembeddings is invertible (Assumption 3.5)
    Needed for d_rep theorems and concept transfer. Stronger than diversity; plausibly holds for random high-dimensional unembeddings.
  • domain assumption Full-rank logit covariance μ_m > 0 (Thm 3.4)
    Needed for the mCCA bound denominator to be finite.
  • domain assumption Concepts linearly encoded in teacher (Def 4.2)
    Theorem 4.3 starts from a teacher that linearly encodes concept h; if untrue, concept-transfer may not hold.
invented entities (1)
  • d_rep — linear identifiability dissimilarity independent evidence
    purpose: A new measure of representational dissimilarity that is zero exactly for linearly equivalent models; used to state Thm 3.9 and Cor 3.10.
    The measure is validated empirically: d_rep correlates with mCCA and linear-probe concept accuracy across three datasets.

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Cite this review

Pith. "Pith review of Logit Distance Bounds Representational Similarity." pith.science (2026). https://pith.science/paper/HVNVB6Z4

@misc{pith2026260215438,
  author       = {Pith},
  title        = {Pith review of: Logit Distance Bounds Representational Similarity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HVNVB6Z4}},
  note         = {Machine review of arXiv:2602.15438}
}
read the original abstract

For a broad family of discriminative models that includes autoregressive language models, identifiability results imply that if two models induce the same conditional distributions, then their internal representations are equal up to an invertible linear transformation. We ask whether an analogous conclusion holds approximately when the distributions are close instead of equal. Building on the observation of Nielsen et al. (2025) that closeness in KL divergence need not imply high linear representational similarity, we study a distributional distance based on logit differences and show that closeness in this distance does yield linear similarity guarantees. Specifically, we define a representational dissimilarity measure based on the models' identifiability class and prove that it is bounded by the logit distance. We further show that, when model probabilities are bounded away from zero, KL divergence upper-bounds logit distance; yet the resulting bound fails to provide nontrivial control in practice. As a consequence, KL-based distillation can match a teacher's predictions while failing to preserve linear representational properties, such as linear-probe recoverability of human-interpretable concepts. In distillation experiments on synthetic and image datasets, logit-distance distillation yields students with higher linear representational similarity and better preservation of the teacher's linearly recoverable concepts.

Figures

Figures reproduced from arXiv: 2602.15438 by the authors.

Figure 1
Figure 1. In the center: The intuition of bounding representational similarity using distributional distance. PΘ is the set of probability distributions parametrized by models in Θ (Eq. (1)) which satisfy Thm. 2.1. These distributions are one-to-one with identifiability classes [(f, g)] in the quotient space Θ/ ∼L [Khemakhem et al., 2020a]. The colored areas in PΘ contain the distributions which are ϵ-close to a reference pf,… view at source ↗
Figure 2
Figure 2. On the left, input data of the Synth dataset (§5), where inputs are colored based on their labels. The remaining plots display the model embeddings of the teacher and student models. We notice that embeddings of points belonging to class 1 are nearest neighbors to those of class 6 and class 2 for the teacher and similarly for the L1-student, but not for the KL student. Here, the KL-student has low linear similarity … view at source ↗
Figure 3
Figure 3. Embeddings of (f, g) in blue.Other coloured dots mark the unembeddings, each be￾longing to a label [PITH_FULL_IMAGE:figures/full_fig_p038_3.png] view at source ↗

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Reference graph

Works this paper leans on

6 extracted references · 4 linked inside Pith

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    ⇐= ”: Assume dlogit(p, p′) = 0. Since dlogit is a metric between probability distributions, this gives usp=p ′. Therefore, Thm. 3.6 gives usd rep((f,g),(f ′,g ′)) = 0. “ =⇒

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