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Revisiting the Platonic Representation Hypothesis: An Aristotelian View

Fabian Gröger, Shuo Wen, Maria Brbić

arXiv:2602.14486v2cs.LGcs.AIcs.CVcs.NE

TL;DR

Representational similarity metrics can be inflated by network width and depth, complicating tests of whether neural-network representations converge. The paper introduces permutation-based null calibration and finds that global convergence largely disappears, while local neighborhood alignment remains across modalities.

  • Problem

    Existing representational similarity metrics are confounded by model width and depth, leaving calibrated evidence for the Platonic Representation Hypothesis unresolved.

  • Method

    The paper introduces a permutation-based null-calibration framework that corrects width-dependent baselines and depth-dependent selection inflation across similarity metrics.

  • Results

    After calibration, global spectral convergence largely disappears, while local neighborhood similarity retains significant cross-modal alignment but local distances do not.

  • Takeaways & Limitations

    The findings motivate the Aristotelian Representation Hypothesis, in which neural networks converge to shared local neighborhood relationships rather than necessarily shared global geometry.

  • Takeaways & Limitations

    The conclusions report calibrated evidence rather than proof of convergence, and the guarantees assume exchangeability or restricted permutations for dependent samples.

Abstract

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The Platonic Representation Hypothesis suggests that representations from neural networks are converging to a common statistical model of reality. We show that the existing metrics used to measure representational similarity are confounded by network scale: increasing model depth or width can systematically inflate representational similarity scores. To correct these effects, we introduce a permutation-based null-calibration framework that transforms any representational similarity metric into a calibrated score with statistical guarantees. We revisit the Platonic Representation Hypothesis with our calibration framework, which reveals a nuanced picture: the apparent convergence reported by global spectral measures largely disappears after calibration, while local neighborhood similarity, but not local distances, retains significant agreement across different modalities. Based on these findings, we propose the Aristotelian Representation Hypothesis: representations in neural networks are converging to shared local neighborhood relationships.

1. Introduction

The paper introduces permutation-based null calibration to remove width- and depth-related inflation in representational similarity measurements. After calibration, global convergence largely disappears, while local neighborhood relationships retain significant cross-modal alignment.

  • The Aristotelian Representation Hypothesis: Neural networks trained with different objectives, data, and modalities converge to shared local neighborhood relationships.The proposed Aristotelian Representation Hypothesis emphasizes which representations are near one another rather than universal distances.
  • Confounders: Embedding width can create positive similarity baselines under independence when dimensionality is large relative to sample size.This dimensionality-driven null inflation systematically distorts interaction-matrix-based similarity metrics.
  • Confounders: Taking the maximum across many layer-pair comparisons inflates reported similarity because the selection process searches a larger space.This depth-related confounder can produce high aggregate scores even without genuine representational relationships.
  • Null calibration: Null calibration transforms any similarity metric into a calibrated score by comparing observed similarity with a permutation-based empirical null distribution.For aggregate statistics, the null distribution uses the same reported summary, such as a maximum over layer pairs, thereby calibrating selection itself.
  • Revisiting the Platonic Representation Hypothesis: After calibration, previously reported convergence in global metrics largely disappears, whereas local neighborhood metrics retain significant cross-modal alignment.The findings suggest that apparent global convergence was primarily driven by width and depth confounders.

2. Related work

Related work spans global and local representational similarity metrics, while highlighting biases that can distort comparisons across models and scales. Prior research motivates calibrated, metric-agnostic evaluation of the Platonic Representation Hypothesis across architectures and modalities.

  • Representational similarity metrics: CCA, SVCCA, and PWCCA compare representation subspaces up to linear transformations, while Procrustes- and shape-based distances use more restricted alignments.The passage identifies Canonical Correlation Analysis (CCA), Singular Vector Canonical Correlation Analysis (SVCCA), and Projection Weighted Canonical Correlation Analysis (PWCCA) as representative metrics.
  • Reliability of representational similarity metrics: Similarity-measure choice can substantially alter conclusions about which models align with the brain, motivating scores calibrated and comparable across metrics and scales.This consequence is emphasized in neuro-AI applications of representational similarity analysis.
  • Reliability of representational similarity metrics: Raw similarity scores can be biased in finite-sample, high-dimensional settings, with distance concentration inflating random k-NN overlap and existing debiasing remaining metric-specific.Debiased CKA has been proposed, but no analogous debiasing method exists for neighborhood-based metrics.
  • Reliability of representational similarity metrics: Prior aggregate-calibration methods had not been applied to representational similarity studies, whereas the proposed framework addresses finite-sample bias and selection inflation in a metric-agnostic way.The framework is presented as a unified calibration approach rather than a correction tied to one similarity metric.
  • The Platonic Representation Hypothesis: The Platonic Representation Hypothesis proposes that scaling drives convergence across architectures and modalities, with convergence reported using both global and local similarity measures.The cited work examines whether differently trained neural networks develop similar representations, including across vision and language.

3. Problem setup

The problem setup compares aligned representation embeddings using spectral, neighborhood, and geometric similarity metrics. It defines a permutation-based null in which pairings are shuffled, because raw similarity requires a metric- and preprocessing-dependent baseline that can confound comparisons.

  • Representation spaces: Representations X and Y are row-wise aligned embeddings for paired inputs, and s(X, Y) quantifies their agreement.X and Y contain n samples in representation spaces of dimensions d_x and d_y, respectively.
  • Metric families: The study considers spectral, neighborhood, and geometric metric families, including CKA, CCA, mKNN, and RSA.These families respectively assess matrix spectra, local topological overlap, and second-order isomorphism.
  • Null hypothesis: Similarity scores are interpreted relative to a null hypothesis asserting no relationship beyond the marginal statistics of X and Y.The null baseline is operationalized by independently permuting Y’s sample indices while holding X fixed.
  • Null hypothesis: Under the null, exchangeability makes the observed pairing statistically indistinguishable from any permutation of correspondences.Formally, the joint distribution of paired samples is invariant under every permutation π of the sample indices.
  • Baseline problem: The null expected similarity need not be zero because its baseline depends on the metric, preprocessing, sample size, and dimension.This baseline therefore acts as a confounding variable in comparative studies.

4. Theoretical motivation: spurious alignment

Raw representational similarity is confounded by representation width and by the size of layer-pair searches. Spectral and geometric metrics acquire width-dependent null baselines, while selection maxima inflate with model depth, making uncalibrated comparisons unreliable.

  • Width confounder: Independent high-dimensional representations can have non-zero null interaction energy, invalidating the intuition that spectral similarity should vanish under independence.This non-vanishing energy underlies the width confounder in spectral-family metrics.
  • Width confounder: O(d/n) is the leading null baseline for spectral metrics, increasing with representation width d at fixed sample size n.The effect persists even when representations share genuine signal.
  • Metric-specific baselines: O(k/n) is the null baseline for neighborhood metrics, which are less confounded than spectral metrics because k ≪d in typical settings.Neighborhood scale k can be fixed across experiments, whereas embedding dimension d is architecture-dependent.
  • Metric-specific baselines: Geometric metrics such as Procrustes inherit the width-dependent O(d/n) baseline because they depend on all-pairs distances or inner products.This contrasts with the O(k/n) baseline of rank-based neighborhood metrics.
  • Depth confounder: Larger layer-pair search spaces inflate raw maximum-alignment scores, so deeper models can appear more aligned purely because M = LALB is larger.The depth confounder operates hierarchically on top of the width confounder, increasing the pool of chance similarities.

5. Representational similarity calibration

The section introduces permutation-based null calibration to correct width- and depth-dependent similarity inflation by converting raw scores into effect sizes with a principled zero point. It extends this calibration to selection-based layer aggregates while providing finite-sample Type-I control under exchangeability.

  • Scalar calibration: Permutation-based null calibration compares observed similarity with scores from permuted sample correspondences, accounting for finite-sample and high-dimensional artifacts.The framework transforms raw similarity scores into an effect size with a principled zero point.
  • Scalar calibration: The permutation p-value is super-uniform under H0, and the rule “scal > 0” gives a finite-sample α-level declaration of similarity above chance.This guarantee holds under Assumption 3.1 and implies p ≤α for the gating rule.
  • Scalar calibration: The calibrated score rescales excess similarity above the null threshold τα to [0, 1], preserving perfect similarity at 1 while adapting to scale-dependent null baselines.For unknown or unbounded maxima, the method uses the unnormalized effect size [s −τα]+ = max(s −τα, 0).
  • Aggregation-aware calibration: Aggregation-aware calibration calibrates the complete selection pipeline, such as a maximum over layer pairs, rather than applying post-selection p-values to individual comparisons.The approach generalizes maxT to arbitrary selection-based aggregates and similarity metrics.
  • Aggregation-aware calibration: The aggregate permutation p-value pagg is super-uniform under H0, while calibrated aggregate magnitude and evidence are reported together with multiplicity correction when needed.The aggregate score uses the null threshold τagg and reports Tcal alongside pagg.

6. Experiments

Controlled experiments show that permutation-based calibration removes width and depth inflation while preserving statistical validity and sensitivity to genuine alignment. After calibration, global convergence largely disappears, whereas local neighborhood agreement remains across modalities, motivating the Aristotelian Representation Hypothesis.

  • Width calibration: Calibration eliminates width-related similarity inflation across metrics, regimes, and noise distributions, including under genuine shared signals.Under the null, calibrated scores stay at zero while raw scores drift with d/n; calibration also corrects width inflation in real networks at fixed n.
  • Statistical guarantees: Calibration controls Type-I error at nominal α = 0.05 and preserves sensitivity, with detection rates increasing rapidly as signal strength grows.The empirical calibration also closely matches analytical CKA bias corrections without metric-specific derivations.
  • Depth calibration: Aggregation-aware calibration removes depth inflation: calibrated max-aggregates remain stable across layer counts, whereas naive entry-wise calibration still inflates.Under H0, raw max-scores are higher at L = 128 than at L = 2 despite no true signal.
  • Platonic Representation Hypothesis: After calibration, global CKA shows no systematic increase with model size, while local neighborhood metrics retain strong cross-modal convergence.The local trend also holds for cycle-kNN and CKNNA and across different α choices.
  • Platonic Representation Hypothesis: Models increasingly agree on which points are neighbors but not on pairwise distances, as CKA-RBF with a small bandwidth shows no calibrated alignment.This supports convergence in local neighbor structure rather than local distances.
  • Platonic Representation Hypothesis: Calibration weakens capability correlations for global metrics but leaves local correlations essentially unchanged, and video–language results reproduce this global-local distinction.For linear CKA, correlation falls from 0.86 to 0.45; for Procrustes distance, from 0.89 to 0.39, while mKNN remains near its original value.

7. Conclusion

The paper introduces a unified null-calibration framework that corrects width- and depth-related distortions in representational similarity metrics. Calibrated experiments support local convergence while finding little evidence for global convergence, with conclusions limited by the lack of a ground-truth scale and exchangeability assumptions.

  • Conclusion: The unified null-calibration framework corrects width-dependent null baselines and depth-dependent selection inflation, yielding effect sizes with principled zero points and valid p-values.The framework is designed to address both artifacts systematically.
  • Relationship to the Platonic hypothesis: The Aristotelian hypothesis is weaker than the Platonic hypothesis: global convergence implies local convergence, but local convergence does not imply global convergence.Experiments support the local form and, after calibration, find little evidence for the global form.
  • Limitations and outlook: The conclusions report calibrated evidence for or against convergence rather than proof because representational similarity lacks a ground-truth scale.This limitation affects how convergence claims should be interpreted.
  • Limitations and outlook: The calibration guarantees assume exchangeability, so grouped or clustered samples require restricted permutations that preserve their dependence structure.Why local neighborhoods converge without global geometric convergence remains an open question.

A. Existing calibration approaches for representational similarity metrics

Table 1 compares prior representational-similarity studies by whether they explicitly debias scores, preserve interpretable bounds, and calibrate selection-based aggregates.

  • A. Existing calibration approaches for representational similarity metrics: The comparison records whether each prior work applies an explicit null correction to its reported similarity.“Debias” denotes explicit null correction.
  • A. Existing calibration approaches for representational similarity metrics: The table evaluates whether corrected similarity scores retain an interpretable upper bound, such as 1 for perfect alignment.“Bounded” refers to preservation of an interpretable upper bound.
  • A. Existing calibration approaches for representational similarity metrics: It also indicates whether methods calibrate selection-based aggregates, including maxima over layer pairs.“Agg-aware” covers calibration of selection-based aggregates.

B. Metrics and score definitions … C.5. The depth confounder

The paper defines spectral, geometric, and neighborhood similarity metrics and establishes that width and depth can inflate uncalibrated scores. Its theoretical results justify permutation calibration as a dependence-robust correction with finite-sample validity.

  • B. Metrics and score definitions; B.1. Preprocessing and basic notation: Representations are centered before similarity computation, and the paper uses CKA, RSA, and mutual k-NN as representative spectral, geometric, and neighborhood metrics.Additional metrics include SVCCA, PWCCA, cycle-kNN, CKNNA, RV coefficient, and Procrustes.
  • B.2. Raw similarity metrics; B.2.1. SPECTRAL METRICS: Linear CKA measures agreement in global second-order geometry, whereas kernel CKA compares kernel-induced similarity patterns and can emphasize local nonlinear structure with small RBF bandwidth.Linear CKA is rotation- and isotropic-rescaling-invariant, while kernel CKA typically lies in [0, 1].
  • B.2.1. SPECTRAL METRICS: CCA-family metrics measure linear subspace alignment, with SVCCA retaining high-variance directions and PWCCA weighting canonical correlations by their importance.The RV coefficient captures global Gram-matrix geometry and coincides exactly with linear CKA on centered representations.
  • B.2.2. GEOMETRIC METRICS; B.2.3. NEIGHBORHOOD METRICS: RSA compares pairwise dissimilarity rankings, Procrustes compares absolute geometry after orthogonal alignment, and neighborhood metrics compare shared nearest-neighbor structure while ignoring exact distances.Cycle-kNN is stricter than mutual k-NN because it requires round-trip neighborhood recognition, while CKNNA combines CKA normalization with mutual-neighbor masking.
  • C. Theoretical Derivations; C.1. Permutation validity, super-uniformity, and gating: Permutation p-values are super-uniform under the exchangeability assumption, providing finite-sample Type-I error control; restricted permutations recover validity when grouped or clustered dependence violates exchangeability.The add-one p-value remains conservative in the presence of ties.
  • C.2. Monotone invariance of rank-based calibration; C.3. Post-selection inflation and aggregation-aware validity: Rank-based calibration is invariant to strictly increasing transformations, and calibrating any measurable aggregation of layer-wise similarities preserves super-uniformity under the same permutation scheme.This covers transformations such as max, row-max, and top-k aggregation.
  • C.4. The width confounder: Under independence, null interaction energy is O(dxdy/n), producing a positive width-dependent baseline when dimensions are comparable to sample size; with genuine signal, sample linear CKA converges to 1 as d →∞ despite population CKA equaling ρ2.CCA, SVCCA, and PWCCA have different width behavior because whitening removes the energy analyzed for spectral metrics, motivating permutation rather than brittle closed-form corrections.
  • C.5. The depth confounder: With M = LALB layer-pair comparisons and sub-Gaussian null tails, the maximum similarity fluctuation is bounded by 3σ√log M without requiring independence.Under i.i.d. or weak dependence, sharper Gumbel asymptotics are available heuristically, but the stated bound is dependence-robust.

C.6. Null Baselines for Neighborhood Metrics

Under independent i.i.d. representations, mutual k-NN neighborhoods have a dimension-independent null baseline determined by random overlap among k-subsets. Their per-anchor overlap is hypergeometric, while aggregate variance is O(1/n) only under an additional cross-anchor independence assumption.

  • Definition C.8: Mutual k-NN overlap compares the k-nearest-neighbor index sets for each sample across two representations.The neighborhoods may be defined using Euclidean or cosine distance, with ties handled by uniform random selection.
  • Proposition C.9: Under i.i.d. sampling, each anchor’s k-NN index set is uniformly distributed over the k-subsets of the other n −1 samples.This requires almost surely distinct distances or uniform random tie-breaking.
  • Theorem C.10: For independent representations, the per-anchor neighborhood intersection is hypergeometric, yielding a mutual k-NN null baseline of k/(n −1).The result follows because the two neighborhood sets are independent uniform k-subsets of the same n −1-element index set.
  • Proposition C.11: With the additional modeling assumption that per-anchor terms are independent, Var[mKNN(X, Y)] = O(1/n).This independence across anchors is not implied by the null hypothesis and gives O(1/n) for fixed k.

D. Implementation … E.5. Width confounder under genuine signal on real networks

The paper specifies permutation-based calibration procedures for scalar and layer-wise similarity comparisons, then validates calibration across synthetic regimes, correction variants, analytical debiasing, and real pretrained networks. These analyses show that calibration removes width- and noise-induced inflation while preserving sensitivity to genuine shared structure.

  • D. Implementation: Scalar null calibration permutes sample indices, returns a permutation p-value, and produces a calibrated score with a principled zero point.The procedure requires only a similarity function and uses a ceiling order statistic rather than an interpolated quantile.
  • D. Implementation: Aggregation-aware calibration applies the same sample permutation across all layers before aggregating layer-wise similarities, while scalar calibration costs O((K+1) Csim).For an LA × LB layer grid, aggregation-aware calibration costs O((K+1) LALB Csim), and null evaluations can run in parallel.
  • E. Additional Experimental Results: Additional analyses evaluate whether calibration remains reliable beyond the Gaussian assumptions used for tractable theory, including heavy-tailed, sparse, and multimodal activation regimes.The appendix extends the main-paper validation with phase diagrams, signal-to-noise heatmaps, correction comparisons, analytical debiasing, and real-network experiments.
  • E.1. Phase diagrams across different noise distributions: Across Gaussian, Student-t, Laplace, and Gaussian-mixture noise, calibrated scores collapse toward zero at high noise while raw scores retain systematic positive bias.At low noise, both raw and calibrated scores correctly indicate high similarity when the shared signal dominates.
  • E.2. SNR heatmaps: Calibrated scores reveal detection boundaries: they concentrate in the low-noise, high-signal regime, decrease monotonically with noise, and collapse to zero when signal is buried.Higher-rank or stronger signals remain detectable across higher noise levels and wider noise ranges.
  • E.3. Comparing calibration approaches: Gated score, null-centered, and ARI-style corrections recover appropriate null baselines across metrics, whereas z-score calibration can artifact under skewed null distributions.This holds for severely inflated CKA, approaching 0.8, and mildly inflated RSA and mKNN, below 0.1.
  • E.4. Comparison with analytical debiasing: Calibrated CKA closely matches debiased CKA, correcting dominant width-induced bias without metric-specific derivation, while dep-cols CKA does not address the experimental bias source.The calibration targets self-similarity diagonal terms and applies to metrics lacking analytical debiasing.
  • E.5. Width confounder under genuine signal on real networks: r = 0.88: on DINOv2 and AugReg ViT pairs with n = 1024 fixed, the permutation threshold τ increases with total representation width dX + dY.Because width varies within model families while training objective, data, architecture, and images remain controlled, the correction’s dependence is attributed to width alone.

E.6. Permutation budget analysis

Permutation calibration involves a runtime–stability tradeoff: threshold estimates stabilize by approximately K = 50, while calibrated scores are more variable below K = 50. The analysis recommends K ≥200 as a precision–efficiency compromise.

  • Tradeoff: Permutation-based calibration trades more stable threshold estimates against higher runtime, requiring guidance on the minimum reliable budget.Computational cost increases with the number of permutations.
  • Stability: Threshold estimates τα stabilize rapidly by approximately K = 50 across all tested metrics, while calibrated scores under H0 vary more at K < 50.Calibrated scores under H0 converge to near-zero by K ≈100–200.
  • Recommendation: K ≥200 is recommended because permutation cost scales linearly with K, providing a favorable tradeoff between precision and efficiency.The recommendation is based on threshold and calibrated-score stability across 50 random seeds.

E.7. Full null drift results

Full null-drift results show that raw representational-similarity scores vary substantially across metric families and exhibit systematic positive bias under null conditions. Permutation calibration removes this spurious drift across Gaussian and heavy-tailed noise, while raw CKA also inflates with dimensionality under alternative generative processes.

  • Gaussian null drift: Null-baseline severity and shape vary substantially across metric families under Gaussian noise.Figure 15 evaluates all metrics, including RSA, the RV coefficient, and Procrustes distance.
  • Gaussian null drift: Spectral metrics show the strongest width drift, whereas neighborhood metrics show the mildest drift.CKA variants and the RV coefficient inflate monotonically; mean CCA peaks near d ≈n, PWCCA saturates, and SVCCA is width-insensitive.
  • Heavy-tailed robustness: Calibration eliminates positive null drift under Student-t noise, whose raw bias is comparable to Gaussian noise.The heavy-tailed experiments use Student-t noise with ν = 3, and calibration adapts without distributional knowledge.
  • Robustness to the generative process: Raw CKA inflates with dimensionality in every tested generative regime, even when representations share inputs but no systematic structure.The robustness experiments use independent random linear maps or MLPs applied to a shared Gaussian input.

E.8. Extended PRH alignment results (image–text) … E.11. Sensitivity to significance level α

Across image–text and video–language experiments, calibrated neighborhood metrics retain cross-modal alignment while calibrated global metrics lose their apparent convergence with model scale. The remaining alignment concerns neighborhood identity rather than exact local distances and is robust to significance-level choice.

  • E.8. Extended PRH alignment results (image–text): Neighborhood metrics retain consistent calibrated alignment across all five vision model families, whereas calibrated spectral metrics show no scaling trend.This holds across self-supervised, supervised, CLIP-aligned, and CLIP-finetuned variants.
  • E.8. Extended PRH alignment results (image–text): All 204 vision–language model pairs are significant at p < 0.05, with most achieving p ≈0.002 for both local and global metrics.The minimum attainable p-value is approximately 0.002 with K = 500 permutations.
  • E.9. Extended video–language alignment results: Video–language results reproduce the image–language pattern: calibrated spectral alignment drops, while calibrated neighborhood alignment remains significant and strengthens with video-encoder capability.The evaluation includes VideoMAE, DINOv2, and CLIP frame-level baselines across BLOOM, OpenLLaMA, and LLaMA families.
  • E.10. Characterizing the locality of cross-modal alignment: The contrast between mKNN and small-σ CKA-RBF indicates agreement on neighborhood identity but not on exact local distances.mKNN captures ordinal neighbor rankings, whereas small-σ CKA-RBF evaluates cardinal distance relationships.
  • E.10. Characterizing the locality of cross-modal alignment: mKNN shows significant calibrated alignment for every tested neighborhood size k ∈{10, 20, 50, 100}, with the strongest scaling trend at small k.At k = 10, the upward capacity trend persists after calibration; at k = 100, the raw trend plateaus for larger models.
  • E.10. Characterizing the locality of cross-modal alignment: At σ = 0.1, CKA-RBF has no significant calibrated alignment for most model pairs, while larger bandwidths show significant alignment with flattened scaling trends.Raw scores near 1.0 at σ = 0.1 collapse to approximately zero after calibration, with p-values exceeding 0.05 for most pairs.
  • E.11. Sensitivity to significance level α: The qualitative conclusions are invariant across α ∈{0.01, 0.05, 0.10}: calibrated global metrics lack convergence trends, while calibrated local metrics retain alignment trends.Stricter α = 0.01 yields slightly lower calibrated scores and permissive α = 0.10 slightly higher scores.
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