Revisiting the Platonic Representation Hypothesis: An Aristotelian View

arxiv:2602.14486

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Representation alignment · 2026-02-16

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2. Part 2
3. Part 3
4. Part 4
5. Part 5
6. Part 6
7. Part 7
8. Part 8

Contents

• Revisiting the Platonic Representation Hypothesis: An Aristotelian View
• Abstract
• Keywords:
• 1 Introduction
• 2 Related work
• Representational similarity metrics.
• Reliability of representational similarity metrics.
• The Platonic Representation Hypothesis.
• 3 Problem setup
• 3.1 Representation spaces and similarity score
• 3.2 The null hypothesis of independence
• Assumption 3.1 (Exchangeability under the null) .
• 3.3 Baseline problem: non-zero null expectations
• 4 Theoretical motivation: spurious alignment
• 4.1 The width confounder
• Proposition 4.1 (Non-vanishing null interaction energy) .
• Neighborhood metrics follow a different regime.
• Proposition 4.2 (Null baseline for neighborhood metrics) .
• 4.2 The depth confounder
• 5 Representational similarity calibration
• 5.1 Null-calibrated similarity
• Corollary 5.1 (Type-I control for calibrated scores) .
• Calibrated score (scalar case).
• 5.2 Aggregation-aware null-calibration
• Consistency of permutations across layers.
• Calibrated score (aggregate case).
• 5.3 Summary
• 6 Experiments
• 6.1 Null-calibration removes width confounder
• 6.2 Null-calibration removes depth confounder
• 6.3 Revisiting the Platonic Representation Hypothesis
• 7 Conclusion
• Relationship to the Platonic hypothesis.
• Limitations and outlook.
• Acknowledgements
• Impact statement
• Appendix A Existing calibration approaches for representational similarity metrics
• Appendix B Metrics and score definitions
• B.1 Preprocessing and basic notation
• B.2 Raw similarity metrics
• B.2.1 Spectral metrics
• Linear Centered Kernel Alignment (CKA) .
• Kernel Centered Kernel Alignment .
• Unbiased Centered Kernel Alignment .
• Canonical Correlation Analysis (CCA) -based similarity.
• Singular Vector Canonical Correlation Analysis (SVCCA) .
• Projection Weighted Canonical Correlation Analysis (PWCCA) .
• RV coefficient.
• B.2.2 Geometric metrics
• Representational Similarity Analysis (RSA) via Spearman correlation of dissimilarity matrices.
• Procrustes distance.
• B.2.3 Neighborhood metrics
• Mutual k k -Nearest Neighbors (mKNN) .
• Cycle- k k NN (bidirectional k k -NN).
• CKA with Neighborhood Alignment (CKNNA).
• Appendix C Theoretical Derivations
• C.1 Permutation validity, super-uniformity, and gating
• Definition C.1 (Super-uniformity) .
• Lemma C.2 (Permutation p p -values are super-uniform) .
• Proof of Lemma C.2 .
• Proof of Corollary 5.1 .
• Restricted permutations under dependence.
• C.2 Monotone invariance of rank-based calibration
• Proposition C.3 (Monotone invariance of rank-based calibration ( Lehmann and Romano, 2005 ) ) .
• Proof.
• C.3 Post-selection inflation and aggregation-aware validity
• Proposition C.4 (Validity for aggregation-aware calibration) .
• Proof of Proposition C.4 .
• C.4 The width confounder
• Proof of Proposition 4.1 .
• Interpretation.
• Width inflation persists under genuine signal ( H 1 H_{1} ).
• Proposition C.5 (Width inflation under genuine signal) .
• Proof.
• CCA-based scores.
• Why we use permutation rather than closed forms.
• C.5 The depth confounder
• Assumption C.6 (Uniform sub-Gaussian right tails and integrability) .
• Proposition C.7 (Maximal inequality, no independence required) .
• Proof.
• Remark.
• C.6 Null Baselines for Neighborhood Metrics
• Definition C.8 (Mutual k k -NN overlap) .
• Proposition C.9 (Uniformity of k k -NN index sets under i.i.d. sampling) .
• Proof.
• Theorem C.10 (Null baseline for mutual k k -NN) .
• Proof.
• Proposition C.11 (Per-anchor variance and generic bounds for mKNN \mathrm{mKNN} under the null) .
• Proof.
• Appendix D Implementation
• Scalar null calibration.
• Aggregation-aware calibration for layer-wise comparisons.
• Computational cost.
• Appendix E Additional Experimental Results
• E.1 Phase diagrams across different noise distributions
• E.2 SNR heatmaps
• E.3 Comparing calibration approaches
• E.4 Comparison with analytical debiasing
• E.5 Width confounder under genuine signal on real networks
• E.6 Permutation budget analysis
• E.7 Full null drift results
• Robustness to the generative process.
• E.8 Extended PRH alignment results (image–text)
• Trend with model scale, before versus after calibration.
• Continuous language-performance axis.
• Statistical significance.
• E.9 Extended video–language alignment results
• E.10 Characterizing the locality of cross-modal alignment
• Experimental setup.
• RBF bandwidth.
• Neighborhood size.
• mKNN across k k values.
• CKA-RBF across bandwidth values.
• Topological versus metric alignment.
• E.11 Sensitivity to significance level α \alpha
• Instructions for reporting errors