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Revisiting the Platonic Representation Hypothesis: An Aristotelian View

Fabian Gröger, Shuo Wen, Maria Brbić

TL;DR

The paper addresses the alleged convergence of neural representations across modalities by showing that width and depth confounders bias traditional similarity metrics. It introduces a permutation-based null-calibration that yields calibrated similarity scores with finite-sample guarantees, correcting for both width-driven null baselines in spectral metrics and depth-driven selection inflation. Applied to cross-modal benchmarks, calibration largely eliminates global convergence signals but reveals robust local neighborhood alignment, supporting an Aristotelian Representation Hypothesis that networks converge to shared local neighborhood structures rather than global distances. This reframes how researchers should assess representation convergence and provides practical, metric-agnostic tools for robust comparisons that can impact transfer learning and neuroscience interpretations, with formal guarantees under $H_0$ via $\tau_\alpha$ thresholds and $p$-values.

Abstract

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.

Revisiting the Platonic Representation Hypothesis: An Aristotelian View

TL;DR

The paper addresses the alleged convergence of neural representations across modalities by showing that width and depth confounders bias traditional similarity metrics. It introduces a permutation-based null-calibration that yields calibrated similarity scores with finite-sample guarantees, correcting for both width-driven null baselines in spectral metrics and depth-driven selection inflation. Applied to cross-modal benchmarks, calibration largely eliminates global convergence signals but reveals robust local neighborhood alignment, supporting an Aristotelian Representation Hypothesis that networks converge to shared local neighborhood structures rather than global distances. This reframes how researchers should assess representation convergence and provides practical, metric-agnostic tools for robust comparisons that can impact transfer learning and neuroscience interpretations, with formal guarantees under via thresholds and -values.

Abstract

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.
Paper Structure (77 sections, 10 theorems, 67 equations, 25 figures, 1 table, 2 algorithms)

This paper contains 77 sections, 10 theorems, 67 equations, 25 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Representational similarity metrics.
  4. Reliability of representational similarity metrics.
  5. The Platonic Representation Hypothesis.
  6. Problem setup
  7. Representation spaces and similarity score
  8. The null hypothesis of independence
  9. Baseline problem: non-zero null expectations
  10. Theoretical motivation: spurious alignment
  11. The width confounder
  12. Neighborhood metrics follow a different regime.
  13. The depth confounder
  14. Representational similarity calibration
  15. Null-calibrated similarity
  16. ...and 62 more sections

Key Result

Proposition 1

Assume the rows are i.i.d. with $\mathbb{E}[\mathbf{x}_i]=\mathbb{E}[\mathbf{y}_i]=0$, $\mathrm{Cov}(\mathbf{x}_i)=\mathbf{I}_{d_x}$, $\mathrm{Cov}(\mathbf{y}_i)=\mathbf{I}_{d_y}$, and $\mathbf{x}_i$ and $\mathbf{y}_i$ are independent. Then

Figures (25)

  • Figure 1: The Aristotelian Representation Hypothesis: Local relations ("who is near whom"), rather than distances between data points, are preserved across different representation spaces. Representation learning algorithms will converge to shared local neighborhood relationships.
  • Figure 2: Null calibration removes width and depth confounders. (a) Width confounder: raw scores exhibit positive null baselines that increase with the ratio of dimension (width) of the spaces and the number of samples; calibration collapses them to zero. (b) Depth confounder: selection-based summaries (max over layers) inflate with search space size; aggregation-aware calibration removes this. (c) After calibration, global metrics lose their convergence trend, while local metrics retain significant alignment.
  • Figure 3: Calibration eliminates spurious similarity across metrics. Raw scores (top) drift with $d/n$; calibrated scores (bottom) collapse to zero. Results for heavy-tailed distributions and additional metrics are in \ref{['app:null-drift-full']}.
  • Figure 4: Statistical guarantees. (Left) Type-I error stays at or below $\alpha$ across configurations. (Right) Power increases rapidly with signal strength; calibration does not sacrifice sensitivity.
  • Figure 5: Aggregation-aware calibration removes depth confounding. Raw max-aggregates of linear scores inflate with layer count under the null; calibrated aggregates are stable and show that naive entry-wise calibration still leads to inflation.
  • ...and 20 more figures

Theorems & Definitions (21)

  • Proposition 1: Non-vanishing null interaction energy
  • Proposition 2: Null baseline for neighborhood metrics
  • Corollary 1: Type-I control for calibrated scores
  • Definition 1: Super-uniformity
  • Lemma 1: Permutation $p$-values are super-uniform
  • proof : Proof of \ref{['lem:perm_super_uniform']}
  • proof : Proof of \ref{['cor:gating_type1']}
  • Proposition 3: Monotone invariance of rank-based calibration lehmann2005testing
  • proof
  • Proposition 4: Validity for aggregation-aware calibration
  • ...and 11 more