Revisiting the Platonic Representation Hypothesis: An Aristotelian View — part 1/8

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# Revisiting the Platonic Representation Hypothesis: An Aristotelian View

###### 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.

###### Keywords:

## 1 Introduction

Quantifying the similarity between neural network representations is central to understanding the geometry of learned representation spaces ( Raghu et al., 2017 ; Nguyen et al., 2021 ) , guiding transfer learning decisions ( Kornblith et al., 2019 ; Neyshabur et al., 2020 ; Gröger et al., 2025 ) , and relating artificial representations to neural measurements in neuroscience ( Schrimpf et al., 2018 ) . The Platonic Representation Hypothesis ( Huh et al., 2024 ) posits that as neural networks scale, representations across different modalities become increasingly similar, suggesting convergence to a shared statistical model of reality. This hypothesis has motivated a growing literature that uses representational similarity to study whether scaling produces universal structure across models ( Huh et al., 2024 ; Maniparambil et al., 2024 ; Tjandrasuwita et al., 2025 ; Zhu et al., 2026 ) . To measure representational similarity across models, different metrics have been proposed, such as Centered Kernel Alignment ( Kornblith et al., 2019 ) , Canonical Correlation Analysis ( Weenink, 2003 ) , Representational Similarity Analysis ( Kriegeskorte et al., 2008 ) , and mutual k k -Nearest Neighbors ( Huh et al., 2024 ) .

In this work, we identify two pervasive confounders that distort representational similarity measurements. The first is the model width : when the embedding dimension increases relative to the sample size, interaction-matrix-based similarity metrics exhibit a systematic positive baseline even when representations are independent. This spurious similarity is a general consequence of dimensionality-driven null inflation: the expected similarity under independence does not vanish but instead depends on both the representation dimensionality and the sample size ( Figure 2 a). As a result, wider models can appear more aligned simply because their representations live in higher-dimensional spaces. The second confounder is the model depth . Many analyses do not compare individual layer pairs, because it is unknown where similarity arises ( Schrimpf et al., 2018 ; Huh et al., 2024 ) . Instead, they search over all pairs and report a summary statistic such as the maximum. This is widespread practice: studies routinely compute similarity over many layer pairs and report an aggregate such as the maximum or best-matching layer ( Raghu et al., 2017 ; Kornblith et al., 2019 ; Schrimpf et al., 2018 ; Huh et al., 2024 ; Kapoor et al., 2025 ) . Taking a maximum over many comparisons inflates the reported score even if there is no similarity, since the expected maximum of independent draws exceeds the mean. This inflation grows with the number of comparisons, so deeper models can appear more aligned simply because more layer pairs are compared ( Figure 2 b). Together, these confounders undermine the comparative use of representational similarity without calibration.

To address these issues, we introduce the null-calibration for representational similarity , a general permutation-based framework that transforms any similarity metric into a calibrated score with a principled null reference, here defined as no relationship ( Figure 2 ). The core idea is to measure how extreme an observed similarity is relative to an empirical null distribution obtained by breaking sample correspondences. For scalar comparisons ( i.e. , width confounder), we estimate a critical threshold from the null distribution and define a calibrated score that is zero when the observed similarity falls below this threshold and rescaled to preserve the maximum at one. For selection-based summaries ( i.e. , depth confounder), we apply aggregation-aware calibration. We compute the null distribution of the same aggregate statistic that is ultimately reported ( e.g. , the maximum over all layer pairs), thereby calibrating the selection step itself.

These observations raise a question: Does the Platonic Representation Hypothesis still hold once similarity is calibrated? We find that, after calibration, the previously reported convergence in global metrics ( Huh et al., 2024 ; Maniparambil et al., 2024 ; Tjandrasuwita et al., 2025 ) largely disappears, suggesting it was driven primarily by width and depth confounders, whereas local neighborhood-based metrics retain significant cross-modal alignment ( Figure 2 c). However, we also observe that the convergence in local distances is not preserved, suggesting that only local neighborhood relationships are aligned. Motivated by these results, we refine the original Platonic Representation Hypothesis and propose the Aristotelian Representation Hypothesis 1 1 1 Calling this refinement Aristotelian : it emphasizes learned representations converging on relations among instances (who is near whom) rather than the idea of convergence toward a globally matching structure. : Neural networks, trained with different objectives on different data and modalities, converge to shared local neighborhood relationships ( Figure 1 ). We name it after the Greek philosopher Aristotle, who was a student of Plato and, in his Categories, established the principles of relatives ( Aristotle, ca. 350 B.C.E ) .

## 2 Related work

##### Representational similarity metrics.

A long line of work compares representation spaces using a variety of similarity measures. Canonical Correlation Analysis (CCA) ( Hotelling, 1992 ) and variants such as Singular Vector Canonical Correlation Analysis (SVCCA) ( Raghu et al., 2017 ) and Projection Weighted Canonical Correlation Analysis (PWCCA) ( Morcos et al., 2018 ) compare subspaces up to linear transformations, while Procrustes- and shape-based distances compare representations up to restricted alignment classes ( Ding et al., 2021 ; Williams et al., 2021 ) . Centered Kernel Alignment (CKA) ( Kornblith et al., 2019 ) has become a dominant tool for comparing deep representations, with kernelized variants extending to nonlinear similarity. Representational Similarity Analysis (RSA) ( Kriegeskorte et al., 2008 ) , originating in neuroscience, compares representational dissimilarity matrices rather than feature bases. Neighborhood-based approaches, such as mutual k k -Nearest Neighbors (mKNN) ( Huh et al., 2024 ) , capture local topological consistency rather than global alignment. However, recent evaluations stress that different metrics encode different invariances and can yield qualitatively different conclusions, motivating more robust reporting practices ( Klabunde et al., 2025a ; Klabunde et al., 2025b ; Ding et al., 2021 ; Harvey et al., 2024 ; Bo et al., 2024 ) . This sensitivity is especially consequential in the neuro-AI literature, where the choice of similarity measure can profoundly change conclusions about which models best align with the brain ( Soni et al., 2024 ) , which is why scores that are calibrated and comparable across metrics and scales are needed.

##### Reliability of representational similarity metrics.

In finite-sample, high-dimensional regimes, raw similarity scores can be systematically biased. Recent works ( Murphy et al., 2024 ; Chun et al., 2025 ) propose debiased CKA , but these corrections are metric-specific . For neighborhood-based metrics, no analogous debiasing methods exist despite distance concentration effects that inflate random k k -NN overlap ( Beyer et al., 1999 ; Aggarwal et al., 2001 ) . Other approaches address confounding from input population structure . For instance, Cui et al. (2022) propose regression-style deconfounding to remove effects of shared input statistics on RSA / CKA . Analogous concerns arise for regression-based predictivity in neuroscience, where high-dimensional fits can inflate alignment scores and may fail to identify genuinely brain-like models ( Schaeffer et al., 2024 ) . A separate reliability issue arises from layer search, where max or top- k k aggregation across many layer pairs introduces multiple-comparison inflation. While resampling-based “maxT” procedures ( Westfall and Young, 1993 ; Nichols and Holmes, 2002 ) can calibrate such aggregates, this has not yet been applied in representational similarity studies. Our calibration framework addresses both finite-sample bias and selection inflation in a unified, metric-agnostic way.

##### The Platonic Representation Hypothesis.

A growing body of work examines whether neural networks trained under different conditions converge toward similar representations. The Platonic Representation Hypothesis ( Huh et al., 2024 ) posits that as models scale, their representations increasingly converge across architectures and even across modalities such as vision and language, with convergence reported under both global and local similarity measures. Follow-up work has examined factors influencing these trends, including model size, training duration, and data distribution ( Raugel et al., 2025 ) , and has explored analogous convergence effects in broader settings such as video models ( Zhu et al., 2026 ) and comparisons to biological vision ( Marcos-Manchón and Fuentemilla, 2025 ) . Prior work attributes convergence to shared factors such as training data distribution, scale, and objective functions ( Raugel et al., 2025 ; Huh et al., 2024 ) . Our calibration-based analysis does not contradict these findings but refines their interpretation, showing that the evidence for convergence depends on whether it is measured by global or local similarity. In this work, we revisit the Platonic Representation Hypothesis using our null-calibration framework that controls for width and depth confounders.

## 3 Problem setup

### 3.1 Representation spaces and similarity score

Let 𝒳 ⊆ ℝ d x \mathcal{X}\subseteq\mathbb{R}^{d_{x}} and 𝒴 ⊆ ℝ d y \mathcal{Y}\subseteq\mathbb{R}^{d_{y}} be two representation spaces, where d x d_{x} and d y d_{y} are the respective space dimensions. For a set of n n input samples, let 𝐗 ∈ ℝ n × d x \mathbf{X}\in\mathbb{R}^{n\times d_{x}} and 𝐘 ∈ ℝ n × d y \mathbf{Y}\in\mathbb{R}^{n\times d_{y}} be the corresponding embeddings in 𝒳 \mathcal{X} and 𝒴 \mathcal{Y} . We assume row-wise alignment such that the i i -th row of 𝐗 \mathbf{X} and 𝐘 \mathbf{Y} correspond to paired inputs. We use a similarity score s ⁡ ( 𝒳 , 𝒴 ) ∈ ℝ s(\mathcal{X},\mathcal{Y})\in\mathbb{R} to quantify the agreement between 𝒳 \mathcal{X} and 𝒴 \mathcal{Y} . In practice, we compute it from 𝐗 , 𝐘 \mathbf{X},\mathbf{Y} and, by a slight abuse of notation, denote it with s ⁡ ( 𝐗 , 𝐘 ) s(\mathbf{X},\mathbf{Y}) .

We consider three families of metrics: (i) spectral: metrics defined on the spectrum of cross-covariance or Gram matrices ( e.g. , CKA , CCA ), (ii) neighborhood: metrics measuring local topological overlap ( e.g. , mKNN ), and (iii) geometric: second-order isomorphism metrics ( e.g. , RSA ). Appendix B provides definitions of the metrics used in this paper.

### 3.2 The null hypothesis of independence

We claim that a similarity score s ⁡ ( 𝐗 , 𝐘 ) s(\mathbf{X},\mathbf{Y}) is uninterpretable without a baseline. To provide this baseline, we define the null hypothesis H 0 H_{0} as the absence of a relationship between 𝐗 \mathbf{X} and 𝐘 \mathbf{Y} beyond their marginal statistics. We operationalize H 0 H_{0} via a permutation group Π n \Pi_{n} acting on sample indices: draw π ∼ Unif ⁡ ( Π n ) \pi\sim\mathrm{Unif}(\Pi_{n}) independently of ( 𝐗 , 𝐘 ) (\mathbf{X},\mathbf{Y}) and evaluate s ⁡ ( 𝐗 , π ⁡ ( 𝐘 ) ) s(\mathbf{X},\pi(\mathbf{Y})) , where π ⁡ ( 𝐘 ) \pi(\mathbf{Y}) permutes the rows of 𝐘 \mathbf{Y} .

###### Assumption 3.1 (Exchangeability under the null) .

Under H 0 H_{0} , the joint distribution of paired samples is invariant to relabeling of correspondences. For any permutation π ∈ Π n \pi\in\Pi_{n} , ℙ H 0 ​ ( 𝐗 , 𝐘 ) = ℙ H 0 ​ ( 𝐗 , π ⁡ ( 𝐘 ) ) \mathbb{P}_{H_{0}}(\mathbf{X},\mathbf{Y})=\mathbb{P}_{H_{0}}(\mathbf{X},\pi(\mathbf{Y})) .

This assumption implies that if no true relationship exists, the observed pairing is statistically indistinguishable from a random shuffling of the data. It allows us to construct an empirical null distribution by holding 𝐗 \mathbf{X} fixed and shuffling the rows of 𝐘 \mathbf{Y} .

### 3.3 Baseline problem: non-zero null expectations

Ideally, under H 0 H_{0} , we desire 𝔼 π ​ [ s ⁡ ( 𝐗 , π ⁡ ( 𝐘 ) ) ] ≈ 0 \mathbb{E}_{\pi}[s(\mathbf{X},\pi(\mathbf{Y}))]\approx 0 . However, for commonly used raw or biased estimators, the expected similarity under the null is not zero , μ 0 ​ ( n , d x , d y ) ≔ 𝔼 π ​ [ s ⁡ ( 𝐗 , π ⁡ ( 𝐘 ) ) ] . \mu_{0}(n,d_{x},d_{y})\,\coloneqq\,\mathbb{E}_{\pi}[s(\mathbf{X},\pi(\mathbf{Y}))]. (1) This baseline μ 0 \mu_{0} is metric- and preprocessing-dependent and can deviate from zero in finite samples. It also varies with sample size and dimension, thus acting as a confounding variable in comparative studies.

## 4 Theoretical motivation: spurious alignment

We motivate and formalize why raw representational similarity metrics fail in cross-scale model comparisons. We identify two distinct sources of confounding: (i) the width confounder driven by representation dimension, and (ii) the depth confounder driven by the number of layers considered when comparing models.

### 4.1 The width confounder

Many spectral-family similarity metrics, e.g. , linear/kernel CKA and the RV coefficient, can be written as functionals of an interaction operator constructed from two representations. One such operator is the (normalized) cross-covariance 𝐂 ~ = 1 n − 1 ​ 𝐗 c ⊤ ​ 𝐘 c ∈ ℝ d x × d y , \widetilde{\mathbf{C}}\;=\;\frac{1}{n-1}\mathbf{X}_{c}^{\top}\mathbf{Y}_{c}\in\mathbb{R}^{d_{x}\times d_{y}}, (2) where 𝐗 c \mathbf{X}_{c} and 𝐘 c \mathbf{Y}_{c} denote row-centered representations ( Section B.1 ).

A common but misleading intuition is that if 𝐗 \mathbf{X} and 𝐘 \mathbf{Y} are independent, then 𝐂 ~ ≈ 𝟎 \widetilde{\mathbf{C}}\approx\mathbf{0} and therefore spectral aggregates should be near zero. In high dimension this fails: the null interaction energy is typically non-zero.

###### Proposition 4.1 (Non-vanishing null interaction energy) .

Assume the rows are i.i.d. with 𝔼 ⁡ [ 𝐱 i ] = 𝔼 ⁡ [ 𝐲 i ] = 0 \mathbb{E}[\mathbf{x}_{i}]=\mathbb{E}[\mathbf{y}_{i}]=0 , Cov ⁡ ( 𝐱 i ) = 𝐈 d x \mathrm{Cov}(\mathbf{x}_{i})=\mathbf{I}_{d_{x}} , Cov ⁡ ( 𝐲 i ) = 𝐈 d y \mathrm{Cov}(\mathbf{y}_{i})=\mathbf{I}_{d_{y}} , and 𝐱 i \mathbf{x}_{i} and 𝐲 i \mathbf{y}_{i} are independent. Then 𝔼 H 0 ​ [ ‖ 𝐂 ~ ‖ F 2 ] = d x ​ d y n − 1 . \mathbb{E}_{H_{0}}\left[\|\widetilde{\mathbf{C}}\|_{F}^{2}\right]\;=\;\frac{d_{x}d_{y}}{n-1}. (3)

Proof. See Section C.4 .