Revisiting the Platonic Representation Hypothesis
An Aristotelian View

The Platonic story

Quantifying the similarity between neural representations is central to understanding learned representation spaces, guiding transfer learning, and connecting artificial representations to neural measurements. The Platonic Representation Hypothesis posits that as neural networks scale, their representations across modalities become increasingly similar, suggesting convergence to a shared statistical model of reality. This claim has motivated follow-up work on scaling, representation alignment, and brain-model comparisons. The trend is consistent across standard metrics, such as CKA, CCA, RSA, mutual $k$-NN, and across model families. The empirical pattern itself is well established. What we re-examine is whether the standard similarity metrics behind it actually support the convergence interpretation drawn from them.

Two confounders distort representational similarity

We identify two pervasive confounders that arise when similarity is measured between neural representations: the model width and the model depth. Both inflate raw scores in a way that grows with model scale, and together they undermine the comparative use of representational similarity without calibration.

The width confounder

Many spectral similarity metrics (linear and kernel CKA, the RV coefficient, CCA/SVCCA/PWCCA, RSA) can be written as functionals of an interaction operator built from two representations, such as the (normalized) cross-covariance $\widetilde{\mathbf{C}} = (n-1)^{-1}\mathbf{X}_{c}^{\top}\mathbf{Y}_{c}$. A natural intuition is that for independent $\mathbf{X}, \mathbf{Y}$ this operator is close to zero and the metric should be too. In high dimensions, this fails. For i.i.d., mean-zero, identity-covariance rows with $\mathbf{X}$ and $\mathbf{Y}$ independent,

and the corresponding CKA null baseline is $\mathcal{O}(d/n)$. Wider models can appear more aligned simply because their representations live in higher-dimensional spaces. The same inflation appears for any interaction-matrix-based similarity, even though the exact form of the null can differ.

Not every metric reacts to width this way. Neighborhood-based similarities (mutual $k$-NN, cycle-$k$-NN, CKNNA) compare a fixed-size set of nearest neighbors instead of an interaction operator in the feature space. Their null depends on the neighborhood size $k$ rather than on the representation dimension $d$.

The depth confounder

When comparing two networks $A$ and $B$ of depths $L_A$ and $L_B$, it is generally unknown at which pair of layers the optimal alignment arises. A common practice is therefore to compute the full $L_A \times L_B$ similarity matrix and report a selection-based summary, most often the maximum across layer pairs. Selection introduces a separate inflation. Under $H_0$, the maximum of $M = L_A L_B$ noisy measurements is inflated above the mean: a uniform sub-Gaussian right-tail bound, which holds for any bounded similarity metric via Hoeffding, gives:

The reported "alignment" therefore grows monotonically with the search space size, so deeper models can appear more aligned simply because more layer pairs are compared, even when nothing is shared between the representations.

Both inflations compound

Wider models inflate the per-pair null. Deeper models additionally inflate any selection-based aggregate over those pairs. The two effects stack, and both come from how the metric is computed at a given model scale, not from any agreement between the representations themselves.

The calibration framework

We operationalize the null hypothesis $H_0$ (no relationship between the two representations) by permuting sample correspondences. For $K$ permutations $\pi_k \in \Pi_n$ drawn i.i.d. uniformly, we form null scores $s^{(k)} = s(\mathbf{X}, \pi_k(\mathbf{Y}))$. Together with the observed score $s_{\mathrm{obs}} = s(\mathbf{X}, \mathbf{Y})$, they define a right-tail critical value $\tau_\alpha$ as the $(1-\alpha)$-quantile of the combined set $\{s_{\mathrm{obs}}, s^{(1)}, \ldots, s^{(K)}\}$. For bounded similarity metrics with known maximum $s_{\max}$ (typically $1$), the max-preserving calibrated score is

$\tau_\alpha$ is the principled zero point. The outer $\max(\cdot, 0)$ enforces $s_{\mathrm{cal}} = 0$ whenever $s_{\mathrm{obs}} \le \tau_\alpha$, i.e. when the observation falls inside the null. The denominator $s_{\max} - \tau_\alpha$ rescales the excess so $s_{\mathrm{cal}}$ lives in $[0, 1]$, with $0$ meaning indistinguishable from chance and $1$ meaning the maximum possible similarity.

The same framework removes both confounders

For the depth confounder, the reported number is a summary of a similarity matrix $\mathbf{S} \in \mathbb{R}^{L_A \times L_B}$, e.g. the maximum across layer pairs $T(\mathbf{S}) = \max_{\ell,\ell'} S_{\ell,\ell'}$. Here the null is the distribution of the same aggregate. For each draw $\pi_k$ we apply the same permutation across all layers, build $\mathbf{S}^{(k)}$, and aggregate to $T^{(k)} = T(\mathbf{S}^{(k)})$. The aggregate critical value is

and the calibrated aggregate keeps the same max-preserving form,

One framework, two uses. Scalar calibration removes the width confounder. Aggregation-aware calibration additionally removes the depth confounder.

Statistical guarantees

The permutation $p$-value is super-uniform under $H_0$: $\mathbb{P}_{H_0}(p \le \alpha) \le \alpha$ for all $\alpha \in [0, 1]$, under a standard exchangeability condition. By the same argument, $\mathbb{P}_{H_0}(s_{\mathrm{obs}} > \tau_\alpha) \le \alpha$, so the gating rule "$s_{\mathrm{cal}} > 0$" is a finite-sample $\alpha$-level declaration of similarity above chance. The same holds for the aggregate case with $T_{\mathrm{obs}}$ and $\tau_\alpha^{\mathrm{agg}}$. Empirically the bound is tight: Type I error tracks the nominal level across a range of $d/n$ ratios, and detection power tracks the strength of any real signal.

Two metric families, two regimes

The width analysis already split the standard similarity metrics into two families with very different null behaviour, and calibration is what makes the split visible. Spectral, interaction-based metrics (linear and RBF CKA, the RV coefficient, CCA/SVCCA/PWCCA, and shape-based measures like RSA) are all functionals of an interaction operator in the high-dimensional feature space, typically the cross-covariance or a Gram-matrix object. Under independent rows with mean zero and identity covariance, the cross-covariance has expected squared Frobenius norm $d_x d_y/(n-1)$, and the resulting CKA null inherits an $\mathcal{O}(d/n)$ scale. Calibration subtracts this baseline. For wide models that means subtracting a piece that grows with $d$, which leaves little of the cross-modal scaling trend behind. The same arithmetic applies to the RV coefficient and to CCA-style projections. The null moves with $d$, and calibration removes most of it.

Neighborhood-based metrics work on a different object. Mutual $k$-NN compares which points are each other's nearest neighbors at a fixed $k$ and asks for set overlap, not for an inner product in the feature space. Its null does not see $d$ directly: $\mathbb{E}_{H_0}[\mathrm{mKNN}_k] = k/(n-1)$. The same regime covers cycle-$k$-NN and the kernel-aware CKNNA, and we test all three in the paper. None of them inflate noticeably with $d$, so after calibration their reported trend on real models is essentially what was already there in the raw score.

The distinction is between families, not between one privileged metric and everything else. Anything whose statistic depends on representations only through fixed-size local neighborhood comparisons (mutual $k$-NN, cycle-$k$-NN, CKNNA, and other neighborhood-overlap measures) sits in the neighborhood regime. Anything that touches the interaction operator (spectral, second-moment, shape, CCA-style projection) sits in the spectral regime. Two families, run on the same models, can therefore report opposite scaling trends, and calibration sharpens that disagreement instead of softening it.

The conclusion deserves care. After calibration we find little empirical evidence of cross-modal global convergence on the val modelset of Huh et al. (2024) and on the video–language panel, and significant evidence of cross-modal local-neighborhood convergence on the same data. This is a statement about what the data shows once the metrics' own scale-dependent bias is removed. It is not a claim that global convergence cannot exist in larger or differently constructed settings. On the val modelset and the video–language panel, the global trend reported in the literature was largely an artefact of width and depth inflation. The local trend was already a real signal.

Revisiting the Platonic Representation Hypothesis

We follow the experimental protocol of Huh et al. (2024): $n = 1024$ image–text pairs from WIT, embeddings from three language model families (Bloomz, OpenLLaMA, LLaMA) and five vision model families (ImageNet-21K, MAE, DINOv2, CLIP, CLIP-finetuned), $204$ vision-language model pairs, layer-wise similarity reported as the maximum over layers. The only change is the calibration step on top.

Global similarity (CKA) Local similarity (mutual k-NN)

CKA (global). Raw scores (dotted) rise with model scale, reproducing the trend cited as evidence of cross-modal convergence. Calibrated scores (solid) flatten.

For global metrics, calibration removes the scaling trend that the original work reported. The uncalibrated CKA grows with model capacity. The calibrated CKA does not. For local metrics, the picture is different: raw and calibrated curves lie close together and both rise with capacity. For local metrics, the raw-score convergence persists after calibration.

The same conclusion holds for other modalities

The framework is metric-agnostic and modality-agnostic. As a further test, we replace the image embeddings with video embeddings from a frozen video encoder and run the same protocol against the same panel of language models.

The same pattern appears in the video–language setting. Global RBF CKA loses most of its reported alignment after calibration, while mutual $k$-NN keeps its scaling trend.

The Aristotelian Representation Hypothesis

After calibration, the global similarity trend reported by spectral metrics like CKA largely flattens, while the local trend reported by neighborhood metrics like mutual $k$-NN does not. The original Platonic Representation Hypothesis is therefore not so much wrong as imprecise: it bundled two trends, only one of which the data still shows after calibration. We refine it to a smaller, but sharper claim about what we observe.

Neural networks, trained with different objectives on different data and modalities, converge to shared local neighborhood relationships.

Why we call it Aristotelian

The name follows the philosophical structure of the refinement. Plato's account of universals places them in a transcendent realm: the particulars we observe are imperfect shadows of one true form. The Platonic Representation Hypothesis fits that picture, with networks approximating a single underlying geometry of the world.

Aristotle, Plato's student, takes the universal out of the transcendent realm and places it in the empirical pattern that holds across particulars. The form of a horse is what each horse has in common, observed in horses, not somewhere else. Categories instead of forms.

Mapped onto representations, the Platonic claim is that capable networks are imperfect projections of one true coordinate system. The refined claim is more modest. What we observe as shared across networks is the categorical structure: which points cluster, which neighborhoods recur. On the coordinates themselves, the calibrated metrics do not show a comparable agreement on the same data.

The reframing changes what one can put to the test. A claim about shared global geometry is difficult to falsify: any specific metric that fails to find it can be dismissed as the wrong measurement. A claim about shared local neighborhoods is operational. For a given neighborhood size $k$, sample size $n$, and representation dimension $d$, one can ask whether the observed agreement exceeds a permutation null at level $\alpha$. On the val modelset, that agreement passes the test across two orders of magnitude in $k$.

We are not making a metaphysical claim about how networks recognise things. The argument is empirical: under the metrics we have and once their scale-dependent bias is removed, what we observe agreeing across networks is local structure. The corresponding global trend, after calibration, is not what the data shows on the same models. Local structure is what we recommend measuring when one talks about "shared representations".

Using calibration in your own work

If you measure representational similarity in your own work (transfer learning, distillation, interpretability, model comparison), the calibration framework is meant to drop into existing pipelines. The Python package is on PyPI:

pip install calibrated-similarity

The recipe is one function call:

from calibrated_similarity import calibrate

result = calibrate(metric=cka, X=feats_a, Y=feats_b, n_permutations=1000, alpha=0.05)
# result.score        -> calibrated similarity in [0, 1]
# result.p_value      -> permutation p-value
# result.tau          -> the (1 - alpha) quantile of the null

The notes below cover the choices you have to make in practice.

Choosing the number of permutations

$K$ permutations give you a null of size $K + 1$ (the observed score counts in the calibrated quantile). The smallest detectable $p$-value is $1/(K+1)$, so $K = 999$ gives $p \ge 0.001$. For headline numbers, $K$ between $1000$ and $10000$ works. For screening many comparisons, $K = 200$ is usually enough to get a stable $\tau_\alpha$.

from calibrated_similarity import calibrate, fdr_threshold

scores = [...] # raw similarities, shape (L, V)
pvalues = [...] # permutation p-values, shape (L, V)

mask = fdr_threshold(pvalues, q=0.05) # cells that pass FDR at q=0.05

aggregated = calibrate(
    metric=cka,
    X=feats_a, Y=feats_b,
    n_permutations=1000,
    aggregator="top_k_mean",  # "max", "rowmax_mean", "topk_mean", ...
    aggregator_k=5,
)