Bridging Critical Gaps in Convergent Learning: How Representational Alignment Evolves Across Layers, Training, and Distribution Shifts
Authors: Chaitanya Kapoor
Sudhanshu Srivastava
Meenakshi Khosla
Venue: NeurIPS 2025 poster ()
Keywords: Artificial Intelligence
Machine Learning
Representational Convergence
Primary area: neuroscience_and_cognitive_science

Abstract: Understanding convergent learning---the degree to which independently trained neural systems---whether multiple artificial networks or brains and models---arrive at similar internal representations---is crucial for both neuroscience and AI. Yet, the literature remains narrow in scope---typically examining just a handful of models with one dataset, relying on one alignment metric, and evaluating networks at a single post-training checkpoint. We present a large-scale audit of convergent learning, spanning dozens of vision models and thousands of layer-pair comparisons, to close these long-standing gaps.  First, we pit three alignment families against one another---linear regression (affine-invariant), orthogonal Procrustes (rotation-/reflection-invariant), and permutation/soft-matching (unit-order-invariant). We find that orthogonal transformations align representations nearly as effectively as more flexible linear ones, and although permutation scores are lower, they significantly exceed chance, indicating a privileged representational basis. Tracking convergence throughout training further shows that nearly all eventual alignment crystallizes within the first epoch---well before accuracy plateaus---indicating it is largely driven by shared input statistics and architectural biases, not by the final task solution. Finally, when models are challenged with a battery of out-of-distribution images, early layers remain tightly aligned, whereas deeper layers diverge in proportion to the distribution shift. These findings fill critical gaps in our understanding of representational convergence, with implications for neuroscience and AI.

Bridging Critical Gaps in Convergent Learning: How Representational Alignment Evolves Across Layers, Training, and Distribution Shifts
Chaitanya Kapoor, Sudhanshu Srivastava, Meenakshi Khosla
Department of Cognitive Science University of California, San Diego
La Jolla, CA 92093
{chkapoor, sus021, mkhosla) Cuesd. edu
Abstract
Understanding convergent learning-thh degree to which independently trained neural systems- -hethhr multiple artificial networks brains and models- -rrivee at similar internal representations-i5 crucial for both neuroscience and AI. Yet, the literature remains narrow scope--ypically examining just handful of models with one dataset, relying on one alignment metric, and evaluating networks at a single post-traine post-training checkpoint. We present large-scale audit of convergent learn- ing, spanning dozens of vision models and thousands of layer-pair comparisons, close these maandinnninng gaps. First, we pit three alignment families against one another-linear regression (affine-invariant) invariant), orthogonal Procrustes (rotation- frefletioo-invariinnt) and permutation/sol maaatiotttttmmamingggggooo Lunit-order-invaiint)) We find that orthogonal transformations align representations nearly effectively more flexible linear ones, and although permutation scores are lower, they signif- icantly exceed chance, indicating a privileged representational basis. Tracking convergence throughout training further shows that nearly all eventual alignment crystallizes within the first epoch-well before accuracy plateaus-indication indicating it is largely driven by shared input statistics and architectural biases, not by the final task solution. Finally, when models are challenged with battery of out-of- distribution images, early layers remain tightly aligned, whereas deeper layers diverge in proportion the distribution shift. These findings fill critical gaps in our understanding of representational convergence, with implications for neuroscience and All
Introduction
Deep Neural Networks (DNNs) are becoming increasingly popular in neuroscience for predicting or as models for reverse-engineering algorithms of neural computa- This congruence invokes the necessity to gain a deep understanding of how DNNs learn to represent information. A core question in this domain is whether independently trained networks converge on similar internal representations--ndd if so, under what conditions and along which dimensions this convergence unfolds? Comparative analysis of model representations helps reverse engineering neural networks by linking architectural components, training objectives, and data inputs to learned representations and, in turn, model behavior.
neural responses 48
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Over the past decade, there has been growing recognition that similar representations emerge diverse models, despite differing in architecture, training procedures, or data modalities. Early work
code is publicty available http://gtttb.cooommuuuullllllltt alignmmmn
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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demonstrated that independent training runs of the same architecture develop a core set of features that align well across networks. For example, early layers in convolutional networks learn Gabor-like Efforts to quantify these similarities have
filters, across a range of architectures and tasks 4 32
employed techniques such as canonical correlation analysis (CCA) and its variants Rd [9] centered kernel alignment (CKA) 311, representational similarity analysis (RSA) 65 and model stitching A. These studies have underscored the high degree of alignment in early layers and noted convergence even in later stages of the network. These findings provide empirical grounding for theories of representational convergence, hinting at the existence of universal principles governing learning, which may also shed light how biological neural circuits process information.
Recent studies have extended this line of inquiry, demonstrating that increasing model capability through increased scale, multitask training, or cross- modssl learning drives representational con- vergence not just within, but also across modalities (37. The Platonic Representation Hypothesis further argues that as models scale and solve more tasks, they are driven to discover universal, modallty-agnosiee representation of reality 2d. Yet, identifying precise conditions of network convergence on similar representations and its implications remain open questions.
To address these gaps, in our work, we examine representational alignment along three key axes:
Across layers:
Prior work has shown that early layers tend to extract general, low-level features-suse as edge detectors in vision networks- while deeper layers develop -ask-specific representations. Studies [31 351 SE
have quantified these changes using methods like CCA and CKA. However, these approaches often rely on single metrics that obscure minimal transformations needed align representations. Understanding the precise nature of these transformations critical for dissecting how representations in different networks relate to each other (e.g., are they similar in information content, representational geometry, or even at the level of single-neuron tuning?). is also unknown whether hierarchical (layer-wise) correspondence holds for other metrics with more restricted invariances than affine transformations.
Across training: A second key question is when convergence emerges. Most studies snap- shot networks only after training has finished, but revealing the mechanisms behind aligmment demands tracking how representations co-evolve during training, not just examining the final state. Conventionally, it is assumed that as different networks optimize on task, their internal representations become more similar, driven by the final task solution. This assumption is the basis for the contravariance principle (9), which posits that when network is pushed to achieve difficult task, there is less room for variation in the final solution, forcing representations to converge. However, the question of when representational convergence occurs during training remains underexplored. Understanding this dynamic can illuminate the roles of initialization, early data statisties, architectural biases, learning dynamics, and the final task solution in shaping alignment. Previous studies have shown a "lower layers learn first" [39 behavior by comparing layers over time on CIFAR- 10 using SVCCA, but little is known about the dynamics of convergence, especially for complex vision networks and using other metrics. More recently, Atanasov et al. [3] described a "silent alignment effect" in which network's 's output aligns with the target function early in training well before the loss falls. Theoretical studies such as g 15 41 present an analytical framework for the temporal dynamics of learning in deep linear networks, distinguishing an early "lazy" phase, characterized by kernel regression dynamics 2 from subsequent "rich" phase characterized by non-linear feature learning. While these models provide valuable insights into training dynamics and convergence in linear regimes, they fall short of fully explaining representational alignment in non non-linear architectures. Though deep linear models provide valuable theoretical traction, further work is needed to elucidate how representational alignment develops and generalizes to the rich, non-lincar regimes typical of contemporary DNNs.
Across distribution shifts: Although many DNNs exhibit highly human-like responses to in-distribution stimuli, there is mounting evidence that their responses can diverge dramatically under out- -ft-----iittttiionnnn (OOD) conditions [3 20 22]. Despite this, the effect of OOD inputs on model-to-modee representational convergence remains poorly understood. Exploring this axis is crucial for rigorously testing the universality of leareed representations.
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Key contributions. In this work, we address these critical gaps by performing large-scale systematic audit of representational convergence along these three axes. First, we employ three alignment metrics- linear regression
Procrustes analysis 45 46 and permutation-ban based each with different restrictions on the freedom of the mapping function, to identify the minimal set of transformations needed to align representations across networks reasonably well for each layer. This approach allows us to dissect how representations relate to each other, whether in terms of information content, representational geometry, single-neuron tuning. Second, we examine the temporal dynamics of convergence during training, revealing that nearly all alignment occurs within the first epoch, challenging the assumption that convergence is tied to task-specific learning. Finally, we explore how changes in input statistics affect representational alignment across layers, demonstrating how OOD inputs differentially affect later versus early layers.
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2 Problem Statement
We consider two representations X; E RRMN,, and X, E RRX,, obtained from different models over . unique stimuli, where N₂ and N,, denote the number of neurons (or units) in each representation, respectively. To systematically identify the minimal transformations needed for alignment, we use three metries that quantify similarities between networks while ignoring nuisance transformations. Among these are (a) the linear regression score which discounts affine transforma- tions, (b) the Procrustes score which discounts rotations and reflections, treating them as nuisance factors and (c) the permutation score which considers the order of units in the representations as arbitrary. These metrics are ordered to reflect progressively less permissive mapping functions (from flexible to strict). When representations have an equal number of neurons (N, N2 N), each of these similarity metrics seeks to minimize their Euclidean distance by optimizing over set of N N N-dimensional mapping matrices M, min|| X; MM 112- Linear regression score imposes
M
constraint M, Procrustes score enforces the matrices to be orthogonal (i.e. M E ((N)) and permutation score requires the matrices to he permutation matrices (W E P(N))). When N₂ * Ny a generalized version of permutation- based alignment called the soft-matching score (29] Here, the mapping matrix M E R NNN, is constrained such that its entries are nonnegative and satisfy Σ No M,, J Vi = I........ No. Σ No M N, Vj .,. These constraints place M in the transportation polytope T(N,, N2,)
Once the optimal mapping matrix M is computed for each metric, we report the alignment using the pairwise correlation Alignment corr (X,, MX,). For asymmetric metries, we report the average alignment score computed in both directions: corr(X,, j. M,), and corr (X, M,X,). where Mj and M2 are the respective transformations. We also report our results using Spearman's rank correlation coefficient in Appendix A2 to address the possibility of being susceptible to high- variance activation dimensions.
By leveraging these wetrics-which progressively relax mapping constraints from strict (soft- (rrrciing//emmuaation)) to flexible (linear regression)-we can dissect the nature of representational alignment across networks, distinguishing between similarity in representational form (captured by the soft-me fatt-matching score, which reduces to the permutation score when N₂ Ng), geometric shape (captured by the Procrustes score), and information content (captured by the linear regression score).
3 Method
Below, we outline our framework for evaluating alignment between different deep convolutional vision models (ResNet18, ResNet50 [25), VGG16, VGG19 [4)). with their training procedures described in Appendix AI
We demonstrate the robustness of all these results on CIFAR100 in
Appendix A4
3.1 Comparing Convolutional Layers
For a given convolutional layer, let the activations represented by E Iexxxxxxx, where m is the number of stimuli, h and w denote the spatial height and width, and e is the number of channels (i.e., convolutional filters). Each convolutional layer produces a feature map whose spatial dimensions are equivariant to translations. That is, circular shift along the spatial dimensions yields an equivalent
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representation (up to a shift). As result, one could compare the full spatial activation patterns between networks by considering an equivalence relation that allows for spatial shifts. However, evaluating an alignment that optimizes over all possible shifts together with another alignment (e.g.- Procrustes) is computationally costly.
Previous work has shown that optimal spatial shift in convolutional layers tend to be close to zero 46 This motivates our simpler approach: rather than collapsing the spatial dimensions by flattening the entire feature map (which would yield X R=)((oww))) we extract single representative value from each channel. In our experiments, we choose the value at the center pixel of each channel, reducing the activation tensor X two-dimensional matrix X' R where each row corresponds to stimulus and each column to a channel. This drastically reduces the computational complexity of computing the optimal mapping. For instance, aligning full, apatiall--flattende representations would incur a runtime of Ommh"
fi c²), whereas our center-pixel approach reduces the problem to aligning an m matrix, resulting in a more tractable complexity.
3.2 Computing Alignment
Alignment was quantified in four regimes: (1) Within-architecture: each layer aligned with its
counterpart in independently initialized instance of the same architecture (trained on CIFAR100 ImageNet); (2) Across architectures: all layer-layer alignments between different architectures; (3)
Across training: analysis repeated after each of the first ten epochs; (4) Under distribution shift: within-architecture alignments tested on ImageNet-trained models using 17 OOD datasets from 20
All scores are averaged over five-fold validation (k- = 5).
4 Results
4.1 Evolution of Convergence Across the Network Hierarchy
How convergence varies with network depth. When comparing representational convergence across the network hierarchy for different seeds of the same architecture, we observe that convergence strongest in the earliest layers and gradually diminishes in deeper layers (Fig. [T] This pattern is consistent across all three metrics and across networks trained on ImageNet. The high alignment early layers likely arises because they capture fundamental,
www-freuency features (e.g., edges, corners, contrast) that are universal across representations 40 S 550 In contrast, deeper layers, while still showing significant alignment (> 0.5), exhibit greater variability due to their sensitivity to specific training conditions and noise. We also contrast this result with randomly initialized (untrained) networks as a baseline. We find that alignment scores across all metrics are consistently lower in untrained networks compared to their trained counterparts. This difference is especially pronounced in deeper layers: for instance, using the Procrustes metric, the mean alignment increases by 145.20 26% in early layers (depth 0.5) and by 493.84% in deeper layers (depth > 0.5) following training. This trend also holds when comparing networks with different architectures, underscoring the robustness of hierarchical convergence across diverse models. Interestingly, similar hierarchical trend is observed in human brain responses to visual stimuli (Appendix A8).
Minimal transformations needed to align representations. Across all layers, find that align- ment scores increase as the mapping functions become more permissive (Permutation + Procrustes + Linear), as expected. However, linear mappings provide only modest improvements over Pro- crustes correlations, indicating that rotational transformations are sufficient to capture the majority of alignment information. This suggests that the added flexibility of linear mappings- such as scaling and shearing does not substantially enhance alignment beyond what is achieved with Procrustes transformations. Importantly, because Procrustes symmetric, this result highlights that alignment reflects a deeper similarity in the geometric structure of representations, rather than merely the ability of one representation to predict another.
Simple permutations achieve significant alignment. Despite the strict constraints imposed by permutation based alignment, Permutation scores achieve surprisingly high alignment levels, indi- cating a strong one-to-one correspondence between individual neurons across network instances- suggesting that convergent learning extends down to the single-neuron level, even without allowing for more flexible transformations.

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Figure I: Representational Convergence Across a Network Hierarchy. We plot the evolution of alignment scores (computed between different seeds of the same network architecture) across the network hierarchy for four vision network architectures trained ImageNet. A consistent downward trend across layers indicates decreasing representational convergence as networks deepen. Alignment consistently follows the order: Linear Procrustes > Permutatton, reflecting the progressively stricter nature of the metries. Lighter shades of the same color denote alignment for random networks. Notably, Procrustes transformations align representations nearly as well as Linear transformations, suggesting that most variability is due rotations rather than more complex transformations. Even Permutation scores-despel their strictness achiive substantial alignment, indicating a strong one- to-one correspondence between neurons across seeds, which points to stable, convergent neuron-level representations, Error bars represent the standard deviation computed across 5-fold cross- volsdavali
Figure 2: Inter-Model Comparisons. We consider all pairs of vision models, and for each pair, compute the alignment scores between every pair of layers using the orthogonal Procrustes (Top) and Soft-Matching (Bottom) metric trained on ImageNet. Gray line plots denote the maximum alignment value for each network over rows (right line) and columns (top line). A common trend that is observed here is the consistent relationships between layers of CNNs trained with different architectures.
To further probe this result and assess the depth of convergent learning, tested the sensitivity permutation alignment to changes in the representational basis.
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Specifically, we applied random rota- tion matrix Q E RRax to the converged basis neural representation, where is the number of neurons in a given layer. The rotation matrix was sampled from Haar distribution via QR de- composition, ensuring that all orthogo- nal matrices were equally likely. We then recomputed the Permutation score after applying this rotation.
Table I: Sensitivity Permutation Scores to Represen- tational Axes. For each ImageNet-trained network we apply a random rotation to the network's unit basis, recom- pute permutation-baase alignment scores, and summarize the results across convolutional layers. Columns report the minimum and maximum alignment scores observed over layers in the native basis and in the rotated basis. The final column gives the percentage change in alignment after rotation. Rotations almost always reduce alignment, indicating the distinguished nature of the representational axes for all layers in networks.
conducted this analysis by taking re- sponse matrices from two identical DC- NNs (initialized with different random seeds) at given convolutional layer, (X1, X2) mxx, where represents the number of stimuli.
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We applied the random rotation Q to one network's 's responses and computed the resulting permutation- based correlation score, Sperm ( Q,)) This process was repeated across all convolutional layers. with the alignment differences summarized in Table 1
These rotations consistently reduced alignment, with drop between 10 202% for ResNet18 across all layers, for instance. This significant decrease highlights that the learned representations are not rotationally invariant and the specific bases in which features are encoded is meaningfully preserved across networks. In other words, convergent learning aligns not just the overall representa- tional structure but also the specific axes along which features are encoded. This observation echoes recent findings by [B0], who report the existence of privileged axes in biological systems as well as the penultimate layer representations of trained artificial networks.
Hierarchical correspondence holds across metrics. Previous studies have shown that for architec- turally identical networks trained from different initializations, the layer most similar to a given layer in another network the corresponding architectural layer [II. However, this finding has primarily been supported using metries invariant to affine transformations (e.g., CCA, SVCCA). Here, we extend this result by showing that stricter metries such as Procrustes and soft-matching scores also reveal the same hierarchical correspondence (Fig. A3 even when comparing networks with different architectures. This suggests that the hierarchical alignment of representations is a fundamental property of neural networks, robust to both architectural differences and the choice of alignment metric. Moreover, our results show that both representational shape (captured by Procrustes) and neuron-level tuning (captured by soft-matching) follow similar alignment patterns, reinforcing the consistency of this hierarchical organization across different levels of representational analysis.
4.2 Evolution of convergence over training
We next explore how representational convergence evolves during the training process. Specifically, we compute Procrustes alignment scores between pairs of networks over training epochs. As shown
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Figure 3: Representational Alignment Through Training Evolution. We visualize the evolution of Procrustes alignment between network pairs during task optimization on ImageNet. Lighter shades indicate earlier epochs, progressively darkening with later epochs. The plots span from epoch 0 (untrained) to epoch 10, with task performance improving over time. Epoch progression can be inferred from the inereasing task performance along the r-axis.
in Fig. 3 striking pattern emerges: the majority of representational convergence oceurs within the before networks task performance peaks. This rapid early convergence suggests that factors independent task optimization drive much of the representational alignment (see also Appendix A7). Shared input statistics, architectural biases, and early training dynamics seem to play a dominant role in shaping representational convergence, overshadowing the influence of the final task-specific solution. This observation challenges prevailing hypotheses, such as the contravariance principle [9] and task generality hypothesis (26). which argue that networks align because only a narrow solution subspace yields high performance. Instead, findings point to inductive biases and early learning processes as the dominant forces shaping representational convergence.
first
Our findings echo those of 17 who showed that most representational re-organization happens within the first few hundred iterations-wal before any substantive task learning. Perturbing weights during this brief window (by re-initializing shuffling them) severely impairs eventual accuracy,

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indicating the formation of early, data-dependent structure. Together, these observations reinforce the view that shared input statistics-nn not task labels are the principal drivers of early representational convergence.
Early convolutional layers show little rraining-induced change on representational similarity (even reducing in some cases). This phenomenon arises because early layers compute near-linear mappings of the input, even in untrained networks, allowing them align well with simple linear transfor- mations. Subsequent training only makes minor adjustments that slightly disrupt initial alignment, though they remain highly similar overall.
4.3 Convergence across distribution shifts
Having established alignment in-distribution images, next probed its robustness to distribution shift. We applied 17 -utto--distribution (OOD) variants from 21 to ImageNet-trained CNNs. Dataset specifies are given in Appendix A5
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Figure 4: Procrustes Score-based Alignment Between Networks Sharing the Same Architecture but Trained With Different Random Seeds, Plotted as Function of Layer Depth. Alignment is measured using within-distribution (WD) stimuli (ImageNet test set) and out-of-distribution (OOD) stimuli, with OOD values averaged across 17 datasets. Error bars represent the standard error computed across the (n = 17) OOD datasets.
Using the Procrustes metric, we compute representational aligmment on these datasets and observed consistent pattern: OOD inputs amplify differences in the later layers of the networks, while early layers maintain comparable alignment levels between in-distribution and OOD stimuli (Fig. Ep We hypothesize that this pattern arises because early layers capture universal features (e.g, edges, textures) that remain nearly identical across distributions, whereas later layers encode task-specific features that are sensitive to distributional shifts, thus amplifying the divergence between models. We also found strong correlation between representational alignment and OOD accuracy: datasets where models maintained higher accuracy show strong alignment and vice versa. This correlation is minimal in early layers, but progressively increases with network depth across all architectures (Fig. 5) analogous trends appear in other networks Fig. A5
These results have several important implications. First, early-layer alignment is remarkably stable- across random initializations and in-distribution and OOD
that these layers encode broadly transferable features that serve as a common scaffold for higher processing. Because this scaffold endures under distribution shift, one could plausibly improve OOD generalisation by fine- tuning only the later layers. Second, these findings inform model-brain comparisons. Though diverse architectures and learning objectives yield similar brain predictivity [12. the amplified divergence of later-layer representations under OOD conditions suggests that OOD stimuli could be especially useful to distinguish and select between models whose representations closely mirror the brain.
4.4 Representational Alignment for Self-Supervised Networks
We next test whether our findings generalize sulf-supervised learning. All analyses were repli- cated on models trained with Momentum Contrast (MoCo) objective 24] (training details in Appendix AT). (1) Across hierarchy: convergence strongest in early layers and tapers with depth (Fig. A); Linear and Procrustes scores remain comparable, suggesting inter-model variability reflects rotations/reflections. (2) Across distribution shifts: early layers maintain alignment for both in-distribution and OOD inputs, while deeper layers diverge (Fig. B), indicating that self-
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Figure 5: Convergence on OOD Inputs. (a) Procrustes alignment vs. task performance of ResNet50 models on each of the 17 datasets for the first convolutional layer (Left) and the penultimate (Right) layer. (b) Correlation between these variables as a function of network depth (normalized by each models depth).
supervised training, like supervised learning, produces stable, general early filters and increasingly Instribution-sensiti higher representations. (3) Across training: most alignment emerges within the
first epoch, well before optimal task performance (Fig. 6 C). (4) Permutation sensitivity: applying
random rotation substantially reduces permutation alignment (Fig. D), confirming convergence to privileged basis regardless of learning paradigm.
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Figure 6: Representational Alignment with Self-Supervised Networks. (A) Representational alignment across layers of pair of MoCo-trained ResNet50 models ImageNet. Error bars indicate standard deviation from 5-fold cross-validation. (B) We plot the Procrustes alignment between MoCo models evaluated in-distribution (ImageNet) and out-o--distribution (Stylized ImageNet (21) stimuli. Error bars show standard error across all (n = 17) OOD datasets. (C) Change in Procrustes alignment score with every training epoch. Colors in the bar-plot indicate the top-1 accuracies during training. (D) Permutation alignment in the native and randomly rotated basis. Each dot corresponds convolutional ResNet50 layer. Bars indicate mean alignment. Rotation reduces mean alignment by 14.87%.
4.5 Representational Convergence in Vision Transformers
We next analyze representational convergence in ViTs. All models and training details are in Appendix AT We analyze the [CLS] (class) token representations at each layer. This is because [CLS] token aggregates the "information content" " of all patch embeddings via the self-attention operator, making it a useful proxy to study representational geometry. In addition, computing representational similarity for (CLS) vectors avoids having to deal with pooling patch similarities into single representative score.

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Figure 7: Representational Convergence in Vision Transformers. (A) We plot the evolution of alignment scores of three metrics (Linear, Procrustes, Permutation) computed between different seeds of the same ViT, which was trained on ImageNet. Error bars denote the standard deviation across 5-fold cross validation. (B) We plot the inter-model orthogonal Procrustes (left) and permutation (right) scores for all layer pairs. Gray line plots denote the maximum alignment value over rows (right line) and columns (top line). (C) We visualize the evolution of the orthogonal Procrustes score different checkpoints, ranging from epochs 0 (untrained) 25. Darker colors correspond to epoch progression. (D) We rotate the converged basis of ViTs by a random rotation matrix and recompute the permutation scores for models trained ImageNet. Each dot represents a layer the VIT. The permutation alignment remains approximately constant in both cases.
Results. Analyses parallel to DCNNs reveal three key findings: (1) Depth-wise convergence: aligmment tapers with network depth across metrics (Fig. A); Procrustes and Linear scores remain comparable, indicating rotattons/reflections explain most inter-seed variability, and layer- wise patterns mirror DCNNs (Fig. B, left). (2) No privileged axes: independent ViT runs do not share a common basis 1ess--peetttiion scores between native and rotated axes are statistically indistinguishable across layers (Fig. D: Appendix A3) (3) Early plateau: alignment stabilizes after the first epoch, except for a [CLS] embedding artifact due uniform positional encoding, indicating late-stage training does not drive convergence (Fig. C).
4.6 Representational Convergence in Language Models
We analyze representation convergence language models using sentence embeddings from the Semantic Textual Similarity Benchmark (STSB) [IO]. Our analyses include two primary comparisons: (i) Same architecture: multiple Pythia-160m instances differing only by random seed, allowing us to analyze variability in representational spaces solely from stochastic training factors (ii) Cross- architecture: Pythia-70m vs. Pythia-160m, to probe the effect of network depth on learned representations.
Methodology. For our representational similarity analysis, we perform two sets of computations: (i) Final Checkpoint: we compute the similarity between every pair of layers for all models at their final checkpoints using the same metries as the case of vision models (Linear, Procrustes. Permutation). (ii) Intermediate Checkpoint: across 154 intermediate checkpoints, we analyze layer representations at normalized depths 0 (beginning), 0.5 (middle), 1 (end) in the network. We track changes in the Procrustes similarity over training.
Results. Comparing representations across language models reveals three key patterns: (1) Metric-
dependent alignment: alignment follows Linear > Procrustes > Permutation (Fig. A); Procrustes and Linear scores are nearly identical, indicating that rotations/rreleections explain most variability
across models, and high alignment persists across layers. (2) Hierarchical correspondence: layers
at similar depths align most closely both within (Fig. g B) and across architectures (Fig. g C). (3)

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Figure 8: Representational Convergence in Language Models. (A) Alignment scores across all layers for Pythia-1600 60m models trained from different seeds evaluated on STSB. Procrustes align representations nearly as well as linear transformations, mimicking the trend observed for vision models. (B) We compute the alignment scores using the Procrustes metric for every pair of layer seed-pair of Pythia-160n models. (C) Same as (B), but for models with different architectures (varying depth). (D) Procrustes alignment between model pairs training checkpoints.
Rapid convergence: alignment emerges early (layers at indices 0, 0.5, 1), peaks around 8-10B tokens, then plateaus slightly declines (Fig. D) even next-token prediction improves up to 300B tokens 6
5 Discussion
We present large-scale account of convergent learning, showing how representational alignment between independently trained networks depends on depth, training time and distribution shifts. Using alignment metrics with varying degrees of transformation invariances for capturing representational similarity, we provide a more nuanced view of convergent learning.
Nonetheless, certain limitations remain. While we show the early emergence of alignment during training, do not precisely quantify when within the first epoch this convergence occurs. Our empirical evidence for early alignment is striking and raises crucial question: what drives such rapid convergence across diverse models? Although a full theoretical account is beyond our current scope, future work using tools like Neural Tangent Kernels (NTKs) [27]. and its extension to CNNs under simplifying assumptions like linearity- maya help elucidate the temporal dynamies of representational convergence. Another limitation stems from the alignment metrics themselves. Although our metrics reveal an early stabilization alignment over training, this could reflect limitations of the metrics rather than the absence of representational changes. Prior work 7 has shown that certain alignment metrics may fail to capture subtle, task-relevant shifts in representations. It remains possible that alternative metrics could reveal gradual alignment changes over training that are invisible to the methods employed here.
Finally, our method for computing alignment uses the center pixel of each feature map, enforcing strict spatial correspondence between representations. This may underestimate alignment, especially in cases where features are slightly shifted spatially. While one could incorporate spatial shifts into alignment computations, they are computationally intensive and beyond the scope of this work. However, prior research has shown that optimal spatial shifts in many convolutional layers are typically negligible [6), justifying our approximation. Nonetheless, scalable methods that account for such spatial variability remain an important direction for future research.
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15


Reviews:
[{"review_id": "j1ceNSKmXb", "reviewer": "Reviewer_yuMg", "summary": "This work seeks to understand how representations in various neural networks converge according to various metrics across datasets, architectures, throughout training, and in out of distribution settings. Through a large scale study, authors reveal that 1) representational convergence occurs even if the mapping function is quite restricted 2) convergence occurs extremely early in training, 3) that OOD stimuli disrupt representational convergence in later layers and 4) convergence occurs even between different CNN architectures.", "questions": "Add Spearman correlation to sanity check results.\n\nAdd randomly initialized baseline models.\n\nWhat produces basis alignment in CNNs as opposed to transformers?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "This study is quite extensive and thoughtful, revealing several notable findings that challenge prevailing wisdom regarding convergence. Most notable among them is that representations tend to converge within the first epoch of training. That said, there are a few opportunities to improve the work, and some sanity checks that might be needed to ensure their validity.\n\nFirst, it seems plausible that the correlation between representations could be driven by the presence of rogue/outlier dimensions (https://arxiv.org/abs/2310.17715). This could especially be the case with the permutation-based alignment score. Outlier dimensions are thought to encode generic task information in extremely high variance dimensions, and so it is possible that such dimensions always encode approximately the same basic information across datasets/architectures/etc. As I understand it, the correlation score is presently simply Pearson correlation, which is susceptible to such outliers. A simple sanity check is to replace Pearson correlation with Spearman correlation, and ensure that all results still hold. \nAdditionally, the results from Figure 1 should include a baseline randomly initialized model to contextualize the observed trends.\n\nWriting: \nThe Difference (%) column in table 1 could be more explicitly described in text.\n\nThe neuroscience framing seems largely unnecessary in this paper, and could very well be distracting to readers unfamiliar with current paradigms in that field. While the compneuro community might benefit from this work, it is unclear that framing paper around the neuroscientific applications is more compelling than immediately (i.e., in the intro) putting this work in conversation with the debates surrounding the platonic representation hypothesis in AI.", "was_revised": true, "final_justification": "The authors ran the relevant extra analyses and baselines during the rebuttal period (random model and Spearman correlation), addressing my major technical concerns with the paper.", "rebuttal": "We would like to thank reviewer **yuMg** for recognizing the novelty of our results! It is indeed quite remarkable that representational convergence emerges within the first epoch itself. \n\n> As I understand it, the correlation score is presently simply Pearson correlation, which is susceptible to such outliers. A simple sanity check is to replace Pearson correlation with Spearman correlation, and ensure that all results still hold. \n\nWe appreciate the reviewers' concern about Pearson correlation. While our use of Pearson correlation follows prior work in representational alignment, we agree that it is susceptible to high-variance outlier dimensions. To directly address this possibility, we conducted an additional series of experiments where we computed alignment scores using Spearman’s rank correlation instead of Pearson’s correlation. Across all models and metrics, we observed that the alignment trends over network depth remained qualitatively unchanged, and all key findings about early emergence of convergence and effect of distribution shifts still held. These results indicate that our conclusions are not driven by a small number of high-variance dimensions. We will include the corresponding figures in the revised supplementary material for completeness. \n\n>  Additionally, the results from Figure 1 should include a baseline randomly initialized model to contextualize the observed trends.\n\n\nWe appreciate this suggestion and have now extended our analysis to include randomly initialized (untrained) networks as a baseline. We find that alignment scores across all metrics are consistently lower in untrained networks compared to their trained counterparts. This difference is especially pronounced in deeper layers: for example, using the Procrustes metric, the mean alignment increases by $145.26\\\\%$ in early layers (depth $< 0.5$) and by $493.84\\\\%$ in deeper layers (depth $> 0.5$) following training. \n\nAdditionally, we observe that in untrained networks, the mean gap between linear and Procrustes alignment is substantially larger $(42.39\\\\% )$ than that seen in trained models $(7.85\\\\%)$. This suggests that training progressively aligns representations in a way that can be well captured by rigid transformations (i.e., rotations/reflections), reducing the need for more flexible affine transformations. Together, these findings reinforce the conclusion that representational convergence is not simply a consequence of initialization biases or mere architectural biases, but is shaped through training. We will include these results in the revised supplementary materials.\n\n> The neuroscience framing seems largely unnecessary in this paper, and could very well be distracting to readers unfamiliar with current paradigms in that field\n\nWe appreciate the reviewer’s perspective. Our original motivation for including the neuroscience framing was to highlight the broader relevance of representational convergence—not only as a phenomenon observed across artificial neural networks, but also as a question of interest in systems neuroscience, where similar analyses are used to compare brain responses across individuals and species. However, we agree that the neuroscience context is not essential to the core contributions of this work, and may distract readers focused on machine learning. In response, we have revised the manuscript to remove neuroscience-specific framing and now focus exclusively on representational convergence between models. \n\n> Writing: The Difference (%) column in table 1 could be more explicitly described in text.\n\nWe have revised the corresponding paragraph to more clearly describe the “Difference $(\\\\%)$” column in **Table 1**. Specifically, we now clarify that this column reports the relative decrease in permutation alignment scores after applying a random rotation to the representational basis, computed as the percentage difference between native and rotated alignment scores for each layer. \n\n> What produces basis alignment in CNNs as opposed to transformers?\n\nWhile a definitive mechanistic account remains an open area of investigation, recent work **[1]** offers compelling evidence that the emergence of basis alignment across CNNs and even between brains and CNNs—may be partly attributable to architectural choices, especially the presence of ReLU nonlinearities. To understand this constraint, consider a representation of post-ReLU activations $\\boldsymbol{x}$. The ReLU operation ensures that all activations in $\\boldsymbol{x}$ are non-negative. If we apply a rotation to these activations, we get $\\boldsymbol{y} = \\boldsymbol{Ux}$, where $\\boldsymbol{U}$ is a rotation matrix. For both $\\boldsymbol{x}$ and $\\boldsymbol{y}$ to be valid post-ReLU activations, they must remain non-negative after the transformation. For this, the matrix $\\boldsymbol{U}$ must be a non-negative matrix. But this means that $\\boldsymbol{U}$ must be a permutation matrix, because every orthogonal matrix with nonnegative entries is necessarily a permutation matrix. This means that $\\boldsymbol{U}$ can only permute (or shuffle) the activation units, rather than performing arbitrary rotations. Thus, the nonlinearity introduced by ReLU disrupts the rotational symmetry of the activation space, potentially explaining why different networks converge to similar bases. In contrast, transformers use GeLU nonlinearities in MLP layers, and notably, the penultimate layer in ViTs often lacks any nonlinearity at all. These architectural choices retain greater rotational freedom in the feature space, which likely explains the lack of axis alignment across transformer runs, as also confirmed by our results. We now cite and briefly summarize this theoretical explanation in the revised manuscript (**Discussion section**) to contextualize the observed architectural difference in axis alignment between CNNs and transformers.\n\n**References**:\n\n**[1]** Privileged representational axes in biological and artificial neural networks (Khosla et al., 2024)"}, {"review_id": "b6iMOu5ZC9", "reviewer": "Reviewer_uReV", "summary": "This paper aims to understand convergent learning and proposes a large-scale audit for it.  The authors conduct an extensive experiment, which demonstrates the usefulness of the proposed method.", "questions": "1. What is the specific expression of the $\\texttt{corr}$ formula in Line 134?\n\n2. The symbol $\\mathcal{O}$ is used repeatedly in Line 163, with different meanings in different places (Line 129).\n\n3. The method part seems to lack a lot of core formula expressions. Should you migrate them from the problem statement?\n\n4. The legend font in all the figures is too small.\n\n5. Is there enough theory to support the effectiveness of the proposed method? Specific analysis is needed.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n1. This paper proposes a comparative method of characterization from three perspectives: layer, training, and distribution shift.\n\n2. Real data analysis demonstrated the impact of the new methods.\n\n**Weaknesses**\n\n1. The main concern of the paper is a presentation. In many places, the meaning of sentences is vague. It is better to elaborate on it and make the sentence shorter.\n\n2. The paper involves numerous symbols and matrix operations, but some symbols lack clear explanations. It is recommended to define each symbol's meaning the first time it appears to avoid ambiguity.\n\n3. All formulas in the article lack numbers, which makes it very inconvenient to quote and explain.\n\n4. The motivation for the proposed method is not well explained, and the novelty of the method seems to be underdeveloped.", "was_revised": true, "final_justification": "After reading the author's response to us and other reviewers, we are confident that the author has addressed most of our concerns appropriately, so we decided to increase our final score.", "rebuttal": "We would like to thank reviewer **uRev** for taking their time to review our manuscript. Below, we have addressed each point in turn, and tried to clarify some major misunderstandings:\n\n**General Remark**\n\nFirst, we respectfully clarify that the novelty of our work lies in its **scale and scope of empirical inquiry**, not in proposing a new algorithm. Specifically, we deliver the first systematic audit of representational convergence across three orthogonal dimensions—layer depth, training trajectory, and distribution shift—spanning CNNs, ViTs, and LLMs.\n\n**Response to Weaknesses**:\n\n1. > The main concern of the paper is a presentation. In many places, the meaning of sentences is vague. It is better to elaborate on it and make the sentence shorter. \n\nWe appreciate the reviewers' concern to help improve our manuscript presentation. Because the comment does not specify which passages are unclear, we are unable to address them individually in the rebuttal. We will, however, undertake a careful language edit for the camera‑ready version and would welcome any concrete pointers the reviewer can provide.\n\n2. > The paper involves numerous symbols and matrix operations, but some symbols lack clear explanations. It is recommended to define each symbol's meaning the first time it appears to avoid ambiguity. \n\nOur manuscript describes mathematical equations in Section 2. We believe all symbols are already defined at first mention in Section 2. the only exception is the transportation polytope $\\mathcal{T}(N_x, N_y)$, for which we cited prior work. In the revision we will **(i)** write out the explicit linear constraints that define $\\mathcal{T}(N_x, N_y)$ and **(ii)** add a one‑page notation table in the appendix for quick reference. If any other expression remains unclear, we would welcome a specific pointer so we can correct it.\n\n3. > All formulas in the article lack numbers, which makes it very inconvenient to quote and explain.\n\nWe have omitted the numbering of equations because no equation was referenced more than once. However, if the reviewer feels that it is necessary to do so for clarity, we can make these changes.\n\n4. Our manuscript does not introduce a new alignment algorithm; instead, it offers a systematic experimental framework to interrogate long‑standing questions about representational convergence. The motivation is stated up front: prior studies typically (i) examine only one metric, (ii) snapshot a single checkpoint, and (iii) ignore distribution shifts—leaving open **(i)** what minimal transformations are sufficient to align representations, **(ii)** when convergence first appears during training, and **(iii)** how it behaves under distribution shift.. We fill these gaps by applying three established metrics side‑by‑side across thousands of layer‑pair comparisons, tracking the entire training trajectory, and evaluating $17$ OOD datasets. This scale and breadth reveal previously unreported phenomena—*e.g.,* rotation suffices, convergence appears in the first epoch, early layers remain aligned under OOD shift—thereby providing new insights rather than a new method.\n\nNext, we address each of the reviewers questions below:\n\n1. $ \\texttt{Alignment} = \\texttt{corr}(\\boldsymbol{X}_j, \\boldsymbol{MX}_i)$ is used to report representational alignment as pearson correlation computed unit-wise across stimuli. After applying the optimal mapping $\\boldsymbol{M}$ to $\\boldsymbol{X}_i$​, we take each corresponding pair of columns—$\\boldsymbol{x}_j(k)$ from $\\boldsymbol{X}_j$ and $\\boldsymbol{Mx}_i(k)$ from $\\boldsymbol{MX}_i$ (where each column is a length‑$\\boldsymbol{M}$ response vector over the same set of stimuli)—and compute the Pearson correlation coefficient $r_k = \\texttt{corr⁡} ⁣(\\boldsymbol{x}_j(k), \\boldsymbol{ Mx}_i(k))$ where $\\texttt{corr}(a, b) = \\frac{\\sum{(a_i - \\bar{a})(b_i - \\bar{b})}}{\\sqrt{\\sum{(a_i - \\bar{a})^2} \\sum{(b_i - \\bar{b})^2}}}$. The reported alignment score is the mean of these per‑unit correlations: $\\texttt{Alignment}(\\boldsymbol{X}_i, \\boldsymbol{X}_j) = \\frac{1}{\\min{(N_x, N_y)}}\\sum r_k$. We have inserted this explicit definition in the revised manuscript.\n\n2. The symbol $\\mathcal{O}$ is used in **L129** as standard notation to describe the standard $n$ dimensional orthogonal group. However, in **L163**, $\\mathcal{O}$ is yet again standard notation to describe computational time complexity (or Big-O notation). To avoid any ambiguity we will use **two distinct fonts: the bold‑face roman capital “$\\mathbf{O}$” for the orthogonal group and the calligraphic “$\\mathcal{O}$” for big-O notation.** \n\n3. Thank you for pointing this out, we will make the text in the legends larger in the revised version of our manuscript.\n\n4. Our study does **not** propose a new algorithmic method that would require novel theoretical guarantees; rather, it is an empirical audit that employs three well‑established alignment metrics (linear regression, orthogonal Procrustes, permutation/soft‑matching). The theoretical properties of these metrics—*e.g.,* invariance classes, optimization uniqueness—are already documented in the literature we cite (Williams et al., 2019; Khosla & Williams,  2024). Our contribution is to apply them side‑by‑side at unprecedented scale and reveal new empirical regularities (minimal gap between procrustes and linear predictivity, first‑epoch convergence, OOD divergence for later layers). Therefore, additional formal analysis is not needed to support the effectiveness of the metrics themselves; the value of our work lies in the comprehensive evidence we present.\n\n**References:**\n\n**[1]** Generalized Shape Metrics on Neural Representations (Williams et al., 2019)\n\n**[2]** Soft Matching Distance: A metric on neural representations that captures single-neuron tuning (Khosla & Williams, 2024)"}, {"review_id": "Vq3UXMFEa5", "reviewer": "Reviewer_CU7z", "summary": "This paper provides a large-scale audit of convergence learning, focusing on three alignment metrics and disentangling the influence of layer position, training phase, and distribution shifts. The main paper provides results for convolutional networks, whereas more networks (self-supervised vision models, vision transformers, and language models) are deferred to the appendix.", "questions": "1) What are the key findings that are shared across all model types? Are there any big differences?\n2) Can you address some of the questions regarding section 4.3 (see the weaknesses).\n3) In Table 1, why do you report min and max alignment scores, and not, for instance, mean and standard deviation? Do you think it better summarizes the results? To me, it feels like I am missing some important information to fully understand what the results look like.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n- The motivation for the paper is clear, and the problem it investigates is important. Thorough studies comparing multiple variables and not just one method evaluated on one dataset can provide deeper insights, objectively compare the methods, and isolate the key factors.\n- The methods are clearly described, and the number of experiments conducted (including the ones in the appendix) is sufficient. All claims and conclusions are supported by the provided figures and tables.\n- The results are put into the context of relevant literature, the authors comment on them well, and provide possible explanations.\n\n**Weaknesses:**\n- I think the biggest weakness is how the results are organized and presented.\n\t- For instance, the main paper contains results only for convolutional models, whereas self-supervised vision and language models are deferred to the appendix. I think the paper could benefit from reorganizing. The key results and conclusions holding across model types could be in the main paper, and the detailed results could be moved to the appendix.\n\t- It is a bit hard to identify the key findings (partially because there is no conclusion section). I found many interesting findings and claims in the results section, but it seems to me that they are not highlighted enough in the other sections, so a reader might easily overlook them unless they read the whole paper carefully.\n- Figures should contain all the information needed to understand them (for example, what each row and column means). It is described in the captions, but one has to read the whole paragraph to find all the information, for instance, the data it was evaluated on.\n-  I struggle to fully understand section 4.3. Some of my questions might just need a clearer explanation, and some are challenging the whole experimental design.\n\t- If I understand it correctly, you use only the Procrustes metric in this part. Why not all three of them?\n\t- In Figure 5, do you compare only the same layers (at the same depth)? Why not all of them? Could it be the case that the networks process the OOD data slightly differently from the WD data, so some of the processing might actually happen at different layers? Also, just from the figure, the difference does not seem to be that big. Did you test the significance of the results?\n\t- I do not understand Figure 6. In a), what is the task performance? Is it layer-specific, or (as I guess) one number for the whole network? It is slightly confusing to plot one information twice. How exactly is b) computed?\n\t- You claim that \"... one could plausibly improve OOD generalisation by fine-tuning only the later layers.\" and \"...OOD stimuli could be especially useful to distinguish and select between models whose representations closely mirror the brain.\" Could you provide more details about this? Have you (or some other work) investigated that?", "was_revised": true, "final_justification": "It is hard to assess the readability and organization of the paper, updated as the authors promised. However, the provided comments and excerpt from the discussion make me believe that these changes will improve the paper and highlight the main contributions of the paper.\nThe authors addressed all my questions. I think the paper is a nice contribution, tackling an important and timely problem. I do not raise the score because I do not think the impact of the paper will be groundbreaking, but I think it is a solid paper and support its acceptance.", "rebuttal": "We would sincerely like to thank reviewer **Cu7z** for their thoughtful comments and recognizing the importance of our work. Below, we address each of their concerns:\n\n> I think the biggest weakness is how the results are organized and presented.\n\nWe agree with the reviewer that the results could be reorganized in the paper to highlight our main contributions. We have now reorganized the manuscript to include all CIFAR-100 results in the appendix instead of the main text. We have also moved our results on self-supervised networks, Vision Transformers and language models to the main text. We also reiterate our novel contributions in the **Discussion section (Sec. 5)** and make our contributions verbose. Specifically, we highlight that our work makes 3 novel contributions. First, no previous work has asked *what is the simplest mapping that still aligns two networks?* We answer this by applying three metrics with progressively weaker invariances (permutation $\\rightarrow$ orthogonal $\\rightarrow$ linear) and find that orthogonal transformations—which preserve geometric structure—align networks nearly as effectively as linear mappings, suggesting simple rotations/reflections drive convergence, not complex affine distortions. This exposes a far tighter structural similarity than earlier CCA‑ or regression‑‑based studies. Second, we document a surprising temporal dynamic: convergence largely occurs in the first epoch, challenging the view that alignment arises solely from prolonged task optimization. Prior work on “early‑phase” dynamics examined *single* networks; no study has shown such *cross‑network* alignment emerging this early, nor replicated it across multiple models and domains (vision/language). Third, by systematically examining the effects of distribution shifts, we find that early layers robustly maintain alignment while later layers diverge significantly under OOD conditions in direct proportion to the distribution shift. To our knowledge, these findings have not been reported before. These findings come from a large‑scale audit  that **(i)** spans CNNs, ViTs, and LLMs; **(ii)** tests three alignment families (permutation, Procrustes, linear regression) under identical protocols; and **(iii)** reports thousands of layer‑pair comparisons—far broader than earlier studies, which typically used a single metric, a handful of models, and a single checkpoint.\n\n> Figures should contain all the information needed to understand them\n\nThank you for pointing this out, we have now added legends to subplots as well as titles such that information about the experimental results need not be necessarily read from their respective captions.\n\n> If I understand it correctly, you use only the Procrustes metric in this part. Why not all three of them?\n\nIndeed, all three alignment metrics—linear regression, orthogonal Procrustes, and permutation/soft-matching—show consistent trends throughout our analyses. We chose to use the Procrustes metric in **Figures 5** and **6** as it offers a principled trade-off between mapping flexibility and interpretability: it discounts only rotations and reflections, and as shown in our earlier results, it achieves alignment scores that are nearly as high as linear regression, indicating that most differences across networks can be accounted for by simple rigid transformations. We have clarified this rationale in the manuscript and included the results for linear predictivity and permutation/soft-matching in the appendix. \n\n> In Figure 5, do you compare only the same layers (at the same depth)? Why not all of them? Could it be the case that the networks process the OOD data slightly differently from the WD data, so some of the processing might actually happen at different layers? Also, just from the figure, the difference does not seem to be that big. Did you test the significance of the results?\n\nWe wanted to maintain consistency with our remaining analyses, by comparing layers at the same depth across two independently trained models. To address the reviewer’s point, we have now computed full layer-by-layer alignment matrices for every OOD dataset. In all cases the strongest alignment still lies on the diagonal, indicating that OOD inputs do not shift processing to different depths; the hierarchical correspondence mirrors that observed with in‑distribution data.\n\nFurther, for significance, we now performed a relative $t$-test comparing WD vs. OOD alignment separately on the “early” layers (normalized depth $< 0.5$) and “late” layers (normalized depth $> 0.5$) across all architectures. The average $p$-values were $0.038$ for early layers and $0.0006$ for late layers, confirming that deeper layers diverge significantly more under distribution shifts. We will include the significance table in the revised manuscript. \n\n\n> I do not understand Figure 6. In a), what is the task performance? Is it layer-specific, or (as I guess) one number for the whole network? It is slightly confusing to plot one information twice. How exactly is b) computed?\n\nIn **Fig. 6a**, each value in the scatter plot represents the overall task performance of the full network on a given OOD dataset **[1]**. Individual points correspond to different OOD datasets, as noted in the figure legend, and each subplot reflects a specific ResNet50-layer indicated in the subplot title. The $y$-axis depicts the Procrustes alignment score between two independently initialized ResNet‑50s at the layer named in the subplot title (*e.g.,* “conv1” or “penultimate”). Our goal with this figure is to demonstrate that early layers stay highly aligned regardless of OOD accuracy, whereas deeper layers’ alignment covaries strongly with the accuracy drop.\n\nFor **Fig. 6b**, we first normalize the layer depths to a common scale $(0-1)$ across all architectures (ResNet18, ResNet50, VGG16 and VGG19) so that their line plots can be directly compared. We then compute for each layer $\\ell$ the correlation $\\rho(\\ell)$ across the $17$ datasets between **(i)** network-level OOD accuracy (dataset) and **(ii)** alignment score ($\\ell$, dataset) as a function of normalized layer depth across each model's entire depth. This analysis demonstrates how the coupling between representational divergence and performance grows along the hierarchy\n\n> You claim that \"... one could plausibly improve OOD generalisation by fine-tuning only the later layers.\" and \"...OOD stimuli could be especially useful to distinguish and select between models whose representations closely mirror the brain.\" Could you provide more details about this? Have you (or some other work) investigated that?\n\nWe apologise for any confusion—those remarks were meant as forward‑looking hypotheses rather than firm conclusions. The direct evidence for \"fine-tuning only later layers improves OOD generalization\" appears to be limited. We inferred the idea from the observations that **(i)** early CNN layers capture generic, transferable features whereas **(ii)** deeper layers become task‑specific, so preserving the former while adapting the latter should in principle help under distribution shift. Likewise, our suggestion that OOD stimuli could better differentiate models was also speculative. Isolated findings already hint at this leverage: Kar et al. (2019) showed that recurrent and feed‑forward CNNs, indistinguishable on simple images, separate when tested on challenging scenes; To our knowledge, however, no work has systematically compared “natural” versus distorted image sets for the explicit purpose of ranking brain‑predictive models. \n\n> What are the key findings that are shared across all model types? Are there any big differences?\n\nAcross every model family we tested, three patterns were universal: **(i)** convergence emerges almost immediately: most of the eventual alignment is in place after the first training epoch; **(ii)** orthogonal maps are nearly as good as full linear maps, indicating that simple rotations/reflections account for most representational differences; and **(iii)** early layers are both highly aligned and robust to distribution shift, whereas deeper layers diverge in proportion to OOD accuracy drops. The first two results also extended to the language domain. The main divergences appear along architectural lines: in vision models alignment falls monotonically with depth, but in language models this trend is not apparent; CNNs exhibit clear “privileged axes” (permutation alignment well above chance), whereas ViTs do not show axis preservation.\n\n> In Table 1, why do you report min and max alignment scores, and not, for instance, mean and standard deviation? Do you think it better summarizes the results? To me, it feels like I am missing some important information to fully understand what the results look like.\n\nWe chose min / max values in **Table 1** because we wanted to know whether *every* layer shows privileged‑axis behavior. Reporting the **minimum** alignment after rotation tells us whether even the *least* aligned layer still exhibits a substantial drop (i.e., no layer is exempt), while the **maximum** establishes how large the effect can be. A mean $\\pm$  SD would blur this worst‑case information; a single high‑variance layer could be “averaged away.” That said, we agree the reader may also want a sense of central tendency, so in the revision we now provide mean $\\pm$ SD across layers in the Appendix and note that the qualitative conclusion—privileged axes at every layer—remains unchanged.\n\n**References:**\n\n**[1]** Partial success in closing the gap between human and machine vision (Geirhos et al., 2021)\n\n**[2]** Evidence that recurrent circuits are critical to the ventral stream's execution of core object recognition behavior (Kar et al., 2019)"}, {"review_id": "nQfvE16wkF", "reviewer": "Reviewer_6h5w", "summary": "This paper investigates convergent learning within neural networks. Here, inter-architectural similarity is evaluated across depth (layers) and convergence (different checkpoints across training), and metric (Procrustes, soft-matching, and linear regression).\n\nThey present a number of findings: 1) While the most flexible, linear regression shows only a minor increase in alignment in all situations (across depth / convergence) over orthogonal Procrustes, suggesting  2) Across most scenarios, representation similarity almost fully manifests within the first epoch of training 3) OOD behavior is observed mostly within the later layers of the network 4) Unlike CNN's, vision transformers do not seem to have a \"privileged axis\" where different instantiations of the same architecture converge to similar representations.\n\nA few side experiments are run, with results such as: a) representation similarity between humans in the visual cortex decrease as information propagate through the visual pathway b) the similarity reported by soft-matching is often decreased when one network's activations are put through an orthogonal transformation, suggesting that actual axes along which information is computed and stored is retained across architectures c) much of their observations for supervised models are mirrored in self-supervised models.", "questions": "- Can you provide justification for taking the center values of the activation maps outside of computational efficiency? I'm quite concerned about how strong of claims can be made with so much of the representation thrown away.\n- How does similarity change if an arbitrary spatial location is chosen for the activation, rather than the center?\n- Would it be possible to evaluate similarity in a limited set of models / layers using the full activation map, compared to just the center channel, to understand the effect of that decision?\n- I may be misrepresenting the novelty in my review, can you state what distinguishes this work from previous?", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n- No complaints about clarity and quality of writing. The paper was easy to follow throughout. \n- In terms of experiments, the specific ones ran are reasonable ones to do to answer their questions of how representation similarity across architectures evolve with a) layer depth (Section 4.1) b) with training time (Section 4.2) c) distributions shifts (Section 4.3). \n- I liked seeing evaluation of across layers, training time, and dataset all within one paper.\n- Large-scale evaluation (across models and layers) provide broad empirical results.\n\n## Weaknesses\n- Choice of center activation: I think choosing the channels at the center of the networks representation might be a problem. I certainly understand the motivation, the activation maps of a CNN--especially early in the network--can be *extremely* large, computing similarity across thousands of layer-layer combinations might well be intractable without some way of reducing dimensionality. But in many cases it may be a massive reduction in the fidelity of the representation. As an example, I believe 224x224x64 is a reasonable early activation map for a ResNet50; if we take the center channel we retain about 0.002% of the data. From this tiny window into the network, strong, generalized claims about the convergent behavior of neural networks are made. The inductive biases of CNN's are all about translation equivariance, local feature detection, and hierarchical feature computation, but it seems like most of that would be discarded by taking the center channel. This may be reasonable choice early in the network layers, but does it hold higher up? [https://arxiv.org/abs/2110.14739](https://arxiv.org/abs/2110.14739) is cited as a possible justification for this, but my reading (e.g., Section 2.4) of this paper doesn't seem to support this conclusion. Further, this paper uses the entire activation map (though it is reshaped for computational convenience).\n- The inductive biases of ViTs seem to encourage entirely distinct modes of computation compared to CNNs, it feels like an oversight to not include analysis across CNNs and ViTs.\n- Novelty: Prior work has explored layer wise similarity and early training dynamics. Methodological novelty is limited, centering mostly on the three axis analysis.\n\nMy major concerns are the use of the center activation and novelty. If use of only the center activate is assumed to be an appropriate choice, I largely have no issues with the methodological approach and experimental results.", "was_revised": true, "final_justification": "I'm maintaining my score of 3 as I'm still concerned about the use of the center activation in the feature map. The authors provided results taking random activations showing little change in results, though these were only for early layers (1-4) and not later ones--I think its reasonable to expect that the model is exploiting *less* spatial information early on (supported by papers I provided showing that use of spatial information builds in later layers).\n\nNonetheless, it should be taken into account that the other reviewers do not take issue in this. Ignoring this concern, I think the paper is fine.", "rebuttal": "We thank reviewer **6h5w** for the thoughtful comments and constructive suggestions. Below, we address each point in turn.\n\n> Choice of center activation\n\nWe appreciate the reviewer’s concern and want to emphasize that comparing convolutional layers is nontrivial due to their spatial structure. Our choice of extracting activations from a single spatial location (the center pixel) was guided by both theoretical and practical considerations: \n\n1. **Translation Equivariance of CNNs:** Convolutional layers are designed to detect the same features across all spatial locations via weight sharing. Each channel corresponds to a filter, and applying the filter across the image produces a spatial activation map. The full activation tensor is thus largely a *repeated sampling of the same feature dimensions at different locations*. With sufficient image samples, the distribution of feature responses at the center pixel is representative of the feature distribution across the entire map. \n\n2. **Empirical Validation in Our Study:** In response to the reviewer’s suggestion, we have now repeated our analyses by sampling random spatial locations in each convolutional layer. The results closely match those obtained using the center pixel, confirming that this choice does not materially affect our findings. We will include these additional analyses in the Appendix of the revised manuscript.\n\n3. **Theoretical Justification (ref. [1], Section 2.4):** As highlighted in **[1]**, the most rigorous way to compare convolutional layers would be to jointly optimize over both channel correspondences and spatial shifts between feature maps. By \"spatial shifts,\" we mean that kernels detecting similar patterns can differ in their preferred spatial offset—for instance, one network’s $3\\times 3$ kernel may respond to a $45^\\circ$ edge centered in its receptive field, while another responds to the same edge slightly top-left. Fully aligning such features requires optimizing over both channels and spatial offsets, but this joint optimization is computationally intractable. Empirically, **[1]** shows that optimal spatial shifts are negligible in ConvNets trained on CIFAR10 (**Figure 3A** in Williams et al. (2019)), motivating our simpler choice to align activations only across channels using a fixed spatial position (center pixel) as a representative sample. \n\n4. **Consistency with prior work:** Our strategy of sampling one spatial position per channel is consistent with earlier convergent‑learning studies. Li et al. **[4]** compute representational alignment by “**sampling a random image *and* a random position within the conv layer**” for each channel, i.e., they reduce every $h\\times w\\times c$ activation map to a single value per channel before alignment. We follow the same spirit—fixing the position to the center pixel for reproducibility and efficiency—while additionally verifying that random‑pixel sampling yields indistinguishable alignment trends.\n\n> The inductive biases of ViTs seem to encourage entirely distinct modes of computation compared to CNNs, it feels like an oversight to not include analysis across CNNs and ViTs\n\nWe agree that understanding how vision transformers and CNNs converge (or fail to) is an important open question. In our work, we just discuss the presence/absence of privileged basis in CNNs vs transformers as a point of divergence. We focused on within‑architectural-family convergence because the methodological assumptions are clear. Extending the analysis across CNNs and ViTs raises non-trivial choices. ViTs operate on token sequences (often summarized by a $\\texttt{[CLS]}$ token); CNNs yield spatial feature maps. There is no obvious one‑token analogue in CNNs, nor a principled way to up‑sample CNN features to match ViT tokens without injecting new invariances. Prior work (*e.g.,* [2]) appears to compare ViTs and CNNs using CKA **[3]** by flattening full feature maps for CNNs and using the $\\texttt{[CLS]}$ token for ViTs (although they don’t spell this out exactly), but this implicitly weights spatial information very differently across architectures. Because a rigorous representational comparison protocol between ViTs and CNNs is still an open research problem and likely deserves a full study of its own, we restricted our claims to effects within CNNs and within ViTs. We now make this scope choice explicit in the manuscript and flag cross‑architectural-family alignment as a promising direction for future work.\n\n> I may be misrepresenting the novelty in my review, can you state what distinguishes this work from previous?\n\nOur work makes 3 novel contributions. First, no previous work has asked *what is the simplest mapping that still aligns two networks?* We answer this by applying three metrics with progressively weaker invariances (permutation $\\rightarrow$  orthogonal $\\rightarrow$  linear) and find that orthogonal transformations—which preserve geometric structure—align networks nearly as effectively as linear mappings, suggesting simple rotations/reflections drive convergence, not complex affine distortions. This exposes a far tighter structural similarity than earlier CCA‑ or regression‑based studies. Second, we document a surprising temporal dynamic: convergence largely occurs in the first epoch, challenging the view that alignment arises solely from prolonged task optimization. Prior work on “early‑phase” dynamics examined single networks; no study has shown such *cross‑network* alignment emerging this early, nor replicated it across multiple models and domains (vision/language). Third, by systematically examining the effects of distribution shifts, we find that early layers robustly maintain alignment while later layers diverge significantly under OOD conditions in direct proportion to the distribution shift. To our knowledge, these findings have not been reported before. These findings come from a large‑scale audit  that (i) spans CNNs, ViTs, and LLMs; (ii) tests three alignment families (permutation, Procrustes, linear regression) under identical protocols; and (iii) reports thousands of layer‑pair comparisons—far broader than earlier studies, which typically used a single metric, a handful of models, and a single checkpoint.\n\n**References:**\n\n**[1]** Generalized Shape Metrics on Neural Representations (Williams et al., 2019)\n\n**[2]** Do Vision Transformers See Like Convolutional Neural Networks (Raghu et al., 2022)\n\n**[3]** Similarity of Neural Network Representations Revisited (Kornblith et al., 2019)\n\n**[4]** Convergent Learning: Do Different Neural Networks Learn the Same Representations (Li et al., 2015)"}]

Author rebuttal:

Meta-review: This paper presents an empirical study of the phenomenon of "representational convergence" across a wide range of neural networks, training trajectories, and distribution shifts. Three alignment metrics (linear regression, orthogonal Procrustes, and permutation/soft-matching) are used across thousands of layer-pair comparisons and multiple architectures, including CNNs, ViTs, and LLMs. The main findings are that: a) orthogonal transformations align representations nearly as  effectively as more flexible linear ones, b) nearly all representational alignment seems to crystallize within the first epoch potentially pointing to shared input statistics and architectural bias playing a role on alignment vs dependence on final task, and c) for out-of-distribution images, early layers remain robustly aligned, whereas deeper layers diverge in proportion to the distribution shift. 

Strengths:  The empirical scope is broad with novel insights into the timing (first-epoch convergence) and structure (orthogonal sufficiency, privileged axes) of representational alignment.  During rebuttal a number of points were addressed (e.g. sensitivity of Spearman correlation, random initialization baselines, random-pixel sampling) to alleviate reviewer concerns.  

Weaknesses:
I share the concern raised by one of the reviewers about the choice of using the center pixel from convolutional feature maps as the basis for the plots. However, the rebuttal showed that via random-pixel sampling provided similar and consistent results.  There were concerns about the presentation and I hope the authors will address them in the revised draft.  The rebuttal discussion was highly engaging and positive.

Reviewer consensus:
Most reviews with the exception of the reviewer who had the methodological concern about the use of center pixel statistics were positive. I share the concern about the methodology and urge the authors to conduct further experiments to clarify their claim.  

Decision::
The paper advances our understanding of representational convergence across neural networks and provides alternative viewpoints to current dialogue on representational alignment. Despite some methodological reservations, I recommend acceptance as a poster as it would be a venue for detailed discussion for the audience with the authors.
Decision: Accept (poster)