Representational Similarity Analysis

Comparing internal representations is a central goal in neuroscience and machine learning, but standard linear alignment metrics (Representational Similarity Analysis, Centered Kernel Alignment, and linear regression) are frequently applied to neural activity coordinates rather than on the underlying features. We show this matters when neural systems operate in superposition, encoding more features than they have neurons via linear compression. Closed-form derivations prove that these metrics depend on the Gram matrices of each system's projection, not on the latent features themselves: alignment thus combines what a system represents with how it is encoded. For those interested in what features two systems share, this is a problem: Two networks can have identical feature content yet appear more dissimilar than networks exhibiting partial feature overlap. This apparent misalignment need not reflect lost information as compressed sensing guarantees sparse features remain recoverable from the compressed activity. We confirm this by training supervised TopK sparse autoencoders that realize solvable compressed sensing by construction, finding alignment on recovered latents restored even when raw-activation alignment remains deflated. We extend the result to unsupervised SAEs trained without ground-truth latents, and to pretrained vision and language model SAEs, where SAE-latent alignment exceeds raw-activation alignment, consistent with superposition in real systems.

Similarity measures are widely used to interpret the representational geometries used by neural networks to solve tasks. Yet, because existing methods compare the extrinsic geometry of representations in state space, rather than their intrinsic geometry, they may fail to capture subtle yet crucial distinctions between fundamentally different neural network solutions. Here, we introduce metric similarity analysis (MSA), a novel method which leverages tools from Riemannian geometry to compare the intrinsic geometry of neural representations under the manifold hypothesis. We show that MSA can be used to i) disentangle features of neural computations in deep networks with different learning regimes, ii) compare nonlinear dynamics, and iii) investigate diffusion models. Hence, we introduce a mathematically grounded and broadly applicable framework to understand the mechanisms behind neural computations by comparing their intrinsic geometries.

Neural network representations are often analyzed as vectors in a fixed Euclidean space. However, their coordinates are not uniquely defined. If a hidden representation is transformed by an invertible linear map, the network function can be preserved by applying the inverse transformation to downstream weights. Representations are therefore defined only up to invertible linear transformations. We study neural representation spaces from this geometric viewpoint and treat them as vector spaces with a gauge freedom under the general linear group. Within this framework, commonly used similarity measures such as cosine similarity become metric-dependent quantities whose values can change under coordinate transformations that leave the model function unchanged. This provides a common interpretation for several observations in the literature, including cosine-similarity instability, anisotropy in embedding spaces, and the appeal of representation comparison methods such as SVCCA and CKA. Experiments on multilayer perceptrons and convolutional networks confirm that inserting invertible transformations into trained models can substantially distort cosine similarity and nearest-neighbor structure while leaving predictions unchanged. These results indicate that analysis of neural representations should focus either on quantities that are invariant under this gauge freedom or on explicitly chosen canonical coordinates.

We propose the Neural Functional Alignment Space (NFAS), a brain-referenced representational framework for characterizing artificial neural networks on equal functional grounds. NFAS departs from conventional alignment approaches that rely on layer-wise features or task-specific activations by modeling the intrinsic dynamical evolution of stimulus representations across network depth. Specifically, we model layer-wise embeddings as a depth-wise dynamical trajectory and apply Dynamic Mode Decomposition (DMD) to extract the stable mode. This representation is then projected into a biologically anchored coordinate system defined by distributed neural responses. We also introduce the Signal-to-Noise Consistency Index (SNCI) to quantify cross-model consistency at the modality level. Across 45 pretrained models spanning vision, audio, and language, NFAS reveals structured organization within this brain-referenced space, including modality-specific clustering and cross-modal convergence in integrative cortical systems. Our findings suggest that representation dynamics provide a principled basis for

Representational similarity metrics typically force all units to be matched, making them susceptible to noise and outliers common in neural representations. We extend the soft-matching distance to a partial optimal transport setting that allows some neurons to remain unmatched, yielding rotation-sensitive but robust correspondences. This partial soft-matching distance provides theoretical advantages -- relaxing strict mass conservation while maintaining interpretable transport costs -- and practical benefits through efficient neuron ranking in terms of cross-network alignment without costly iterative recomputation. In simulations, it preserves correct matches under outliers and reliably selects the correct model in noise-corrupted identification tasks. On fMRI data, it automatically excludes low-reliability voxels and produces voxel rankings by alignment quality that closely match computationally expensive brute-force approaches. It achieves higher alignment precision across homologous brain areas than standard soft-matching, which is forced to match all units regardless of quality. In deep networks, highly matched units exhibit similar maximally exciting images, while unmatched units show divergent patterns. This ability to partition by match quality enables focused analyses, e.g., testing whether networks have privileged axes even within their most aligned subpopulations. Overall, partial soft-matching provides a principled and practical method for representational comparison under partial correspondence.

For a broad family of discriminative models that includes autoregressive language models, identifiability results imply that if two models induce the same conditional distributions, then their internal representations are equal up to an invertible linear transformation. We ask whether an analogous conclusion holds approximately when the distributions are close instead of equal. Building on the observation of Nielsen et al. (2025) that closeness in KL divergence need not imply high linear representational similarity, we study a distributional distance based on logit differences and show that closeness in this distance does yield linear similarity guarantees. Specifically, we define a representational dissimilarity measure based on the models' identifiability class and prove that it is bounded by the logit distance. We further show that, when model probabilities are bounded away from zero, KL divergence upper-bounds logit distance; yet the resulting bound fails to provide nontrivial control in practice. As a consequence, KL-based distillation can match a teacher's predictions while failing to preserve linear representational properties, such as linear-probe recoverability of human-interpretable concepts. In distillation experiments on synthetic and image datasets, logit-distance distillation yields students with higher linear representational similarity and better preservation of the teacher's linearly recoverable concepts.

The Platonic Representation Hypothesis suggests that representations from neural networks are converging to a common statistical model of reality. We show that the existing metrics used to measure representational similarity are confounded by network scale: increasing model depth or width can systematically inflate representational similarity scores. To correct these effects, we introduce a permutation-based null-calibration framework that transforms any representational similarity metric into a calibrated score with statistical guarantees. We revisit the Platonic Representation Hypothesis with our calibration framework, which reveals a nuanced picture: the apparent convergence reported by global spectral measures largely disappears after calibration, while local neighborhood similarity, but not local distances, retains significant agreement across different modalities. Based on these findings, we propose the Aristotelian Representation Hypothesis: representations in neural networks are converging to shared local neighborhood relationships.

Representational similarity methods compare the geometries of neural representations, but they do not measure how consistently the geometry of a single representation is recovered from subsets of its feature coordinates. We call this property geometric stability and introduce Shesha, which estimates it by correlating representational dissimilarity matrices from complementary random feature subsets. Shesha is not invariant to orthogonal rotations: representations with identical Gram matrices, and therefore identical linear CKA, can have different geometric stability. Controlled transformations further separate the quantities. Across $2{,}463$ encoder configurations spanning seven domains, similarity and stability are positively associated across non-PCA transformations ($ρ=+0.75$) but negatively associated under PCA-coordinate compression ($ρ=-0.47$). We further evaluate 170 pretrained vision models across six datasets. DINOv2 combines strong transfer performance with bottom-quartile stability on five of six datasets, showing that transferability and feature-split stability need not coincide. Across random feature subsets, the marginal relationship between Shesha and linear-probe variability is dataset-dependent; after controlling for task alignment with LogME, higher Shesha is associated with lower variability on five of six datasets. These results identify geometric stability as a basis-dependent property that complements representational similarity and task alignment.

Studying the shape of text embedding spaces enhances our understanding of model behavior and how geometry affects downstream task performance. While a range of topological and geometric metrics have been explored, they are typically analyzed in isolation, and the connections between them remain largely unknown. In this paper, we present Unified Topological Signatures (UTS) for characterizing the structure of embedding spaces. We evaluate our method across a wide range of text embedding models and retrieval datasets, finding that it outperforms individual metrics in downstream performance prediction. Through several experiments, we demonstrate that the signatures characterize embedding spaces more accurately, improve the prediction of model-specific properties and biases, and reveal novel insights into representational similarity that challenge the global claims of the Platonic Representation Hypothesis.

Centered kernel alignment (CKA) is a popular metric for comparing representations, determining equivalence of networks, and conducting neuroscience research. However, CKA does not account for the underlying manifold and relies on numerous heuristics that cause it to behave differently at different scales of data. In this work, we propose Manifold-approximated Kernel Alignment (MKA), which incorporates manifold geometry (i.e., local neighborhood) into the alignment task. We derive a theoretical framework for MKA. Our construction removes representation-dependent degree weighting, yields a closed-form alignment score, and gives each row a probabilistic interpretation as a local Markov transition distribution. We further establish kernel- and score-level stability with respect to the neighborhood size. We perform empirical evaluations on synthetic datasets and real-world examples to characterize and compare MKA to its contemporaries. In controlled synthetic experiments, MKA is consistently less sensitive to hyperparameters and captures the selected topology-preserving relationships. On the large-scale benchmark, MKA achieves its strongest aggregate performance in vision, is competitive on graphs, and is weaker than CKA in several NLP evaluations. Our findings suggest that manifold-aware kernel alignment provides a robust foundation for measuring representations and is potentially useful when local neighborhood geometry is important, with possible applications in representation learning.

It was recently observed that the representations of different models that process identical or semantically related inputs tend to align. We analyze this phenomenon using the Information Imbalance, an asymmetric rank-based measure that quantifies the capability of a representation to predict another, providing a proxy of the cross-entropy which can be computed efficiently in high-dimensional spaces. By measuring the Information Imbalance between representations generated by DeepSeek-V3 processing translations, we find that semantic information is spread across many tokens, and that semantic predictability is strongest in a set of central layers of the network, robust across six language pairs. We measure clear information asymmetries: English representations are systematically more predictive than those of other languages, and DeepSeek-V3 representations are more predictive of those in a smaller model such as Llama3-8b than the opposite. In the visual domain, we observe that semantic information concentrates in middle layers for autoregressive models and in final layers for encoder models, and these same layers yield the strongest cross-modal predictability with textual representations of image captions. Our results support the hypothesis of semantic convergence across languages, modalities, and architectures, while showing that directed predictability between representations varies strongly with layer-depth, model scale, and language.