Neural Representation Geometry

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18 papers in the last four weeks, up 125% on the four weeks before. 0.2% of all new papers.

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Latest papers 182

May 1, 2026cs.NE

Geometric and dynamical analysis of attractor boundaries and storage limits in kernel Hopfield networks

High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit strong storage capabilities, but the dynamical and geometric mechanisms underlying their stability remain poorly understood. This paper investigates the global geometry of attractor basins and the mechanisms governing the storage limit in KLR-trained Hopfield networks. We combine empirical evaluations using random sequences and real-world image embeddings (CIFAR-10) with morphing experiments and statistical Signal-to-Noise Ratio (SNR) analysis. Our experiments show that the network achieves a storage capacity for random sequences up to

P/N \approx 16

, while maintaining stable retrieval for structured data at effective loads near

P/N \approx 20

. Morphing analysis indicates that attractors on the "Ridge of Optimization" are separated by sharp, phase-transition-like boundaries, characterized by steep effective potential barriers and critical slowing down. Furthermore, by comparing an SNR analysis with a geometric reference point inspired by Cover's theorem, we show that the practical storage limit is governed primarily not by a lack of geometric separability in the feature space, but by the loss of dynamical stability against crosstalk noise. These findings suggest that KLR networks function as highly localized exemplar-based memories that operate near the onset of dynamical collapse, providing a useful perspective on the design of robust, large-scale retrieval systems.

Akira Tamamori

Faculty of Information Science, Aichi Institute of Technology 1247 Yachigusa, Yakusa-cho, Toyota-shi, Aichi 470-0392, Japan

Dynamical Systems Neural Representation Geometry Hopfield Networks Associative Memory

Apr 30, 2026cs.LG

Do Sparse Autoencoders Capture Concept Manifolds?

Sparse autoencoders (SAEs) are widely used to extract interpretable features from neural network representations, often under the implicit assumption that concepts correspond to independent linear directions. However, a growing body of evidence suggests that many concepts are instead organized along low-dimensional manifolds encoding continuous geometric relationships. This raises three basic questions: what does it mean for an SAE to capture a manifold, when do existing SAE architectures do so, and how? We develop a theoretical framework that answers these questions and show that SAEs can capture manifolds in two fundamentally different ways: globally, by allocating a compact group of atoms whose linear span contains the entire manifold, or locally, by distributing it across features that each selectively tile a restricted region of the underlying geometry. Empirically, we find that SAEs suboptimally recover continuous structures, mixing the global subspace and local tiling solutions in a fragmented regime we call dilution. This explains why manifold structure is rarely visible at the level of individual concepts and motivates post-hoc unsupervised discovery methods that search for coherent groups of atoms rather than isolated directions. More broadly, our results suggest that future representation learning methods should treat geometric objects, not just individual directions, as the basic units of interpretability.

Usha Bhalla, Thomas Fel, Can Rager +9

aHarvard University · bNortheastern University · cTechnion IIT +1

Neural Representation Geometry Sparse Autoencoders Neural Network Interpretability Manifold Learning Sparse Coding

Apr 29, 2026cs.CL

Semantic Structure of Feature Space in Large Language Models

We show that the geometric relations between semantic features in large language models' hidden states closely mirror human psychological associations. We construct feature vectors corresponding to 360 words and project them on 32 semantic axes (e.g. beautiful-ugly, soft-hard), and find that these projections correlate highly with human ratings of those words on the respective semantic scales. Second, we find that the cosine similarities between the semantic axes themselves are highly predictive of the correlations between these scales in the survey. Third, we show that substantial variance across the 32 semantic axes lies on a low-dimensional subspace, reproducing patterns typical of human semantic associations. Finally, we demonstrate that steering a word on one semantic axis causes spillover effects on the model's rating of that word on other semantic scales proportionate to the cosine similarity between those semantic axes. These findings suggest that features should be understood not only in isolation but through their geometric relations and the meaningful subspaces they form.

Austin C. Kozlowski, Andrei Boutyline

Knowledge Lab · University of Chicago · Department of Sociology +1

Neural Representation Geometry LLM Interpretability

Apr 28, 2026cs.LG

Monitoring Neural Training with Topology: A Footprint-Predictable Collapse Index

Representational collapse, where embeddings become anisotropic and lose multi-scale structure, can erode downstream performance long before performance metrics react. We propose an online, topology-aware monitor for evolving neural representations that couples Modular Morse Homology Maintenance (MMHM) with a composite Collapse Index (CI). Instead of rebuilding complexes each epoch, we apply sparse edits at a fixed scale and maintain a discrete Morse matching, yielding fast, incremental updates. Across LLM fine-tuning and temporal KGE training, CI provides a low-latency early-warning signal suitable for in-training interventions. Code and experimental scripts will be released publicly

Alexander Kalinowski

SUNY Empire

Representation Collapse Neural Representation Geometry Neural Network Training Dynamics Topological Data Analysis

Apr 27, 2026cs.IR

Geometric Analysis of Self-Supervised Vision Representations for Semantic Image Retrieval

Content-based image retrieval (CBIR) systems enable users to search images based on visual content instead of relying on metadata. The text domain has benefited from vector search of representations created with unsupervised methods such as BERT. However, modern self-supervised learning methods for vision are mostly not reported in CBIR-related literature, instead relying on supervised models or multi-modal methods that align text and vision. We evaluate how the representations learned by modern self-supervised learning methods for vision perform under typical retrieval stacks that leverage vector databases and nearest neighbor search. Our evaluation reveals that the latent space geometry impacts approximate nearest neighbor (ANN) indexing. Specifically, highly anisotropic representations with high skewness produced by several modern SSL methods degrade the performance of partition-based and hashing-based search, even if their own linear probe or K-NN accuracy is not affected. In contrast, representations with higher isotropy and local purity better satisfy the distance-based assumptions of ANN indexes, leading to improved semantic retrieval performance.

Esteban Rodríguez-Betancourt, Edgar Casasola-Murillo

Posgrado en Computación e Informática Universidad de Costa Rica · Escuela de Ciencias de la Computación Universidad de Costa Rica

Neural Representation Geometry Self-Supervised Visual Representation Learning Approximate Nearest Neighbor Search

Apr 27, 2026cs.AI

Representational Curvature Modulates Behavioral Uncertainty in Large Language Models

In autoregressive large language models (LLMs), temporal straightening offers an account of how the next-token prediction objective shapes representations. Models learn to progressively straighten the representational trajectory of input sequences across layers, potentially facilitating next-token prediction via linear extrapolation. However, a direct link between this trajectory and token-level behavior has been missing. We provide such a link by relating contextual curvature-a geometric measure of how sharply the representational trajectory bends over recent context-to next-token entropy. Across two models (GPT-2 XL and Pythia-2.8B), contextual curvature is correlated with entropy, and this relationship emerges during training. Perturbation experiments reveal selective dependence: manipulating curvature through trajectory-aligned interventions reliably modulates entropy, while geometrically misaligned perturbations have no effect. Finally, regularizing representations to be straighter during training modestly reduces token-level entropy without degrading validation loss. These results identify trajectory curvature as a task-aligned representational feature that influences behavioral uncertainty in LLMs.

Jack King, Evelina Fedorenko, Eghbal A. Hosseini

Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 · Google DeepMind

Neural Representation Geometry Autoregressive Language Modeling LLM Uncertainty Estimation Representation Geometry in Language Models

Apr 22, 2026cs.NE

Quantization robustness from dense representations of sparse functions in high-capacity kernel associative memory

High-capacity associative memories based on Kernel Logistic Regression (KLR) achieve strong retrieval performance but typically require substantial computational resources. This paper investigates the compressibility of KLR Hopfield networks to clarify the geometric principles underlying their robust representations. We present a geometric interpretation based on spontaneous symmetry breaking and Walsh analysis, and examine it through compression experiments involving quantization and pruning. The experiments reveal a clear asymmetry: the network remains robust under low-precision quantization while exhibiting strong sensitivity to pruning. We interpret this behavior through a "sparse function, dense representation" principle, in which a sparse input mapping is implemented through a dense bimodal parameterization. These findings suggest a practical route toward hardware-efficient kernel associative memories and provide insight into the geometric principles underlying robust representation in neural systems.

Akira Tamamori

Faculty of Information Science, Aichi Institute of Technology 1247 Yachigusa, Yakusa-cho, Toyota-shi, Aichi 470-0392, Japan

Neural Representation Geometry Hopfield Networks Associative Memory Neural Network Quantization

Apr 22, 2026cs.LG

Rethinking Intrinsic Dimension Estimation in Neural Representations

The analysis of neural representation has become an integral part of research aiming to better understand the inner workings of neural networks. While there are many different approaches to investigate neural representations, an important line of research has focused on doing so through the lens of intrinsic dimensions (IDs). Although this perspective has provided valuable insights and stimulated substantial follow-up research, important limitations of this approach have remained largely unaddressed. In this paper, we highlight a crucial discrepancy between theory and practice of IDs in neural representations, theoretically and empirically showing that common ID estimators are, in fact, not tracking the true underlying ID of the representation. We contrast this negative result with an investigation of the underlying factors that may drive commonly reported ID-related results on neural representation in the literature. Building on these insights, we offer a new perspective on ID estimation in neural representations.

Rickmer Schulte, David Rügamer

LMU Munich, MCML

Intrinsic Dimensionality Neural Representation Geometry Neural Network Interpretability

Apr 22, 2026cs.LG

ILDR: Geometric Early Detection of Grokking

Grokking describes a delayed generalization phenomenon in which a neural network achieves perfect training accuracy long before validation accuracy improves, followed by an abrupt transition to strong generalization. Existing detection signals are indirect: weight norm reflects parameter-space regularization and consistently lags the transition, while GrokFast's slow gradient EMA, used without gradient amplification, is unstable across seeds with standard deviation exceeding mean lead time. We propose the Inter/Intra-class Distance Ratio (ILDR), a geometric metric computed on second-to-last layer representations as the ratio of inter-class centroid separation to intra-class scatter. ILDR provides an early detection signal: it rises and crosses a threshold at 2.5 times its baseline before the grokking transition appears in validation accuracy, indicating early geometric reorganization in representation space. Grounded in Fisher's linear discriminant criterion, ILDR requires no eigendecomposition and runs in O(|C|^2 + N). It is evaluated exclusively on held-out data, making it robust to memorization effects. Across modular arithmetic and permutation group composition (S5), ILDR leads the grokking transition by 9 to 73 percent of the training budget, with lead time increasing with task algebraic complexity. Over eight random seeds, ILDR leads by 950 +/- 250 steps with a coefficient of variation of 26 percent, and post-grokking variance drops by 1696 times, consistent with a sharp phase transition in representation space. Using ILDR as an early stopping trigger reduces training by 18.6 percent on average. Optimizer interventions triggered at the ILDR threshold demonstrate bidirectional control over the transition, suggesting ILDR tracks representational conditions underlying generalization rather than a downstream correlate.

Shreel Golwala

Virginia Tech

Neural Network Generalization Early Stopping Neural Representation Geometry Grokking

Apr 20, 2026cs.CV

Geometric Decoupling: Diagnosing the Structural Instability of Latent

Latent Diffusion Models (LDMs) achieve high-fidelity synthesis but suffer from latent space brittleness, causing discontinuous semantic jumps during editing. We introduce a Riemannian framework to diagnose this instability by analyzing the generative Jacobian, decomposing geometry into \textit{Local Scaling} (capacity) and \textit{Local Complexity} (curvature). Our study uncovers a \textbf{

Geometric Decoupling"}: while curvature in normal generation functionally encodes image detail, OOD generation exhibits a functional decoupling where extreme curvature is wasted on unstable semantic boundaries rather than perceptible details. This geometric misallocation identifies

Geometric Hotspots" as the structural root of instability, providing a robust intrinsic metric for diagnosing generative reliability.

Yuanbang Liang, Zhengwen Chen, Yu-Kun Lai

School of Computer Science and Informatics, Cardiff University, Cardiff, United Kingdom

Neural Representation Geometry Latent Diffusion Models Generative Modeling Riemannian Geometry

Apr 20, 2026cs.LG

Semantic Step Prediction: Multi-Step Latent Forecasting in LLM Reasoning Trajectories via Step Sampling

Semantic Tube Prediction (STP) leverages representation geometric to regularize LLM hidden-state trajectories toward locally linear geodesics during fine-tuning, thereby greatly improving data efficiency. The original STP recipe samples random token sub-spans, which is compatible with the base large language model (LLM) training architecture. Inspired by STP, we are interested to investigate whether the sampling position can further enhance the semantic structure of multi-step reasoning, and hence affect its geometric impact. We applied STP at consecutive semantic reasoning step boundaries and achieved 168x more accurate multi-step latent prediction than frozen baselines on ProcessBench (3,400 samples), compared to only 4x for the random-token STP. Probing the latent manifold with a learned non-linear predictor reveals that STP-shaped trajectories are smooth curves, not straight lines: a 3-layer MLP reduces prediction error by a further 3-12x over linear extrapolation on step-boundary models. Removing the language modeling loss yields trajectories that are 2x more MLP-predictable than the combined loss, revealing a tradeoff between generation quality and geometric purity. Our results identify sampling position as the critical variable in geometric regularization and establish multi-step latent prediction MSE as a new evaluation metric for this class of methods.

Yidi Yuan

Home Team Science and Technology Agency

Neural Representation Geometry Language Model Post-Training

Apr 20, 2026q-bio.NC

The Umwelt Representation Hypothesis: Rethinking Universality

Recent studies reveal striking representational alignment between artificial neural networks (ANNs) and biological brains, leading to proposals that all sufficiently capable systems converge on universal representations of reality. Here, we argue that this claim of Universality is premature. We introduce the Umwelt Representation Hypothesis (URH), proposing that alignment arises not from convergence toward a single global optimum, but from overlap in ecological constraints under which systems develop. We review empirical evidence showing that representational differences between species, individuals, and ANNs are systematic and adaptive, which is difficult to reconcile with Universality. Finally, we reframe ANN model comparison as a method for mapping clusters of alignment in ecological constraint space rather than searching for a single optimal world model.

Victoria Bosch, Rowan Sommers, Adrien Doerig +1

Institute of Cognitive Science, University of Osnabrück, Osnabrück, Germany. · Department of Psychology and Education, Freie Universität Berlin, Berlin, Germany. · Bernstein Center for Computational Neuroscience, Berlin, Germany

Neural Representation Geometry

Apr 19, 2026cs.LG

ATLAS: Constitution-Conditioned Latent Geometry and Redistribution Across Language Models and Neural Perturbation Data

Constitution-conditioned post-training can be analysed as a structured perturbation of a model's learned representational geometry. We introduce ATLAS, a geometry-first program that traces constitution-induced hidden-state structure across charts, models, and substrates. Instead of treating the relevant unit as a single behaviour, neuron, vector, or patch, ATLAS tests a local chart whose tangent structure, occupancy distribution, and behavioural coupling can be measured under system change. On Gemma, the anchored source-local chart captures 310 / 320 reviewed source rows and all 84 / 84 reviewed score-flip rows, but compact exact-patch sufficiency does not close, so the exportable unit is the broader source-defined family. Freezing that family, we re-identify a target-local realisation in an unadapted Phi model, where the fully adjudicated confirmatory contrast separates with AUC 0.984 and mean gap 5.50. In held-out ALM8 mouse frontal-cortex perturbation data, the same source-defined family receives support across 5/5 folds, with mean held-out AUC 0.72 and mean fold gap 4.50. A multiple-choice analysis provides the main boundary: nearby target-local signals can appear without source-faithful closure. The resulting correspondence is not coordinate identity, site identity, or a target-side mediation theorem. It is geometric recurrence under redistribution: written constitutions can induce recoverable latent geometry whose organisation remains detectable across model and substrate changes while its local coordinates, occupancy, and behavioural expression shift.

Gareth Seneque, Lap-Hang Ho, Nafise Erfanian Saeedi +2

Australian Broadcasting Corporation

LLM Alignment Neural Representation Geometry Transfer Learning

Apr 17, 2026math.OC

A Wasserstein Geometric Framework for Hebbian Plasticity

We introduce the Tan-HWG framework (Hebbian-Wasserstein-Geometry), a geometric theory of Hebbian plasticity in which memory states are modeled as probability measures evolving through Wasserstein minimizing movements. Hebbian learning rules are formalized as Hebbian energies satisfying a sequential stability condition, ensuring well-posed fiberwise JKO updates, optimal-transport realizations, and an energy descent inequality. This variational structure induces a fundamental separation between internal and observable dynamics. Internal memory states evolve along Wasserstein geodesics in a latent curved space, while observable quantities, such as effective synaptic weights, arise through geometric projection maps into external spaces. Simplicial projections recover classical affine schemes (including exponential moving averages and mirror descent), while revealing synaptic competition and pruning as geometric consequences of mass redistribution. Hilbertian projections provide a geometric account of phase alignment and multi-scale coherence. Classical neural networks appear as flat projections of this curved dynamics, while the framework naturally accommodates richer distributional representations, including structural weights and embedding memories, and their spectral extensions in complex internal spaces. Under mild Lipschitz regularity assumptions, including a quasi-stationary "sleep-mode" regime, we establish the existence of continuous-time limit curves. This yields a variational formulation of memory consolidation as a perturbed Wasserstein gradient flow. The framework thus provides a unified geometric foundation for synaptic plasticity, representation dynamics, and context-dependent computation.

Ulrich Tan

Neural Representation Geometry Hebbian Learning Wasserstein Gradient Flows Memory Consolidation

Apr 16, 2026cs.LG

Gating Enables Curvature: A Geometric Expressivity Gap in Attention

Multiplicative gating is widely used in neural architectures and has recently been applied to attention layers to improve performance and training stability in large language models. Despite the success of gated attention, the mathematical implications of gated attention mechanisms remain poorly understood. We study attention through the geometry of its representations by modeling outputs as mean parameters of Gaussian distributions and analyzing the induced Fisher--Rao geometry. We show that ungated attention operator is restricted to intrinsically flat statistical manifolds due to its affine structure, while multiplicative gating enables non-flat geometries, including positively curved manifolds that are unattainable in the ungated setting. These results establish a geometric expressivity gap between ungated and gated attention. Empirically, we show that gated models exhibit higher representation curvature and improved performance on tasks requiring nonlinear decision boundaries whereas they provide no consistent advantage on tasks with linear decision boundaries. Furthermore, we identify a structured regime in which curvature accumulates under composition, yielding a systematic depth amplification effect.

Satwik Bathula, Anand A. Joshi

Department of Electrical and Computer Engineering University of Southern California

Information Geometry Neural Representation Geometry Self-Attention Gated Attention Transformer Attention Attention Mechanisms+1

Apr 15, 2026cs.LG

A Complete Symmetry Classification of Shallow ReLU Networks

Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as reverse engineering," or parameter identifiability", has led to the natural question of parameter space symmetries\textemdash the study of distinct parameters in neural architectures which realize the same function. Indeed, the quotient space obtained by identifying parameters giving rise to the same function, called the \textit{neuromanifold}, has been shown in some cases to have rich geometric properties, impacting optimization dynamics. Thus far, techniques towards complete classifications have required the analyticity of the activation function, notably excising the important case of ReLU. Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case.

Pranavkrishnan Ramakrishnan

Shallow Neural Networks Neural Representation Geometry Parameter Identifiability ReLU Neural Networks

Apr 8, 2026math.NA

Neural parametric representations for thin-shell shape optimisation

Shape optimisation of thin-shell structures requires a flexible, differentiable geometric representation suitable for gradient-based optimisation. We propose a neural parametric geometry representation (NRep) for shells based on a neural network with periodic activation functions. The NRep is defined using a multi-layer perceptron (MLP), which maps the parametric coordinates of mid-surface vertices to their physical coordinates. A structural compliance optimisation problem is posed to optimise the shape of a thin-shell parameterised by the NRep subject to a volume constraint, with the network parameters as design variables. The resulting shape optimisation problem is solved using a gradient-based optimisation algorithm. Benchmark examples with classical solutions and comparisons with the free-form deformation method demonstrate that the proposed NRep is capable of representing shell geometries with local geometric features using a small set of network parameters. The robustness of the approach has been demonstrated with different initial geometries, boundary conditions and neural network hyperparameters. The approach also exhibits potential for complex lattice-skin structures, owing to the compact and expressive geometry representation afforded by the NRep.

Xiao Xiao, Fehmi Cirak

School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China. · Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, United Kingdom.

Neural Representation Geometry

Apr 3, 2026cs.LG

Understanding Latent Diffusability via Fisher Geometry

Diffusion models often degrade in latent spaces, yet the formal causes remain poorly understood. We quantify latent-space diffusability via the rate of change of the Minimum Mean Squared Error (MMSE) along the diffusion trajectory. Our framework decomposes this MMSE rate into contributions from Fisher Information (FI) and Fisher Information Rate (FIR). We show that isometric embeddings preserve intrinsic FI and establish quantitative intrinsic-FI bounds for a broader class of bi-Lipschitz encoders with controlled weak volume distortion, whereas FIR is governed by the interplay between encoder and data geometries. Our analysis separates four geometric contributions in local stability bounds for Gaussian-smoothed FIR: dimensional compression, tangential distortion, high-frequency encoder curvature, and curvature of data manifold. Experiments across diverse autoencoding architectures provide qualitative support for the geometric mechanisms identified by the theory and show that empirical FI and FIR track several measures of generation quality and latent-space geometry in the settings tested. We establish FI and FIR as a comprehensive analytical framework for understanding latent diffusability.

Jing Gu, Morteza Mardani, Wonjun Lee +2

School of Mathematics, University of Minnesota · NVIDIA · Department of Mathematics, The Ohio State University +1

Information Geometry Neural Representation Geometry Latent Diffusion Models Diffusion Models

Mar 31, 2026cs.LG

Similarity of Neural Network Representations in Superposition

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.

Sunny Liu, Habon Issa, André Longon +4

Cold Spring Harbor Laboratory · UC San Diego · Anthropic +2

Feature Superposition Representational Similarity Analysis Neural Representation Geometry Compressed Sensing Neural Encoding

Mar 1, 2026cs.CV

Differential privacy representation geometry for medical image analysis

Differential privacy (DP)'s effect in medical imaging is typically evaluated only through end-to-end performance, leaving the mechanism of privacy-induced utility loss unclear. We introduce Differential Privacy Representation Geometry for Medical Imaging (DP-RGMI), a framework that interprets DP as a structured transformation of representation space and decomposes performance degradation into encoder geometry and task-head utilization. Geometry is quantified by representation displacement from initialization and spectral effective dimension, while utilization is measured as the gap between linear-probe and end-to-end utility. Across over 594,000 images from four chest X-ray datasets and multiple pretrained initializations, we show that DP is consistently associated with a utilization gap even when linear separability is largely preserved. At the same time, displacement and spectral dimension exhibit non-monotonic, initialization- and dataset-dependent reshaping, indicating that DP alters representation anisotropy rather than uniformly collapsing features. Correlation analysis reveals that the association between end-to-end performance and utilization is robust across datasets but can vary by initialization, while geometric quantities capture additional prior- and dataset-conditioned variation. These findings position DP-RGMI as a reproducible framework for diagnosing privacy-induced failure modes and informing privacy model selection.

Soroosh Tayebi Arasteh, Marziyeh Mohammadi, Sven Nebelung +1

Lab for AI in Medicine, RWTH Aachen University, Aachen, Germany · Department of Diagnostic and Interventional Radiology, University Hospital RWTH Aachen, Aachen, Germany · INDA Institute, RWTH Aachen University, Aachen, Germany

Neural Representation Geometry Healthcare Privacy-Preserving ML Differential Privacy Medical Image Representation Learning

Feb 27, 2026cs.CV

Compositional Generalization Requires Linear, Orthogonal Representations in Vision Embedding Models

Compositional generalization, the ability to recognize familiar parts in novel contexts, is a defining property of intelligent systems. Although modern models are trained on massive datasets, they still cover only a tiny fraction of the combinatorial space of possible inputs, raising the question of what structure representations must have to support generalization to unseen combinations. We formalize three desiderata for compositional generalization under standard training (divisibility, transferability, stability) and show they impose necessary geometric constraints: representations must decompose linearly into per-concept components, and these components must be orthogonal across concepts. This provides theoretical grounding for the Linear Representation Hypothesis: the linear structure widely observed in neural representations is a necessary consequence of compositional generalization. We further derive dimension bounds linking the number of composable concepts to the embedding geometry. Empirically, we evaluate these predictions across modern vision models (CLIP, SigLIP, DINO) and find that representations exhibit partial linear factorization with low-rank, near-orthogonal per-concept factors, and that the degree of this structure correlates with compositional generalization on unseen combinations. As models continue to scale, these conditions predict the representational geometry they may converge to. Code is available at https://github.com/oshapio/necessary-compositionality.

Arnas Uselis, Andrea Dittadi, Seong Joon Oh

Tübingen AI Center, University of Tübingen · Helmholtz · Technical University of Munich

Neural Representation Geometry Visual Representation Learning Linear Representation Hypothesis Orthogonal Representation Learning Geometric Representation Learning Compositional Representation Learning+1

Feb 24, 2026cs.CL

Disentangling Geometry, Performance, and Training in Language Models

Geometric properties of Transformer weights, particularly the unembedding matrix, have been widely useful in language model interpretability research. Yet, their utility for estimating downstream performance remains unclear. In this work, we systematically investigate the relationship between model performance and the unembedding matrix geometry, particularly its effective rank. Our experiments, involving a suite of 108 OLMo-style language models trained under controlled variation, reveal several key findings. While the best-performing models often exhibit a high effective rank, this trend is not universal across tasks and training setups. Contrary to prior work, we find that low effective rank does not cause late-stage performance degradation in small models, but instead co-occurs with it; we find adversarial cases where low-rank models do not exhibit saturation. Moreover, we show that effective rank is strongly influenced by pre-training hyperparameters, such as batch size and weight decay, which in-turn affect the model's performance. Lastly, extending our analysis to other geometric metrics and final-layer representation, we find that these metrics are largely aligned, but none can reliably predict downstream performance. Overall, our findings suggest that the model's geometry, as captured by existing metrics, primarily reflects training choices rather than performance.

Atharva Kulkarni, Jacob Mitchell Springer, Arjun Subramonian +1

University of Southern California · 2Carnegie Mellon Univer · University of California, Los Angeles.

LLM Evaluation Neural Representation Geometry

Feb 16, 2026cs.LG

Symmetry in language statistics shapes the geometry of model representations

The internal representations learned by language models consistently exhibit striking geometric structure: calendar months organize into a circle, historical years form a smooth one-dimensional manifold, and cities' latitudes and longitudes can be decoded using a linear probe. To explain this neural code, we first show that language statistics exhibit translation symmetry (for example, the frequency with which any two months co-occur in text depends only on the time interval between them). We prove that this symmetry governs these geometric structures in high-dimensional word embedding models, and we analytically derive the manifold geometry of word representations. These predictions empirically match large text embedding models and large language models. Moreover, the representational geometry persists at moderate embedding dimension even when the relevant statistics are perturbed (e.g., by removing all sentences in which two months co-occur). We prove that this robustness emerges naturally when the co-occurrence statistics are controlled by an underlying latent variable. Our results indicate that these representational manifolds originate in the statistical symmetries of natural language.

Dhruva Karkada, Daniel J. Korchinski, Andres Nava +2

1UC Berkeley · 2EPFL · 3Johns Hopkins +1

Representation Learning Neural Representation Geometry Word Embeddings Representation Geometry in Language Models

Feb 16, 2026cs.LG

Revisiting the Platonic Representation Hypothesis: An Aristotelian View

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.

Fabian Gröger, Shuo Wen, Maria Brbić

1EPFL · University of Basel · 3HSLU

Representational Similarity Analysis Neural Representation Geometry Platonic Representation Hypothesis

Feb 5, 2026cs.LG

Multi-Way Representation Alignment

The Platonic Representation Hypothesis suggests that independently trained neural networks converge to increasingly similar latent spaces. However, current strategies for mapping these representations are inherently pairwise, scaling quadratically with the number of models and failing to yield a consistent global reference. In this paper, we study the alignment of

M \ge 3

models. We first adapt Generalized Procrustes Analysis (GPA) to construct a shared orthogonal universe that preserves the internal geometry essential for tasks like model stitching. We then show that strict isometric alignment is suboptimal for retrieval, where agreement-maximizing methods like Canonical Correlation Analysis (CCA) typically prevail. To bridge this gap, we finally propose Geometry-Corrected Procrustes Alignment (GCPA), which establishes a robust GPA-based universe followed by a post-hoc correction for directional mismatch. Extensive experiments demonstrate that GCPA consistently improves any-to-any retrieval while retaining a practical shared reference space.

Akshit Achara, Tatiana Gaintseva, Mateo Mahaut +5

King’s College London, London, United Kingdom · Queen Mary University of London, London, United Kingdom · Universitat Pompeu Fabra, Barcelona, Spain +5

Neural Representation Geometry Latent Space Alignment

Jan 27, 2026cs.LG

The Geometric Mechanics of Contrastive Representation Learning: Alignment Potentials, Entropic Dispersion, and Cross-modal Divergence

While InfoNCE underlies modern contrastive learning, its geometric mechanisms remain under-characterized beyond the canonical alignment--uniformity decomposition. We develop a measure-theoretic framework in which representation measures evolve on a fixed embedding manifold. In the large-batch limit, we prove value and gradient consistency, linking the stochastic objective to explicit deterministic energy landscapes and revealing a geometric bifurcation between unimodal and symmetric multimodal regimes. In the unimodal case, the intrinsic energy is strictly convex and admits a unique Gibbs equilibrium, showing that entropy acts as a tie-breaker within the aligned basin. In the multimodal case, the intrinsic geometry becomes cross-coupled and contains a persistent negative symmetric divergence term: each modality's marginal reshapes the effective landscape of the other, allowing strong pairwise alignment to coexist with a persistent modality gap. Controlled synthetic experiments and analyses of pretrained CLIP representations support these predictions. Overall, our results shift the analytical lens from pointwise discrimination to population geometry, showing that pairwise alignment alone is insufficient to control cross-modal marginal structure.

Yichao Cai, Zhen Zhang, Yuhang Liu +1

Australian Institute for Machine Learning, Adelaide University · Responsible AI Research Centre

Cross-Modal Learning Contrastive Learning Neural Representation Geometry Cross-Modal Alignment Multimodal Contrastive Learning

Jan 14, 2026cs.LG

Geometric Stability: The Missing Axis of Representations

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.

Prashant C. Raju

Representational Similarity Analysis Neural Representation Geometry Visual Representation Learning Geometric Representation Learning

Jan 5, 2026cs.LG

Output Embedding Centering for Stable LLM Pretraining

Pretraining of large language models is not only expensive but also prone to certain training instabilities. A specific instability that often occurs at the end of training is output logit divergence. The most widely used mitigation strategies, z-loss and logit soft-capping, merely address the symptoms rather than the underlying cause of the problem. In this paper, we analyze the instability from the perspective of the output embeddings' geometry and identify anisotropic embeddings as its source. Based on this, we propose output embedding centering (OEC) as a new mitigation strategy, and demonstrate that it suppresses output logit divergence. OEC can be implemented in two different ways: as a deterministic operation called

μ

-centering, or a regularization method called

μ

-loss. Our experiments show that both variants outperform z-loss in terms of training stability, while being on par with logit soft-capping. This holds true both in the presence and the absence of weight tying. As a secondary result, we find that

μ

-loss is significantly less sensitive to regularization hyperparameter tuning than z-loss.

Felix Stollenwerk, Anna Lokrantz, Niclas Hertzberg

AI Sweden

Language Model Pretraining Neural Representation Geometry

Dec 23, 2025cs.CL

Probing Spectrum-Like Organization of States of Mind in Transformer Representation Spaces

We investigate whether graded states of mind form spectrum-like structure in transformer representation spaces. To do so, we construct a dataset of 636 short natural-language sentences annotated with both a continuous score from

-5

to

5

and one of seven ordered tiers, ranging from collapsed or scarcity-driven expressions to more coherent, reflective, and integrative ones. We evaluate five frozen transformer representations: four sentence-embedding models and one decoder-only residual-stream representation. Across all representations, simple probes reliably recover both the continuous score and the discrete tier labels, and permutation tests show that performance significantly exceeds shuffled-label baselines. Additional analyses reveal a consistent geometric pattern: UMAP projections show low-to-high organization, confusion matrices concentrate errors between neighboring tiers, and directional ablation identifies a prominent score-aligned component. These results suggest that transformer representations contain statistically significant, spectrum-like organization aligned with the annotated state-of-mind structure. The annotations are used only as an operational framework for representation analysis, not as a clinical or diagnostic measure.

Sophie Zhao

School of Computer Science Georgia Institute of Technology

Transformer Interpretability Neural Representation Geometry

Nov 20, 2025cs.CV

TetraSDF: Analytic Isosurface Extraction with Multi-resolution Tetrahedral Grid

Extracting an explicit surface that exactly matches the zero-level set of a neural signed distance function (SDF) remains challenging. Sampling-based isosurfacing methods such as Marching Cubes introduce discretization error. In contrast, continuous piecewise affine (CPWA) analytic approaches typically require plain ReLU MLPs, which limits the ability to learn high-frequency SDFs in practice. We present TetraSDF, an analytic isosurface extraction framework for SDFs that retains the expressiveness of grid-based encoders while enabling exact zero-level set extraction, by representing the SDF with a ReLU MLP composed with a multi-resolution tetrahedral positional encoder. Our positional encoder's barycentric interpolation preserves a global CPWA structure, allowing us to track ReLU linear regions within an encoder-induced polyhedral complex. We further introduce a fixed analytic input preconditioner derived from the encoder's metric to reduce directional bias, thereby stabilizing training. Across multiple benchmarks, TetraSDF matches or surpasses existing grid-based encoders in SDF reconstruction accuracy, while faithfully recovering the network's zero-level set as a triangle mesh.

Seonghun Oh, Youngjung Uh, Jin-Hwa Kim

Yonsei University, Republic of Korea · NAVER AI Lab, Republic of Korea · Seoul National University, Republic of Korea

3D Mesh Generation Neural Representation Geometry Signed Distance Functions 3D Reconstruction