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 11, 2026cs.LG

The Geometric Wall: Manifold Structure Predicts Layerwise Sparse Autoencoder Scaling Laws

Sparse autoencoders (SAEs) operationalise the linear representation hypothesis: they reconstruct model activations as sparse linear combinations of interpretable dictionary atoms, on the implicit assumption that activation space is well approximated by a globally linear structure. Their reconstruction error varies sharply across layers in ways that existing scaling laws, fitted at single layers, do not explain. We argue that this variation is the empirical trace of a geometric mismatch: where the activation manifold is curved and its intrinsic dimension varies across layers, no sparse linear dictionary can match it uniformly, and the SAE's width-sparsity scaling becomes a layer-dependent function of manifold structure rather than a single universal law. We conduct the first cross-layer SAE scaling study, fitting and regressing on 844 residual-stream Gemma Scope SAE checkpoints across 68 layers of Gemma 2 2B and 9B. Stage 1 fits a per-layer scaling-law surface; Stage 2 regresses the fitted parameters and the derived per-layer width exponents on four layerwise geometric summaries. We find that manifold geometry predicts the per-layer width exponent in both models, and that the same regression coefficients learnt on one model predict the other model's per-layer exponents under cross-model transfer, indicating a transferable geometric law. At the showcase layers where richer width grids permit identification of the asymptotic floor, we find that the fitted floor tracks the layerwise geometric ordering: higher curvature and intrinsic dimension correspond to higher floor, consistent with the irreducible second-order residual that any sparse linear approximation of a curved manifold must leave behind. SAEs thus encounter not a finite-resource ceiling but a geometry-dependent wall, set by the manifold they are trying to reconstruct.

Eslam Zaher, Maciej Trzaskowski, Quan Nguyen +1

ARC Training Centre for Information Resilience (CIRES) · School of Mathematics and Physics, University of Queensland · Institute for Molecular Bioscience, University of Queensland +2

Neural Representation Geometry Sparse Autoencoders

May 11, 2026cs.LG

Flag Varieties: A Geometric Framework for Deep Network Alignment

Alignment, the tendency of adjacent weight matrices in deep networks to develop compatible subspace orientations, underlies gradient flow, Neural Collapse, and representation similarity across architectures. Despite extensive empirical documentation, these phenomena have resisted unified theoretical treatment: existing explanations are post-hoc, each fitted to a specific observation with whatever mathematics is at hand. We reverse this direction by deriving the mathematical structure that layerwise alignment inherently demands. Using geometric invariant theory, we prove that alignment geometry has a canonical closed, polystable stratum given by a flag variety, and that subspace intersection dimension is its unique reparameterization-invariant observable, establishing that subspace metrics are not empirical conventions but mathematical necessities. This unified framework yields two dynamical consequences: ridge regularization drives subspace alignment at an exponential rate set by weight decay, whereas nonlinear activations induce a commutator obstruction to exact basis alignment, generically present in nonlinear networks and absent in linear ones. Together these give a geometric explanation of the Level-2/3 hierarchy in Neural Collapse from first principles rather than post-hoc analysis. The commutator magnitude and head subspace overlap further serve as weight-space windows into internal alignment structure, requiring no forward passes. Experiments on multilayer perceptrons, residual networks, and pretrained language models support the proposed diagnostics and delineate their scope.

Jingchuan Xiao, Xinyi Sui, Cihan Ruan

Department of Mathematics and Computer Studies, Mary Immaculate College, Ireland · Department of Computer Science and Engineering, Santa Clara University, USA

Neural Representation Geometry Neural Collapse Neural Network Training Dynamics

May 10, 2026cs.LG

SEMASIA: A Large-Scale Dataset of Semantically Structured Latent Representations

Latent representations learned by neural networks often exhibit semantic structure, where concept similarity is reflected by geometric proximity in embedding space. However, comparing such spaces across models remains difficult: changes in architecture, pretraining data, objective, or random seed can yield embeddings with similar content but incompatible geometry. This latent space alignment problem is central to interpretability, transfer and multimodal learning, federated systems, and semantic communication; however, progress remains limited by the lack of large-scale, model-diverse, and metadata-rich benchmarks. To address this gap, we introduce SEMASIA, a large-scale collection of latent representations extracted from approximately 1,700 pretrained vision models across eight standard image-classification benchmarks. SEMASIA pairs embeddings with structured metadata describing architectures, training regimes, pretraining sources, and model scale. We demonstrate three applications of the resource. First, we analyze the conceptual organization of individual latent spaces, showing consistent prototype-like clustering and hierarchical semantic neighborhoods across models and datasets. Second, we benchmark supervised alignment mappings between latent spaces using reconstruction error and downstream task performance. Third, we perform a large-scale regression analysis of how pretraining-data complexity, specialization, transfer learning, augmentation, and model scale relate to geometric and probing properties of embeddings. By coupling representational scale with standardized metadata, SEMASIA provides a reproducible foundation for studying latent geometry, evaluating alignment methods, and developing next-generation heterogeneous and interoperable AI systems.

Mario Edoardo Pandolfo, Enrico Grimaldi, Lorenzo Marinucci +4

Dept. Computer, Control and Management Engineering, Sapienza University of Rome, Rome, Italy. · National Inter-University Consortium for Telecommunications (CNIT), Parma, Italy. · Dept. of Statistical Sciences, Sapienza University of Rome, Rome, Italy. +1

Neural Representation Geometry Visual Representation Learning Latent Space Alignment

May 10, 2026cs.CV

HyNeuralMap: Hyperbolic Mapping of Visual Semantics to Neural Hierarchies

Understanding the intricate mappings between visual stimuli and neural responses is a fundamental challenge in cognitive neuroscience. While current approaches predominantly align images and functional magnetic resonance imaging (fMRI) responses in Euclidean space, this geometry often struggles to preserve fine-grained semantic relationships and latent hierarchical structures across visual and neural modalities. To overcome this, we propose HyNeuralMap, a framework that employ hyperbolic Lorentz model to map visual semantics into a shared, cross-subject neural hierarchy. By leveraging the negative curvature of hyperbolic space as an inductive bias, the proposed framework better captures hierarchical semantic organization and cross-subject neural similarities. Specifically, visual and neural embeddings are jointly optimized through hyperbolic geometric alignment, where geodesic distances preserve semantic proximity and hierarchical relationships more effectively than Euclidean embeddings. Experiments demonstrate that HyNeuralMap consistently outperforms state-of-the-art Euclidean baselines in both multi-label semantic prediction and cross-modal retrieval tasks. This confirms hyperbolic geometry's superiority for cross-modal semantic alignment and hierarchical modeling, providing a new avenue for vision-neural representation learning.

Zihan Ma, Tian Xia, Kexin Wang +3

School of Electronic Information (School of Artificial Intelligence), the Xi’an Key Laboratory of Radiomics and Intelligent Perception, Northwest University, Xi’an 710127, China

Neural Representation Geometry Cross-Modal Alignment Hyperbolic Representation Learning

May 10, 2026cs.AI

The Wittgensteinian Representation Hypothesis: Is Language the Attractor of Multimodal Convergence?

Understanding why independently trained neural networks from different modalities converge toward shared representations, and where this convergence leads, remains an open question in representation learning. All existing evidence relies on symmetric similarity measures, which can detect convergence but are structurally blind to its direction. We introduce directional convergence analysis using cycle-kNN, an asymmetric alignment measure, applied across dozens of independently trained unimodal models spanning point clouds, vision, and language. We uncover a consistent directional asymmetry: non-language modalities move toward the neighborhood structure of language significantly more than the reverse, and this pattern holds across all model families and scales--yet is entirely invisible to symmetric measures. Mechanistic analysis traces the directionality to feature density asymmetry, whereby language representations occupy the most compact regions of representational space. The Information Bottleneck framework provides a principled interpretation: optimization under compression drives representations toward discrete, compositional structures characteristic of language. We formalize this as the Wittgensteinian Representation Hypothesis: the semantic structure of language is the asymptotic attractor of multimodal representation convergence.

Zhaoyang Zhang, Run Shao, Dongyue Wu +4

1Central South University · 4Ant Group · 2Huazhong University of Science and Technology +1

Neural Representation Geometry Cross-Modal Alignment Multimodal Understanding

May 9, 2026cs.CV

Investigating Anisotropy in Visual Grounding under Controlled Counterfactual Perturbations

Visual Grounding benchmarks assume that the object described by a referring expression is always present in the image, and grounding models are therefore rarely evaluated under semantically mismatched captions. In such cases, models frequently exhibit approximation behavior, producing a plausible bounding box that satisfies only part of the expression (\eg, preserving the original object while ignoring modified contextual cues). Because mismatched captions represent realistic edge cases, this behavior compromises reliability and raises concerns from an explainability perspective. Identifying its underlying causes is thus essential for improving model faithfulness and interpretability. Adopting a mechanistic interpretability viewpoint, this work examines whether embedding anisotropy contributes to counterfactual failures. A similarity-controlled counterfactual caption generation protocol is introduced to systematically perturb object or contextual components within predefined embedding similarity intervals, enabling a fine-grained analysis of grounding behavior as a function of alignment. Experiments on two Transformer-based models with markedly different embedding geometries (BERT-based TransVG and CLIP-based SwimVG) reveal no meaningful correlation between cosine similarity and approximation. These findings suggest that anisotropy alone does not account for counterfactual errors, and that robustness requires investigating finer-grained geometric properties of the embedding space.

Gabriele Lombardo, Luigi Maiorana, Liliana Lo Presti +1

University of Palermo Palermo, Italy

Neural Representation Geometry VLM Interpretability Visually Grounded Reasoning Visual Grounding

May 9, 2026cs.LG

A Geometric Perspective on Next-Token Prediction in Large Language Models: Three Emerging Phases

We investigate the geometry of predictive information across the layers of large language models (LLMs). We repurpose representation lenses-learned affine maps trained to predict the next token from intermediate residual streams-as geometric diagnostic tools. Rather than asking what the model predicts at each layer, we ask where predictive information resides and how it evolves across depth. We define at each layer a predictive readout subspace as the dominant k-dimensional singular subspace of such a map on the d-dimensional residual stream (where k is a resolution parameter), and track its trajectory on the Grassmann manifold as a similarity profile across layers. The profile is well described by unimodal distributions exhibiting a rise, near-plateau, and descent; varying k from 1% to 50% of d traces a Pareto frontier between visibility and energy retention, yet the same structure emerges at all scales. Across eight models from two families (Qwen2.5 and OLMo2, 1B-32B), we identify three geometric phases. Updates are approximately orthogonal to the residual stream throughout; what distinguishes the phases is their effect on the effective rank, which expands, stabilizes, and concentrates. In the first, Seeding Multiplexing, feed-forward memories and attention layers seed a candidate set in superposition in family-specific proportions, with the final token rising as leading candidate from 20% to 35% of positions across this phase. In the second, Hoisting Overriding, updates override existing subspaces to concentrate the candidate distribution without expanding the rank. In the third, Focal Convergence, high-energy low-rank updates write the winner into a form aligned with the unembedding direction. Phases 1 and 3 grow slowly with model depth, while Phase 2 expands linearly. The additional capacity of deeper LLMs is largely absorbed by candidate disambiguation.

Gianfranco Lombardo, Giuseppe Trimigno, Stefano Cagnoni

Department of Engineering and Architecture University of Parma Parma, IT 43125

Neural Representation Geometry LLM Interpretability Mechanistic Interpretability Next-Token Prediction Sequence Modeling

May 9, 2026cs.LG

When Is Rank-1 Steering Cheap? Geometry, Granularity, and Budgeted Search

Activation steering offers a lightweight way to control LLMs without retraining, but its effectiveness varies sharply across concepts. Prior work often reads this variability as evidence that many concepts are not captured by a single steering direction. We argue instead that much of it reflects search difficulty: a useful rank-1 intervention often exists, but finding it can be expensive. We formalize rank-1 steering as a budget-constrained optimization over intervention layer and coefficient. Across concepts and model families, prompt-boundary directional alignment predicts where effective interventions occur, enabling geometry-guided search that reaches high utility with substantially fewer evaluations, reducing the trials needed to recover 95% of best-found utility by 39.8% on average across three model families. To explain why some concepts remain expensive even under better search, we introduce concept granularity, a measure of directional heterogeneity across contrastive contexts. Granularity distinguishes concepts whose difference vectors share a stable global direction from those where prompts agree locally within each input but the utility-maximizing direction rotates systematically across inputs. Higher granularity is associated with slower convergence and lower best-found performance (Pearson

r{=}0.44

with trials-to-95%,

r{=}{-}0.46

with best-found utility, both

p<0.001

). We present GRACE, a Granularity- and Representation-Aware Concept Engineering framework that uses activation geometry to diagnose the dominant source of steering difficulty, select the appropriate remedy, and allocate optimization effort efficiently. Our results shift the frame from "when does rank-1 fail?" to "when is rank-1 cheap and stable?", turning activation geometry from a descriptive tool into an actionable prior for LLM control.

John T. Robertson, Jianing Zhu, Haris Vikalo +1

The University of Texas at Austin

Neural Representation Geometry Activation Steering

May 9, 2026cs.LG

Bilinear autoencoders find interpretable manifolds

Sparse autoencoders have become a standard tool for uncovering interpretable latent representations in neural networks. Yet salient concepts often span manifolds that current linear methods cannot capture without post hoc analysis. This paper uses quadratic latents to close this gap: we implement these with bilinear autoencoders, which decompose activations into low-rank quadratic forms, compose linearly in weight space, and admit input-independent geometric analysis. This qualitative difference in what concepts quadratic latents can detect challenges the standard linear representation hypothesis. Our experiments and visualisations show that multi-dimensional geometries are highly prevalent and that composite latents capture them well, systematically improving reconstruction error in language models. Furthermore, we show that autoencoders with varying geometric priors recover the same input subspace despite their dictionary entries being distinct. Practically, these models serve as an unsupervised tool for manifold discovery, which we demonstrate through an interactive online visualizer for Qwen 3.5. This is a step toward nonlinear but mathematically tractable latent representations whose composition is expressive and interpretable by design.

Thomas Dooms, Ward Gauderis, Geraint Wiggins +1

University of Antwerp · Vrije Universiteit Brussel · University of Antwerp, sqIRL

Neural Representation Geometry Autoencoders LLM Interpretability Manifold Learning

May 8, 2026cs.LG

The Propagation Field: A Geometric Substrate Theory of Deep Learning

Modern deep learning treats neural networks primarily as endpoint functions from inputs to outputs. Inspired by the shift from force to geometry in physics, we ask whether a network should instead be understood through the geometry of its internal propagation. We define a neural propagation field as the collection of hidden-state trajectories and local Jacobian operators across depth. Endpoint losses constrain only the boundary behavior of this field, leaving its interior geometry underdetermined. We show that endpoint-equivalent models can differ by orders of magnitude in trajectory and Jacobian structure, and introduce observable field metrics such as path sensitivity, solver consistency, and trajectory/Jacobian retention. In controlled teacher-flow and PDE systems, endpoint fitting fails to recover the underlying propagation law. In real multi-path tasks, field-aware objectives improve unseen-path generalization, OOD robustness, and calibration when aligned with the observation structure, but can collapse when over-constrained. In continual learning, field-preservation regularization complements replay and distillation: on Split CIFAR-100, DER++ with field preservation improves average accuracy, backward transfer, and field-retention metrics. These results identify propagation-field quality as a measurable and trainable property of neural networks beyond endpoint performance.

Xingrui Gu

1EECS, University of California, Berkeley, Berkeley, CA.

Neural Network Generalization Neural Representation Geometry Jacobian Regularization Neural Network Training Dynamics

May 8, 2026cs.LG

The Geometric Structure of Models Learning Sparse Data

The manifold hypothesis (MH) is often used to explain how machine learning can overcome the curse of dimensionality. However, the MH is only applicable in regimes where the training data provides a sufficiently dense sample of the underlying low-dimensional data manifold, or where such a low-dimensional manifold is conceivably present. We describe the regimes where the MH is not applicable as sparse. In this paper, we demonstrate that models succeed in the sparse regime by exploiting a highly structured local geometry, a property we formalize as normal alignment. We prove that normal-aligned classifiers -- whose input-output Jacobians are rank-one and align perfectly with the training data -- minimize the training objective under norm constraints and achieve maximal local robustness under a non-zero Jacobian constraint. For continuous piecewise-affine deep networks, normal alignment manifests geometrically as centroid alignment within the network's induced power diagram partition and results from the feature-learning regime. Motivated by these theoretical insights, we introduce GrokAlign, a regularization strategy that actively induces normal alignment. We demonstrate that GrokAlign significantly accelerates the training dynamics of deep networks relevant to the grokking phenomenon. Furthermore, we apply the principle of normal alignment to Recursive Feature Machines (RFMs) to introduce Recursive Feature Alignment Machines (RFAMs). We show that RFAMs exhibit greater adversarial robustness compared to RFMs when trained on tabular data.

Thomas Walker, T. Mitchell Roddenberry, Ahmed Imtiaz Humayun +2

Rice University · Rice University† · Now at Google Research +1

Neural Representation Geometry Statistical Learning Theory Data-Efficient Learning

May 8, 2026cs.CV

Is Class Signal Clustered or Routed in Task-Induced Implicit Neural Representation Weight Spaces?

Implicit neural representations (INRs) encode images as neural-network weights, making image classification a problem of weight-space classifiability. A natural geometric hypothesis is that classifier feedback should make image-specific weights cluster by class in the shared-anchor coordinate. We test this hypothesis in the SIREN-based Meta Weight Transformer (MWT) regime, where end-to-end training meta-learns a shared initialization and inner-loop update schedule for fitting image-specific SIRENs. We find that this prediction fails. Exposed weight-space geometry and supervised clustering pressure do not reliably track trained-reader accuracy; clustering can even make local neighborhoods more class-consistent while making the trained reader worse. Crucially, the reader constructs rather than inherits class-aligned geometry: token-flow diagnostics show that class-aligned neighborhoods become strongly predictive of trained-reader accuracy only after late reader interactions, not in the input coordinate. We further identify the native SIREN bias column in the augmented weight token as a low-dimensional, sample-dependent causal readout route for the trained reader; targeted controls rule out generic scalar-column and marginal-distribution artifacts. The diagnosis motivates interventions that strengthen reader routing, add an explicit bias route, or use denser inner-loop fitting; under the lane-specific training conventions used here, route-directed variants often outperform the shared-anchor baseline but interact non-additively. Task-induced INR weights are classifiable not because they form raw geometric clusters, but because their class signal is routed through the reader.

Xinyi Guo, Mingyi He, Haobin Ding +7

South China Normal University · Beijing University of Chemical Technology · Tsinghua University +1

Neural Representation Geometry Implicit Neural Representations Neural Network Interpretability Weight-Space Analysis Image Classification

May 8, 2026cs.CL

Mean-Pooled Cosine Similarity is Not Length-Invariant: Theory and Cross-Domain Evidence for a Length-Invariant Alternative

Mean-pooled cosine similarity is the default metric for comparing neural representations across languages, modalities, and tasks. We establish that this metric is not length-invariant: under the anisotropy that characterizes modern transformer representations, mean-pooled cosine grows monotonically in sequence length, independent of representational content. Empirically, on HumanEvalPack across four code LLMs, the length ratio alone explains

R^2 = 0.52

--

0.75

of cross-language "Python proximity," while AST depth and shared-token fraction add less than 3% of explained variance beyond length. Substituting Centered Kernel Alignment (CKA) reduces explained variance by 83% and reverses the sign of the length coefficient (

β_{\mathrm{len}}: +0.86 \to -0.37

). The same pattern holds in Mistral-7B on parallel WMT pairs (

R^2 = 0.23

EN-FR,

R^2 = 0.33

EN-DE for cosine;

R^2 < 0.01

for CKA). In CLIP ViT-B/32, mean-pooling reduces the length effect relative to EOS-pooling (

R^2: 0.21 \to {<}0.01

), as predicted by the theory's dependence on anisotropy. We argue that length-invariant metrics such as CKA should be the default for cross-representation comparisons, and that recent claims of cross-lingual representational convergence built on mean-pooled cosine warrant re-examination.

Sibayan Mitra, Dhruv Kumar

BITS Pilani · Birla Institute of Technology and Science, Pilani, India.

Neural Representation Geometry Representation Geometry in Language Models

May 8, 2026cs.LG

Geometric Asymmetry in MoE Specialization: Functional Decorrelation and Representational Overlap

Mixture-of-Experts (MoE) architectures achieve scalable capacity through sparse routing, yet the geometric structure of expert specialization remains poorly understood. We introduce a unified Jacobian-PCA-Grassmann framework for analyzing MoE layers in both function space and representation space. Across pretrained MoE Transformers (Mistral, Qwen), we find a consistent structural asymmetry: experts exhibit strong functional decorrelation (consistently low, near-zero cross-expert Jacobian alignment) while their routed representations occupy distinct but partially overlapping subspaces. This indicates that functional decorrelation and representation overlap coexist rather than coincide in MoE specialization. Controlled routing experiments further indicate that routing sparsity appears to be a key factor shaping this geometry: top-k routing induces sharper functional separation and larger subspace divergence, whereas fully soft routing yields more entangled expert structure. Together, these results suggest a geometric interpretation in which MoE layers may be viewed as implementing locally decorrelated operators over overlapping submanifolds on a shared representation manifold, and provide a general diagnostic framework for studying conditional computation in modern Transformer architectures.

Feilong Liu

Transformer Interpretability Neural Representation Geometry Sparse Mixture-of-Experts Mixture-of-Experts Models

May 7, 2026cs.LG

A Generalized Singular Value Theory for Neural Networks

Building on the abstract Generalized Singular Value Decomposition (GSVD) theory of Brown et al. [2025], we prove that most modern neural architectures admit a generalized SVD representation in which they are left-invertible before a final linear layer, with no change in input-output behavior. Furthermore, the left-invertible nonlinear portion of the input-output behavior can be made to be \emph{norm preserving}, meaning that perturbations in the left-invertible ``embedding'' (the activations prior to the final linear layer in this representation) correspond proportionally to changes in the input, i.e., distance in feature space can be calibrated directly to distance in input space. We provide a data-driven algorithm for estimating this representation from trained models and propose a model architecture that naturally facilitates the decomposition. We then provide a proof-of-concept that the learned representation can be used to identify adversarial perturbations to model inputs, and develop the theory necessary for future applications to areas such as model bias and invertibility.

Brian Charles Brown, Robert Bridges, David Grimsman +2

AI Sweden Gothenburg, Sweden

Neural Representation Geometry Geometric Representation Learning Singular Value Decomposition

May 7, 2026cs.LG

Towards Metric-Faithful Neural Graph Matching

Graph Edit Distance (GED) is a fundamental, albeit NP-hard, metric for structural graph similarity. Recent neural graph matching architectures approximate GED by first encoding graphs with a Graph Neural Network (GNN) and then applying either a graph-level regression head or a matching-based alignment module. Despite substantial architectural progress, the role of encoder geometry in neural GED estimation remains poorly understood. In this paper, we develop a theoretical framework that connects encoder geometry to GED estimation quality for two broad classes of neural GED estimators: graph similarity predictors and alignment-based methods. On fixed graph collections, where the doubly-stochastic metric

d_{\mathrm{DS}}

is comparable to GED, we show that graph-level bi-Lipschitz encoders yield controlled GED surrogates and improved ranking stability; for matching-based estimators, node-level bi-Lipschitz geometry propagates to encoder-induced alignment costs and the resulting optimized alignment objective. We instantiate this perspective using FSW-GNN, a bi-Lipschitz WL-equivalent encoder, as a drop-in replacement in representative neural GED architectures. Across representative baselines and benchmark datasets, the resulting geometry-aware variants significantly improve GED prediction and ranking metrics. A faithfulness case study of untrained encoders, together with ablations and transfer experiments, supports the view that these gains arise from improved representation geometry, positioning encoder geometry as a useful design principle for neural graph matching.

Jyotirmaya Shivottam, Subhankar Mishra

1National Institute of Science Education and Research (NISER), Bhubaneswar, India · 2Homi Bhabha National Institute, Mumbai, India

Graph Neural Networks Neural Representation Geometry Graph Matching

May 7, 2026cs.LG

Invariant Features in Language Models: Geometric Characterization and Model Attribution

Language models exhibit strong robustness to paraphrasing, suggesting that semantic information may be encoded through stable internal representations, yet the structure and origin of such invariance remain unclear. We propose a local geometric framework in which semantically equivalent inputs occupy structured regions in latent space, with paraphrastic variation along nuisance directions and semantic identity preserved in invariant subspaces. Building on this view, we make three contributions: (1) a geometric characterization of invariant latent features, (2) a contrastive subspace discovery method that separates semantic-changing from semantic-preserving variation, and (3) an application of invariant representations to zero-shot model attribution. Across models and layers, empirical results support these contributions. Invariant structure emerges in specific depth regions, semantic displacement lies largely outside the nuisance subspace, and representation-level interventions indicate a causal role of invariant components in model outputs. Invariant representations also capture model-specific geometric patterns, enabling accurate attribution. These findings suggest that semantic invariance can be viewed as a local geometric property of latent representations, offering a principled perspective on how language models organize meaning.

Agnibh Dasgupta, Abdullah Tanvir, Xin Zhong

Department of Computer Science University of Nebraska Omaha, Omaha, NE, USA

Neural Representation Geometry Invariant Representation Learning Representation Geometry in Language Models

May 7, 2026cs.LG

Topological Signatures of Grokking

We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase in both the maximum and total persistence of first homology (

H_1

). Persistence diagrams reveal the emergence of a dominant long-lived topological feature together with increasingly structured secondary features, reflecting the underlying cyclic structure of the task. Compared to existing spectral and geometric diagnostics -- specifically, Fourier analysis and local intrinsic dimension -- persistent homology provides a unified geometric and topological characterization of representation learning, capturing both local and global multi-scale structure. Ablations across data regimes and control settings show that these topological transitions are tied to generalization rather than memorization. Our results suggest that persistent homology offers a principled and interpretable framework for analyzing how neural networks internalize latent structure during training.

Yifan Tang, Qiquan Wang, Inés García-Redondo +1

Imperial College London · Queen Mary University of London · University of Fribourg

Neural Network Generalization Neural Representation Geometry Grokking

May 7, 2026cs.LG

Normalized Architectures are Natively 4-Bit

Training large language models at 4-bit precision is critical for efficiency. We show that nGPT, an architecture that constrains weights and hidden representations to the unit hypersphere, is inherently more robust to low-precision arithmetic. This removes the need for interventions-such as applying random Hadamard transforms and performing per-tensor scaling calculations-to preserve model quality, and it enables stable end-to-end NVFP4 training. We validate this approach on both a 1.2B dense model and hybrid (Mamba-Transformer) MoE models of up to 3B/30B parameters. We trace this robustness to the dot product: while quantization noise remains largely uncorrelated in both standard and normalized architectures, the signal behaves differently. In nGPT, the hypersphere constraint enhances weak positive correlations among the element-wise products, leading to a constructive accumulation of the signal across the hidden dimension while the noise continues to average out. This yields a higher effective signal-to-noise ratio and a flatter loss landscape, with the effect strengthening as the hidden dimension grows, suggesting increasing advantages at scale. A reference implementation is available at https://github.com/anonymous452026/ngpt-nvfp4

Maxim Fishman, Brian Chmiel, Ron Banner +2

∧NVIDIA · ◦Department of Electrical and Computer Engineering - Technion, Haifa, Israel

LLM Quantization Neural Representation Geometry 4-Bit Quantization Neural Network Robustness LLM Training

May 6, 2026cs.LG

Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior

Neural representations carry rich geometric structure; but does that structure causally shape behavior? To address this question, we intervene along paths through activation space defined by different geometries, and measure the behavioral trajectories they induce. In particular, we test whether interventions that respect the geometry of activation space will yield behaviors close to those the model exhibits naturally. Concretely, we first fit an activation manifold

M_h

to representations and a behavior manifold

M_y

to output probability distributions. We then test the link

M_h \leftrightarrow M_y

via interventions: we find that steering along

M_h

, which we term manifold steering, yields behavioral trajectories that follow

M_y

, while linear steering -- which assumes a Euclidean geometry -- cuts through off-manifold regions and hence produces unnatural outputs. Moreover, optimizing interventions in activation space to produce paths along

M_y

recovers activation trajectories that trace the curvature of

M_h

. We demonstrate this bidirectional relationship between the geometry of representation and behavior across tasks and modalities. In language models, we use reasoning tasks with cyclic and sequential geometries as well as in-context learning tasks with more complex graph geometries. In a video world model, we use a task with geometry corresponding to physical dynamics. Overall, our work shows that geometry in neural representation is not merely incidental, but is in fact the proper object for enabling principled control via intervention on internals. This recasts the core problem of steering from finding the right direction to finding the right geometry.

Daniel Wurgaft, Can Rager, Matthew Kowal +13

Stanford University · University College London · Northeastern University +2

Neural Representation Geometry Manifold Learning Causal Interventions in Language Models Activation Steering

May 6, 2026cs.LG

Training-Time Batch Normalization Reshapes Local Partition Geometry in Piecewise-Affine Networks

Batch normalization (BN) is central to modern deep networks, but its effect on the realized function during training remains less understood than its optimization benefits. We study training-time BN in continuous piecewise-affine (CPA) networks through the geometry of switching hyperplanes and the induced affine-region partition. Conditioned on a mini-batch, we show that BN defines for each neuron a reference hyperplane through the batch centroid, and that breakpoint-switching hyperplanes are parallel translates whose offsets are expressed in batch-standardized coordinates and are independent of the raw bias. This yields an exact criterion for when a switching hyperplane intersects a local

\ell_\infty

window and motivates a local region-density functional based on exact affine-region counts. Under explicit sufficient conditions, we show that BN increases expected local partition refinement in ReLU and more general piecewise-affine networks, and that this mechanism transfers locally through depth inside parent affine regions where the upstream representation map is an affine embedding. These results provide a function-level geometric account of training-time BN as a batch-conditional recentering mechanism near the data.

Xuan Qi, Yi Wei, Fanqi Yu +3

AI for Good Istituto Italiano di Tecnologia, Genoa, Italy Department of Computer Science University of Verona, Verona, Italy

Neural Representation Geometry

May 6, 2026cs.LG

A geometric relation of the error introduced by sampling a language model's output distribution to its internal state

GPT-style language models are sensitive to single-token changes at generation points where the predicted probability distribution is spread across multiple tokens. Viewing this sensitivity as a geometric property, we derive an

\mathfrak{so}(n)

-valued 1-form that depends only on the geometry of the token embeddings. Despite this purely geometric origin, we show that its curvature is semantically meaningful: On chess reasoning tasks, the curvature couples to the world model of an off-the-shelf instruction-tuned model, with transformations clustering by board region and respecting piece importance. Our findings suggest that token space geometry directly reflects how models internally represent problems.

Albert F. Modenbach

Department of Mathematics, King’s College London, United Kingdom.

Neural Representation Geometry LLM Interpretability Representation Geometry in Language Models

May 5, 2026cs.LG

Task Vector Geometry Underlies Dual Modes of Task Inference in Transformers

Transformers are effective at inferring the latent task from context via two inference modes: recognizing a task seen during training, and adapting to a novel one. Recent interpretability studies have identified from middle-layer representations task-specific directions, or task vectors, that steer model behavior. However, a lack of rigorous foundations hinders connecting internal representations to external model behavior: existing work fails to explain how task-vector geometry is shaped by the training distribution, and what geometry enables out-of-distribution (OOD) generalization. In this paper, we study these questions in a controlled synthetic setting by training small transformers from scratch on latent-task sequence distributions, which allows a principled mathematical characterization. We show that two inference modes can coexist within a single model. In-distribution behavior is governed by Bayesian task retrieval, implemented internally through convex combinations of learned task vectors. OOD behavior, by contrast, arises through extrapolative task learning, whose representations occupy a subspace nearly orthogonal to the task-vector subspace. Taken together, our results suggest that task-vector geometry, training distributions, and generalization behaviors are closely related.

Hao Yan, Haolin Yang, Yiqiao Zhong

University of Wisconsin–Madison · University of Chicago

Transformer Interpretability Neural Representation Geometry Task Vectors OOD Generalization In-Context Learning

May 4, 2026cs.LG

Beyond Activation Alignment: The Geometry of Neural Sensitivity

Activation-alignment measures such as Representational Similarity Analysis (RSA), Canonical Correlation Analysis (CCA), and Centered Kernel Alignment (CKA) are widely used to compare biological and artificial neural representations. Recent theoretical work interprets many of these methods as assessing agreement between optimal linear readouts over broad families of global tasks. However, agreement at the level of global readouts does not determine how a system uses local stimulus evidence. Specifically, representations may align in activation space yet differ in their sensitivity to small perturbations. To address this challenge, we introduce a complementary framework based on local decodable information, which focuses on a representation's ability, under noise, to discriminate small perturbations within a specified stimulus-coordinate subspace. Building on Fisher information and local representation geometry, we summarize each representation using the expected projected pullback/Fisher metric over that subspace. This formulation induces a second-moment family of local discrimination tasks, for which the resulting operator provides a minimal, complete dataset-level summary of expected discriminability. We compare these regularized signatures using a log-spectral distance on the manifold of symmetric positive definite (SPD) matrices, yielding the Spectral Riemannian Alignment Score (S-RAS) and a uniform multiplicative certificate over the corresponding family of lifted task values. Empirically, this framework enables the recovery of corresponding layers across independently trained artificial neural networks, supports transferable class-conditional probes, reveals controlled dissociations between standard and robust training, and uncovers stimulus-coordinate family effects across mouse visual cortex using the Allen Brain Observatory static gratings dataset.

Amirhossein Yavari, Farnaz Zamani Esfahlani

Stephenson School of Biomedical Engineering, University of Oklahoma · Data Science and Analytics Institute, University of Oklahoma

Neural Representation Geometry Neural Decoding Visual Neuroscience Fisher Information Representation Alignment

May 4, 2026cs.LG

Geometric and Spectral Alignment for Deep Neural Network II

This paper develops the angular and static-channel component of Geometric and Spectral Alignment for residual Jacobian chains. Starting from Cartan-coordinate rigidity and fitted effective-rank windows, we study how dominant singular subspaces are transported across adjacent layers and how the resulting finite matrices can be displayed in physical channel coordinates. The main results are deterministic, margin-verified results. We bound the error between full interface transport and its dominant-window truncation, add fitted-tail errors so that empirical spectra can be certified against the Gibbs--Cartan tail model, and distinguish source-mode incidence from fully physical input-output channel incidence. Given row groups and active supports, the Physical Alignment Matrix decomposes orthogonally as core plus overlap plus noise. Active-column gaps, pairwise overlap margins, and noise bounds combine into a static certificate radius under which the full transport and the truncated transport induce the same active supports, pairwise incidence graph, SRS sets, hub columns, and core/overlap/noise masks. The finer SC/SA/ST labels of the Invariant Channel Mapping require additional row-energy and profile-correlation margins, stated as explicit perturbation tests. The empirical section reports the matrices and block-energy heatmaps that measure these certificate quantities across CNNs, language models, and vision/diffusion backbones. The figures are interpreted as finite-dimensional measurements; complete membership in the Physical GSA certificate domain requires checking the numerical margin protocol stated in Section 10.

Ziran Liu, Wei Wang, Jinhao Wang +5

Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai 200433, China · Futurewei Technologies, Inc., San Jose, CA 95131 · Dept. of Computer Science and Engineering, Santa Clara University, Santa Clara, CA 95050 +1

Neural Representation Geometry

May 2, 2026cs.LG

Concepts Whisper While Syntax Shouts: Spectral Anti-Concentration and the Dual Geometry of Transformer Representations

We test whether the causal inner product of \citet{park2024linear} -- defined by the unembedding covariance

Σ

-- enables cross-lingual concept transport. Across 17 models and 4 language pairs, a matched-spectrum randomization test finds that Whitened Causal Alignment is indistinguishable from spectral regularization alone (

p = 0.95

). However, this failure reveals a broader phenomenon: anti-concentration is observed in residual-stream difference-of-means vectors across five architecture families (

p < 10^{-33}

) and supported by SAE features (e.g.,

p = 4.5 \times 10^{-19}

) and linear probes on Gemma and Llama. We discover a \emph{dual geometry}: activation-space concept directions anti-concentrate in the spectral tail, while static unembedding-row contrasts \emph{concentrate} in high-variance directions (

p < 10^{-4}

). Split-injection causal interventions support the functional basis on Gemma and Llama (Cohen's

d

up to

1.80

), and POS-tag probing across 8 models shows syntax preferentially encodes in the high-variance subspace in 6 of 8 architectures (

p < 0.013

), with the Qwen~2.5 family showing a significant reversal consistent with architecture-specific spectral structure. These results suggest transformers may rotate semantic content into spectrally quiet regions during contextualized processing, encoding concepts where they can be manipulated with reduced grammatical disruption.

Pratyush Acharya, Nuraj Rimal, Habish Dhakal

Transformer Interpretability Representation Learning Neural Representation Geometry Cross-Lingual Representation Alignment Neural Encoding Cross-Lingual Transfer

May 1, 2026cs.LG

Spectral Graph Sparsification Preserves Representation Geometry in Graph Neural Networks

Spectral graph sparsification is a classical tool for reducing graph complexity while preserving Laplacian quadratic forms. In graph neural networks (GNNs), sparsification is often used to accelerate computation while maintaining predictive performance. In this work, we study a complementary representation-level question: does sparsification preserve the geometry of learned embeddings? For polynomial-filter GNNs, we prove that any

ε

-spectral sparsifier induces

O(ε)

perturbations in polynomial graph filters, multilayer hidden representations, and their Gram matrices. These guarantees imply stability of squared pairwise distances, class means, and covariance structure in embedding space. We further establish finite-time training stability: under smoothness and boundedness assumptions, gradient descent on dense and sparsified graphs produces weight trajectories whose separation grows at most proportionally to the sparsification distortion. Empirically, effective-resistance sparsification validates the predicted perturbation chain on synthetic graphs and preserves hidden representation geometry on real datasets. In our experiments, the gram matrix and training dynamics show low divergence even under substantial sparsification, consistent with the predicted stability under spectral sparsification. Hidden Gram preservation strongly predicts neighborhood preservation and class-centroid stability across FashionMNIST, Cora, and Paul15. Together, these results show that spectral sparsification preserves not only graph operators, but also the representation geometry that supports downstream use of GNN embeddings for interpretability.

Sanjukta Krishnagopal

Department of Computer Science, University of California Santa Barbara

Graph Neural Networks Neural Representation Geometry Graph Sparsification Spectral Methods

May 1, 2026cs.LG

Diffusion Operator Geometry of Feedforward Representations

Neural networks transform data through learned representations whose geometry affects separation, contraction, and generalization. Recent work studies this geometry using discrete curvature on neighborhood graphs, suggesting Ricci-flow-like behavior across layers. We develop a smooth operator-theoretic alternative for feedforward representation snapshots. Each feature cloud induces a Gaussian-kernel diffusion Markov operator, and transport, spectral, label-boundary, and local-scale observables are derived from this single object via Bakry-Emery

Γ

-calculus. In a balanced Gaussian class-conditional snapshot model with shared covariance, the population operator has closed-form class affinities, leakage, and coarse spectra, all controlled by pairwise regularized Mahalanobis separations

c_\varepsilon^{(a,b)}

. We also prove that the resulting operator observables vary smoothly under feature perturbations, while hard neighborhood-graph diagnostics can change discontinuously. Synthetic experiments validate the closed-form Gaussian bridge, while learned MNIST experiments show that the same operator observables track training, width, and perturbation stability. Together, these results give a stable operator-geometric framework for analyzing feedforward representation geometry.

Kanishka Reddy

Department of Applied Mathematics University of Washington Seattle, WA

Neural Representation Geometry Feedforward Neural Networks

May 1, 2026cs.CL

Controlled Paraphrase Geometry in Sentence Embedding Space: Local Manifold Modeling and Latent Probing

The paper studies the local geometry of embedding clouds induced by \emph{controlled local classes of semantically close sentences}. The central question is how controlled paraphrase-like semantic variation is organized in sentence embedding space and whether this local structure can be explicitly modeled by low-degree fitted carriers. We introduce a local geometric modeling scheme based on affine, quadratic, and cubic fitted models. We also use a surface-based latent probing procedure that constructs synthetic latent points in a reduced local PCA space with respect to the fitted carrier. The procedure is intended as an offline method for representation-space analysis, local manifold modeling, and geometry-aware latent probing. Generated latent points are evaluated using criteria that measure consistency with the fitted surface, preservation of neighborhood structure, agreement with the empirical distribution, stability of Hessian-based second-order shape descriptors, and stability of fitted-model coefficients. Experiments on controlled sets of semantically close sentences show that nonlinear local models describe embedding clouds more accurately than affine models. Surface-based generation provides strong fitted-geometry fidelity, including surface consistency, Hessian-based shape consistency, and coefficient consistency. Downstream experiments show that geometric validity of synthetic latent points does not automatically translate into improved classification performance. The results support explicit local geometric modeling of sentence embedding space and highlight the need to distinguish geometric validity from discriminative utility. As a resource contribution, we introduce \textbf{CoPaGE-300K}, a controlled template-based dataset of semantically close sentence variants with slot-level annotations and precomputed sentence embeddings.

Leonid Bedratyuk

Representation Learning Neural Representation Geometry Sentence Embeddings Manifold Learning

May 1, 2026cs.LG

Aitchison Embeddings for Learning Compositional Graph Representations

Representation learning is central to graph machine learning, powering tasks such as link prediction and node classification. However, most graph embeddings are hard to interpret, offering limited insight into how learned features relate to graph structure. Many networks naturally admit a role-mixture view, where nodes are best described as mixtures over latent archetypal factors. Motivated by this structure, we propose a compositional graph embedding framework grounded in Aitchison geometry, the canonical geometry for comparing mixtures. Nodes are represented as simplex-valued compositions and embedded via isometric log-ratio (ILR) coordinates, which preserve Aitchison distances while enabling unconstrained optimization in Euclidean space. This yields intrinsically interpretable embeddings whose geometry reflects relative trade-offs among archetypes and supports coherent behavior under component restriction; we consider both fixed and learnable ILR bases. Across node classification and link prediction, our method achieves competitive performance with strong baselines while providing explainability by construction rather than post-hoc. Finally, subcompositional coherence enables principled component restriction: removing and renormalizing subsets preserves a well-defined geometry, which we exploit via subcompositional dimensionality removal to probe how archetype groups influence representations and predictions.

Nikolaos Nakis, Chrysoula Kosma, Panagiotis Promponas +2

Human Nature Lab, Yale University, New Haven, USA · Universit´e Paris Saclay, Universit´e Paris Cit´e, ENS Paris Saclay, CNRS, SSA, INSERM, Centre Borelli, Gif-sur-Yvette, France · Yale Institute for Network Science, Yale University, New Haven, USA +2

Neural Representation Geometry Graph Representation Learning Compositional Representation Learning