Neural Representation Geometry

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Neural representation geometry asks how learned features are organized by distances, directions, neighborhoods, manifolds, and topology, and how this structure supports prediction, memory, transfer, or behavior. Researchers compare models and brains using methods such as CKA, Procrustes alignment, and relative representations, and characterize spaces with covariance spectra, intrinsic-dimension estimates, persistent homology, and Riemannian geometry; sparse autoencoders and interventions probe feature structure and causal roles. Work spans vision and language models, contrastive embeddings, world models, and neural recordings, with evaluations including retrieval, out-of-distribution transfer, compositional generalization, and brain alignment. A central tension is that measured geometry can change under function-preserving coordinate transformations, while similarity or geometric patterns alone do not establish shared computation, causal importance, or better performance; robust, task-aware measures remain an open need.

Latest papers

259 papers37 papers in the last four weeks, up 118% on the four weeks before. 0.2% of all new papers.

Feb 24, 2026

Geometric Analysis of Speech Representation Spaces: Topological Disentanglement and Confound Detection

Speech-based clinical tools are increasingly deployed in multilingual settings, yet whether pathological speech markers remain geometrically separable from accent variation remains unclear. Systems may misclassify healthy non-native speakers or miss pathology in multilingual patients. We propose a four-metric clustering framework to evaluate geometric disentanglement of emotional, linguistic, and pathological speech features across six corpora and eight dataset combinations. A consistent hierarchy emerges: emotional features form the tightest clusters (Silhouette 0.250), followed by pathological (0.141) and linguistic (0.077). Confound analysis shows pathological-linguistic overlap remains below 0.21, which is above the permutation null but bounded for clinical deployment. Trustworthiness analysis confirms embedding fidelity and robustness of the geometric conclusions. Our framework provides actionable guidelines for equitable and reliable speech health systems across diverse populations.

Bipasha Kashyap, Pubudu N. Pathirana

Networked Sensing & Biomedical Engineering (NSBE) Research lab, School of Engineering, Deakin University, Australia

Representation Disentanglement Clinical Speech Processing Neural Representation Geometry Demographic Bias in ASR

Feb 24, 2026

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 Language Model Performance Prediction

Feb 22, 2026

Partial Soft-Matching Distance for Neural Representational Comparison with Partial Unit Correspondence

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

Chaitanya Kapoor, Alex H. Williams, Meenakshi Khosla

Center for Neural Science, New York University, New York City, NY, 10003 · Center for Computational Neuroscience, Flatiron Institute, New York City, NY, 10003

Representational Similarity Analysis Neural Representation Geometry Latent Space Alignment

Feb 21, 2026

Toward Manifest Relationality in Transformers via Symmetry Reduction

Transformer models contain substantial internal redundancy arising from coordinate-dependent representations and continuous symmetries, in model space and in head space, respectively. While recent approaches address this by explicitly breaking symmetry, we propose a complementary framework based on symmetry reduction. We reformulate representations, attention mechanisms, and optimization dynamics in terms of invariant relational quantities, eliminating redundant degrees of freedom by construction. This perspective yields architectures that operate directly on relational structures, providing a principled geometric framework for reducing parameter redundancy and analyzing optimization.

J. François, L. Ravera

Mons University (UMONS), 20 Place du Parc, 7000 Mons, Belgium. · Grupo de Investigación en Física Teórica, Universidad Católica De La Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile.

Transformer Transformer Attention Invariant Representation Learning Neural Representation Geometry

Feb 17, 2026

Relative Geometry of Neural Forecasters: Linking Accuracy and Alignment in Learned Latent Geometry

Neural networks can accurately forecast complex dynamical systems, yet how they internally represent underlying latent geometry remains poorly understood. We study neural forecasters through the lens of representational alignment, introducing anchor-based, geometry-agnostic relative embeddings that remove rotational and scaling ambiguities in latent spaces. Applying this framework across seven canonical dynamical systems - ranging from periodic to chaotic - we reveal reproducible family-level structure: multilayer perceptrons align with other MLPs, recurrent networks with RNNs, while transformers and echo-state networks achieve strong forecasts despite weaker alignment. Alignment generally correlates with forecasting accuracy, yet high accuracy can coexist with low alignment. Relative geometry thus provides a simple, reproducible foundation for comparing how model families internalize and represent dynamical structure.

Deniz Kucukahmetler, Maximilian Jean Hemmann, Julian Mosig von Aehrenfeld +4

Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany · School of Embedded Composite Artificial Intelligence (SECAI), Dresden/Leipzig, Germany · Leipzig University, Leipzig, Germany +2

Neural Representation Geometry Latent Space Alignment Dynamical Systems

Feb 16, 2026

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 Latent Space Geometry

Feb 16, 2026

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

Permutation Testing Representational Similarity Analysis Neural Representation Geometry Platonic Representation Hypothesis

Feb 11, 2026

Attention Geometry

Attention normalizes learned scores into a Markov operator. In the continuum, the local sector becomes diffusion on a learned Riemannian manifold, but attention natively bonds distant points, creating nonlocal structure such that no tensor geometry can describe. The Bakry-Emery calculus, built directly from the transition operator, extends geometry to these potentially stratified spaces, giving a coordinate-free account of what vision and language transformers learn.

Julio Candanedo

Transformer Attention Neural Representation Geometry

Feb 5, 2026

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, 2026

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, 2026

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, 2026

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, 2025

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, 2025

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

Oct 28, 2025

A Unified Geometric Space for Topological Alignment Between Transformer-Based Models and Human Brain Networks

Whether artificial neural networks organize information comparably to the human brain remains unclear. Prior brain--AI alignment studies are constrained by specific inputs and tasks, limiting cross-modal comparison. Here we introduce a brain--model topological alignment space, mapping Transformer attention topology onto human intrinsic connectivity networks (ICNs) to enable task-free, modality-agnostic comparison. Analyzing 151 Transformer-based models with 62,480 attention head graphs, we observe a continuous arc-shaped distribution reflecting varying alignment. Models optimized for global semantics aligned with higher-order ICNs, while local-detail models aligned with sensory ICNs. Non-intuitive findings include reduced alignment in DINOv2 compared to its predecessors and a counterintuitive scaling inversion in distilled DeiT models, while fine-tuning and instruction tuning had limited effect. Alignment scores showed no significant correlation with ImageNet accuracy (r = 0.266, p = 0.156). This work offers a quantitative framework for comparing the organizational principles of artificial and biological systems.

Silin Chen, Yuzhong Chen, Caiwei Wang +9

The Clinical Hospital of Chengdu Brain Science Institute, MOE-K Lab for NeuroInformation, Brain-Apparatus Communication Institute, School of Life Science and Technology, University of Electronic Science and Technology of China, Chengdu. 611731, China. · School of Automation, Northwestern Polytechnical University, Xi’an 710072, China. · Department of Biomedical Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA. +1

Neural Representation Geometry Transformer Attention Brain Network Analysis

Oct 27, 2025

Manifold Approximation leads to Robust Kernel Alignment

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

Mohammad Tariqul Islam, Du Liu, Deblina Sarkar

MIT

Kernel Methods Representational Similarity Analysis Neural Representation Geometry

Jul 25, 2025

DINO-SLAM: DINO-informed RGB-D SLAM for Neural Implicit and Explicit Representations

This paper presents DINO-SLAM, a DINO-informed design strategy to enhance implicit (Neural Radiance Field -- NeRF) and explicit representations (Gaussian Splatting -- GS) in SLAM systems through the more comprehensive semantics understanding enabled by DINO. This latter alone, however, lacks proper 3D geometry understanding, allowing only for marginal improvements. Therefore, we rely on a Scene Geometry Encoder (SGE) to enrich DINO features into geometry-aware DINO features (geoDINO), to better understand those geometric relationships that vanilla DINO features fail to capture. Building upon it, we propose two foundational paradigms for NeRF and GS SLAM systems integrating geoDINO features. Compared to state-of-the-art methods, our DINO-informed pipelines achieve superior performance on the Replica, ScanNet, and TUM datasets.

Ziren Gong, Xiaohan Li, Fabio Tosi +4

University of Bologna, Italy · Faculty of Dentistry, The University of Hong Kong, China · Rawmantic AI, China +1

Simultaneous Localization and Mapping Neural Representation Geometry 3DGS SLAM Visual SLAM 3D Scene Representation

Apr 23, 2024

The Geometry of the Set of Equivalent Linear Neural Networks

We characterize the geometry of the set of all weight vectors for which a fully-connected linear neural network computes the same linear transformation

W

. Said another way, the set of all sequences of matrices (of predetermined sizes) whose product is

W

. This set of weight vectors is an algebraic variety in the Euclidean weight space, called the fiber of

W

under the matrix multiplication map. Sometimes this variety is a manifold, but usually not. The rank stratification of a fiber is a natural partition of the fiber into smooth manifolds of various dimensions called strata. We derive the dimensions of these strata and characterize how they are connected to each other. They satisfy the frontier condition: if a stratum intersects the closure of another stratum, then the former stratum is a subset of the closure of the latter stratum. This subset relationship can be expressed as a partial order with a single minimal element, a stratum that is included in the closure of every other stratum. Hence the strata can be arranged in a hierarchy represented by a directed acyclic graph that represents that partial order; we give an algorithm that computes that graph. A key result describes the relationship between this partial order and the ranks of certain matrices in the network. We derive the tangent and normal subspaces of the strata, and we construct elegant bases for those subspaces. Each stratum represents a different pattern of information flow through the network, expressed as a barcode, borrowed from the study of persistent homology. The topology of a stratum depends solely on its barcode. So does its geometry, up to a linear transformation in weight space. Connections among the strata are best understood through "barcode moves," which can be expressed as simple modifications of the barcodes.

Jonathan Richard Shewchuk, Sagnik Bhattacharya

Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, California 94720

Deep Linear Networks Weight-Space Analysis Neural Representation Geometry

Sep 30, 2022

Disentanglement with Biological Constraints: A Theory of Functional Cell Types

Neurons in the brain are often finely tuned for specific task variables. Moreover, such disentangled representations are highly sought after in machine learning. Here we mathematically prove that simple biological constraints on neurons, namely nonnegativity and energy efficiency in both activity and weights, promote such sought after disentangled representations by enforcing neurons to become selective for single factors of task variation. We demonstrate these constraints lead to disentanglement in a variety of tasks and architectures, including variational autoencoders. We also use this theory to explain why the brain partitions its cells into distinct cell types such as grid and object-vector cells, and also explain when the brain instead entangles representations in response to entangled task factors. Overall, this work provides a mathematical understanding of why single neurons in the brain often represent single human-interpretable factors, and steps towards an understanding task structure shapes the structure of brain representation.

James C. R. Whittington, Will Dorrell, Surya Ganguli +1

Stanford & Oxford · UCL · Stanford +1

Disentangled Representation Learning Neural Representation Geometry