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.

Apr 15, 2026

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 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 12, 2026

NSFL: A Post-Training Neuro-Symbolic Fuzzy Logic Framework for Boolean Operators in Neural Embeddings

Standard dense retrievers lack a native calculus for multi-atom logical constraints. We introduce Neuro-Symbolic Fuzzy Logic (NSFL), a framework that adapts formal t-norms and t-conorms to neural embedding spaces without requiring retraining. NSFL operates as a first-order hybrid calculus: it anchors logical operations on isolated zero-order similarity scores while actively steering representations using Neuro-Symbolic Deltas (NS-Delta) -- the first-order marginal differences derived from contextual fusion. This preserves pure atomic meaning while capturing domain reliance, preventing the representation collapse and manifold escape endemic to traditional geometric baselines. For scalable real-time retrieval, Spherical Query Optimization (SQO) leverages Riemannian optimization to project these fuzzy formulas into manifold-stable query vectors. Validated across six distinct encoder configurations and two modalities (including zero-shot and SOTA fine-tuned models), NSFL yields mAP improvements up to +81%. Notably, NSFL provides an additive 20% average and up to 47% boost even when applied to encoders explicitly fine-tuned for logical reasoning. By establishing a training-free, order-aware calculus for high-dimensional spaces, this framework lays the foundation for future dynamic scaling and learned manifold logic.

Vladi Vexler, Ofer Idan, Gil Lederman +1

Huawei Tel-Aviv Research Center

Information Retrieval Neuro-Symbolic Reasoning Dense Retrieval Fuzzy Logic Neural Representation Geometry

Apr 8, 2026

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 5, 2026

The Geometric Alignment Tax: Tokenization vs. Continuous Geometry in Scientific Foundation Models

Foundation models for biology and physics optimize predictive accuracy, but their internal representations systematically fail to preserve the continuous geometry of the systems they model. We identify the root cause: the Geometric Alignment Tax, an intrinsic cost of forcing continuous manifolds through discrete categorical bottlenecks. Controlled ablations on synthetic dynamical systems demonstrate that replacing cross-entropy with a continuous head on an identical encoder reduces geometric distortion by up to 8.5x, while learned codebooks exhibit a non-monotonic double bind where finer quantization worsens geometry despite improving reconstruction. Under continuous objectives, three architectures differ by 1.3x; under discrete tokenization, they diverge by 3,000x. Evaluating 14 biological foundation models with rate-distortion theory and MINE, we identify three failure regimes: Local-Global Decoupling, Representational Compression, and Geometric Vacuity. A controlled experiment confirms that Evo 2's reverse-complement robustness on real DNA reflects conserved sequence composition, not learned symmetry. No model achieves simultaneously low distortion, high mutual information, and global coherence.

Prashant C. Raju

Neural Representation Geometry Geometric Representation Learning

Apr 5, 2026

Geometric Limits of Knowledge Distillation: A Minimum-Width Theorem via Superposition Theory

Knowledge distillation compresses large teachers into smaller students, but performance saturates at a loss floor that persists across training methods and objectives. We argue this floor is geometric: neural networks represent far more features than dimensions through superposition, and a student of width

d_S

can encode at most

d_S \cdot g(α)

features, where

g(α) = 1/((1-α)\ln\frac{1}{1-α})

is a sparsity-dependent capacity function. Features beyond this budget are permanently lost, yielding an importance-weighted loss floor. We validate on a toy model (48 configurations, median accuracy >93%) and on Pythia-410M, where sparse autoencoders measure

F \approx 28{,}700

features at

α\approx 0.992

(critical width

d_S^* \approx 1{,}065

). Distillation into five student widths confirms the predicted monotonic floor ordering. The observed floor decomposes into a geometric component and a width-independent architectural baseline (

R^2 = 0.993

). Linear probing shows coarse concepts survive even 88% feature loss, revealing the floor arises from aggregate loss of fine-grained features in the importance distribution's long tail. Our results connect representation geometry to distillation limits and provide a practical tool for predicting distillation performance from SAE measurements alone.

Nilesh Sarkar, Dawar Jyoti Deka

Erd˝os AI Labs

Knowledge Distillation Neural Representation Geometry Representation Learning

Apr 3, 2026

Retina gap junctions support the robust perception by warping neural representational geometries along the visual hierarchy

Deep Neural Networks (DNNs) are vulnerable to elaborately designed adversarial noise, although they have achieved extraordinary success in many tasks. Compared with DNNs, the human visual system is highly robust. However, it is unclear how the human visual system defends against adversarial attacks, especially the role of the early visual system and its influence on the brain manifold. Due to retina gap junctions being crucial for the denoising function in the early visual system, we combine a retina gap junction-based filter, G-filter, with DNN as an abstract human visual system model called the biological hybrid model. We adopt this model to study the defense performance of retina gap junctions and their impact on the brain manifold. Compared with other defense methods, the biological hybrid model is more robust and can be further improved by introducing noise during training. Next, we analyze the manifold and its decision boundary of the biological hybrid model from a geometry perspective. The results show that the biological hybrid model has a unique 2D decision boundary with high nonlinearity and a lower curvature of the decision boundary of the manifold compared to other defense methods. The transforming manifold may account for the high robustness of the biological hybrid model. Finally, to dissect G-filter and clarify its internal mechanism, we borrow the Neural Ordinary Differential Equation (ODE) concept and rewrite G-filter into an equivalent recurrent neural network. The results show that the decision boundary of the model's manifold will gradually change with time and eventually reach a steady state, which is modulated by gap junction conductance, revealing the influence of retina gap junctions on the brain manifold is a gradually evolving process.

Yang Yue, Shenjian Zhang, Yonghong Tian +2

School of Computer Science, Peking University, China

Adversarial Robustness Neural Representation Geometry Visual Neuroscience

Apr 3, 2026

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 Latent Space Geometry

Mar 31, 2026

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 31, 2026

The Geometry of Polynomial Group Convolutional Neural Networks

We study polynomial group convolutional neural networks (PGCNNs) for an arbitrary finite group

G

. In particular, we introduce a new mathematical framework for PGCNNs using the language of graded group algebras. This framework yields two natural parameterizations of the architecture, based on Hadamard and Kronecker products, related by a linear map. We compute the dimension of the associated neuromanifold, verifying that it depends only on the number of layers and the size of the group. Furthermore, we show the general fiber of both parameterizations is trivial up to the regular group action and rescaling. Hence both parametrization maps are identifiable.

Yacoub Hendi, Daniel Persson, Magdalena Larfors

Uppsala University · Chalmers University of Technology

Convolutional Neural Networks Polynomial Neural Networks Equivariant Neural Networks Parameter Identifiability Neural Representation Geometry

Mar 30, 2026

Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds

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

N Alex Cayco-Gajic, Arthur Pellegrino

Gatsby Unit, University College London

Neural Representation Geometry Representational Similarity Analysis Neural Network Interpretability

Mar 28, 2026

The Geometric Cost of Normalization: Affine Bounds on the Bayesian Complexity of Neural Networks

LayerNorm and RMSNorm impose fundamentally different geometric constraints on their outputs - and this difference has a precise, quantifiable consequence for model complexity. We prove that LayerNorm's mean-centering step, by confining data to a linear hyperplane (through the origin), reduces the Local Learning Coefficient (LLC) of the subsequent weight matrix by exactly

m/2

(where

m

is its output dimension); RMSNorm's projection onto a sphere preserves the LLC entirely. This reduction is structurally guaranteed before any training begins, determined by data manifold geometry alone. The underlying condition is a geometric threshold: for the codimension-one manifolds we study, the LLC drop is binary -- any non-zero curvature, regardless of sign or magnitude, is sufficient to preserve the LLC, while only affinely flat manifolds cause the drop. At finite sample sizes this threshold acquires a smooth crossover whose width depends on how much of the data distribution actually experiences the curvature, not merely on whether curvature exists somewhere. We verify both predictions experimentally with controlled single-layer scaling experiments using the wrLLC framework. We further show that Softmax simplex data introduces a "smuggled bias" that activates the same

m/2

LLC drop when paired with an explicit downstream bias, proved via the affine symmetry extension of the main theorem and confirmed empirically.

Sungbae Chun

Department of Mathematics, Universität Bonn

Layer Normalization Neural Representation Geometry

Mar 28, 2026

On the Spectral Geometry of Cross-Modal Representations: A Functional Map Diagnostic for Multimodal Alignment

We study cross-modal alignment between independently pretrained vision (DINOv2) and language (all-MiniLM-L6-v2) encoders using the functional map framework from computational geometry, which represents correspondence between representation manifolds as a compact linear operator between graph Laplacian eigenbases. While the framework underperforms Procrustes alignment and relative representations for cross-modal retrieval across all supervision budgets, it reveals a structural property of multimodal representations. We find that the Laplacian eigenvalue spectra of the two encoders are quantitatively similar (normalized spectral distance 0.043), indicating that independently trained models develop manifolds of comparable intrinsic complexity. However, the functional map exhibits near-zero diagonal dominance (mean below 0.05) and large orthogonality error (70.15), showing that the eigenvector bases are effectively unaligned. We term this decoupling the spectral complexity--orientation gap: models converge in how much structure they capture but not in how they organize it. This gap defines a boundary condition for spectral alignment methods and motivates three diagnostic quantities : diagonal dominance, orthogonality deviation, and Laplacian commutativity error for characterizing cross-modal representation compatibility.

Krisanu Sarkar

Indian Institute of Technology Bombay, Mumbai, India

Functional Maps Cross-Modal Alignment Vision-Language Alignment Neural Representation Geometry

Mar 27, 2026

The Geometry of Forgetting

Why do we forget? Why do we remember things that never happened? The conventional answer points to biological hardware. We propose a different one: geometry. Here we show that high-dimensional embedding spaces, subjected to noise, interference, and temporal degradation, reproduce quantitative signatures of human memory with no phenomenon-specific engineering. Power-law forgetting (

b = 0.460 \pm 0.183

, human

b \approx 0.5

) arises from interference among competing memories, not from decay. The identical decay function without competitors yields

b \approx 0.009

, fifty times smaller. Time alone does not produce forgetting in this system. Competition does. Production embedding models (nominally 384--1{,}024 dimensions) concentrate their variance in only

{\sim}16

effective dimensions, placing them deep in the interference-vulnerable regime. False memories require no engineering at all: cosine similarity on unmodified pre-trained embeddings reproduces the Deese--Roediger--McDermott false alarm rate (

0.583

versus human

{\sim}0.55

) with zero parameter tuning and no boundary conditions. We did not build a false memory system. We found one already present in the raw geometry of semantic space. These results suggest that core memory phenomena are not bugs of biological implementation but features of any system that organizes information by meaning and retrieves it by proximity.

Sambartha Ray Barman, Andrey Starenky, Sophia Bodnar +2

Sentra, 235 2nd Street, San Francisco, CA 94105, USA · Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Associative Memory Cognitive Modeling Neural Representation Geometry

Mar 24, 2026

The Coordinate System Problem in Persistent Structural Memory for Neural Architectures

We introduce the Dual-View Pheromone Pathway Network (DPPN), an architecture that routes sparse attention through a persistent pheromone field over latent slot transitions, and use it to discover two independent requirements for persistent structural memory in neural networks. Through five progressively refined experiments using up to 10 seeds per condition across 5 model variants and 4 transfer targets, we identify a core principle: persistent memory requires a stable coordinate system, and any coordinate system learned jointly with the model is inherently unstable. We characterize three obstacles -- pheromone saturation, surface-structure entanglement, and coordinate incompatibility -- and show that neither contrastive updates, multi-source distillation, Hungarian alignment, nor semantic decomposition resolves the instability when embeddings are learned from scratch. Fixed random Fourier features provide extrinsic coordinates that are stable, structure-blind, and informative, but coordinate stability alone is insufficient: routing-bias pheromone does not transfer (10 seeds, p>0.05). DPPN outperforms transformer and random sparse baselines for within-task learning (AULC 0.700 vs 0.680 vs 0.670). Replacing routing bias with learning-rate modulation eliminates negative transfer: warm pheromone as a learning-rate prior achieves +0.003 on same-family tasks (17 seeds, p<0.05) while never reducing performance. A structure completion function over extrinsic coordinates produces +0.006 same-family bonus beyond regularization, showing the catch-22 between stability and informativeness is partially permeable to learned functions. The contribution is two independent requirements for persistent structural memory: (a) coordinate stability and (b) graceful transfer mechanism.

Abhinaba Basu

National Institute of Electronics and Information Technology (NIELIT)

Memory-Augmented Neural Networks Transfer Learning Neural Representation Geometry

Mar 21, 2026

Neural collapse in the orthoplex regime

When training a neural network for classification, the feature vectors of the training set are known to collapse to the vertices of a regular simplex, provided the dimension

d

of the feature space and the number

n

of classes satisfies

n\leq d+1

. This phenomenon is known as neural collapse. For other applications like language models, one instead takes

n\gg d

. Here, the neural collapse phenomenon still occurs, but with different emergent geometric figures. We characterize these geometric figures in the orthoplex regime where

d+2\leq n\leq 2d

. The techniques in our analysis primarily involve Radon's theorem and convexity.

James Alcala, Rayna Andreeva, Vladimir A. Kobzar +4

Department of Mathematics, University of Southern California, Los Angeles, CA · School of Informatics, University of Edinburgh, Edinburgh, Scotland · Department of Mathematics, The Ohio State University, Columbus, OH +3

Neural Collapse Neural Representation Geometry

Mar 17, 2026

Predictive Statistics Shape Emergent World Representations of Grid Walkers

Next-token predictors often appear to develop internal representations of the latent world and its rules. The probabilistic nature of these models suggests a deep connection between the structure of the world and the geometry of probability distributions. In order to understand this link more precisely, we use a minimal stochastic process as a controlled setting: constrained random walks on a two-dimensional lattice that must reach a fixed endpoint after a predetermined number of steps. Optimal prediction of this process solely depends on a sufficient vector determined by the walker's position relative to the target and the remaining time horizon; in other words, the probability distributions are parametrized by the world's grid geometry. We train decoder-only transformers and recurrent networks on prefixes sampled from the exact distribution of these walks and compare their hidden activations to sufficient statistics of prediction, by measuring alignment and linear readability across layers. We find that the transformer's computation factors into two stages: the first attention block extracts the sufficient statistic from the input, and later layers transform it into the next-step predictive geometry. Across constraint variants the post-attention representation is universal: a shared world-state of the lattice that can be read directly as a world model, traced to the predictive geometry of the data. Later layers then specialize it to each variant's next-step distribution. Recurrent networks reach the same Bayes-optimal loss but do not isolate this world-state as a separate stage, showing that the world-model geometry also depends on architecture. Although demonstrated in a toy system, the results suggest that the geometry of the predictive distribution is a useful lens on how neural networks internalize the structure of their data.

Sasha Brenner, Thomas R. Knösche, Nico Scherf

Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany · Leipzig University, Germany · ScaDS.AI, Dresden/Leipzig, Germany

Predictive Representation Learning World Model Learning Neural Representation Geometry Transformer Interpretability

Mar 12, 2026

NBAvatar: Neural Billboards Avatars with Realistic Hand-Face Interaction

We present NBAvatar - a method for realistic rendering of head avatars handling non-rigid deformations caused by hand-face interaction. To this end, we introduce a novel hybrid implicit-explicit representation for animated avatars by combining the training of explicit oriented planar primitives with implicit neural rendering. Such a combination of representations in the end-to-end pipeline enables NBAvatar to handle temporally and pose-consistent geometry, along with fine-grained appearance details provided by the neural rendering technique. To enable joint optimization of different representations we propose a geometry-aware training scheme that allows our hybrid representation to surpass existing approaches in terms of novel-view and novel-pose rendering quality. Specifically, NBAvatar achieves up to 53% LPIPS reduction compared to Gaussian-based avatar methods, while also improving PSNR and SSIM, and achieves higher structural similarity compared to the state-of-the-art hand-face interaction method InteractAvatar.

David Svitov, Mahtab Dahaghin, Pietro Morerio +1

Università degli Studi di Genova, Genoa, Italy · Istituto Italiano di Tecnologia (IIT), Genoa, Italy

Human Avatar Animation Neural Rendering Neural Representation Geometry

Mar 12, 2026

Context-dependent manifold learning: A neuromodulated constrained autoencoder approach

Many physical systems exhibit a low-dimensional structure that varies with external parameters: link lengths in a robot, forcing constants in a fluid, or Reynolds numbers in a flow shift the underlying manifold while preserving its intrinsic dimension. Constrained AutoEncoders (cAEs) learn such manifolds through an idempotent encoder-decoder projection, a property that unconstrained autoencoders cannot match and that is essential whenever the model is applied iteratively. However, the standard strategies for making a cAE context-dependent, namely concatenating the context to the input or affinely modulating hidden activations, break the encoder-decoder idempotency, sacrificing the projection guarantee precisely in the setting where it would be most valuable. To restore this guarantee under context variation, we developed the Neuromodulated Constrained Autoencoder (NcAE), which modulates the activation slope and bias of a cAE through a context-driven hyper-network. This paper presents the NcAE, its theoretical foundation, and its empirical validation. We prove that for every context, including contexts unseen at training time, the reconstruction map remains an idempotent projection, the topology of the learned manifold is invariant, and context perturbations induce smooth changes in the manifold. We evaluated our approach on a 16-DoF pendulum with context-dependent coupling and the Lorenz96 system across a bifurcation. The NcAE matched or exceeded the best of six baselines on reconstruction, idempotency, and latent-geometry metrics, while being the only architecture that preserves geometric consistency by construction. The NcAE thereby provides a stable, geometry-preserving coordinate system across families of physical regimes.

Jérôme Adriaens, Gustave Bainier, Guillaume Drion +1

Neuroengineering Lab, Department of Electrical Engineering and Computer Science, University of Liège · University of Liège, Liège, Belgium

Autoencoders Manifold Learning Neural Representation Geometry

Mar 7, 2026

Vision-Guided MPPI for Agile Drone Racing: Navigating Arbitrary Gate Poses via Neural Signed Distance Fields

Autonomous drone racing requires the tight coupling of perception, planning, and control under extreme agility. However, recent approaches typically rely on precomputed spatial reference trajectories or explicit 6-DoF gate pose estimation, rendering them brittle to spatial perturbations, unmodeled track changes, and sensor noise. Conversely, end-to-end learning policies frequently overfit to specific track layouts and struggle with zero-shot generalization. To address these fundamental limitations, we propose a fully onboard, vision guided optimal control framework that enables reference-free agile flight through arbitrarily placed and oriented gates. Central to our approach is Gate-SDF, a novel, implicitly learned neural signed distance field. Gate-SDF directly processes raw, noisy depth images to predict a continuous spatial field that provides both collision repulsion and active geometric guidance toward the valid traversal area. We seamlessly integrate this representation into a sampling-based Model Predictive Path Integral (MPPI) controller. By fully exploiting GPU parallelism, the framework evaluates these continuous spatial constraints across thousands of simulated trajectory rollouts simultaneously in real time. Furthermore, our formulation inherently maintains spatial consistency, ensuring robust navigation even under severe visual occlusion during aggressive maneuvers. Extensive simulations and real-world experiments demonstrate that the proposed system achieves high-speed agile flight and successfully navigates unseen tracks subject to severe unmodeled gate displacements and orientation perturbations. Videos are available at https://zhaofangguo.github.io/vision_guided_mppi/

Fangguo Zhao, Hanbing Zhang, Zhouheng Li +2

College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China

Autonomous Drone Racing Aerial Robotics Vision-Based Robot Control Neural Representation Geometry MPPI Control

Mar 6, 2026

Gauge Freedom and Metric Dependence in Neural Representation Spaces

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

Jericho Cain

Physics Department, Portland Community College, Portland, OR, USA

Neural Representation Geometry Representational Similarity Analysis

Mar 2, 2026

Intrinsic Task Symmetry Drives Generalization in Algorithmic Tasks

Grokking, the sudden transition from memorization to generalization, is characterized by the emergence of low-dimensional representations, yet the mechanism underlying this organization remains elusive. We propose that intrinsic task symmetries primarily drive grokking and shape the geometry of the model's representation space. We identify a consistent three-stage training dynamic underlying grokking: (i) memorization, (ii) symmetry acquisition, and (iii) geometric organization. We show that generalization emerges during the symmetry acquisition phase, after which representations reorganize into a structured, task-aligned geometry. We validate this symmetry-driven account across diverse algorithmic domains, including algebraic, structural, and relational reasoning tasks. Building on these findings, we introduce a symmetry-based diagnostic that anticipates the onset of generalization and propose strategies to accelerate it. Together, our results establish intrinsic symmetry as the key factor enabling neural networks to move beyond memorization and achieve robust algorithmic reasoning.

Hyeonbin Hwang, Yeachan Park

KAIST · Sejong University

Neural Network Generalization Grokking Algorithmic Generalization Neural Representation Geometry Neural Network Memorization Algorithmic Reasoning

Mar 2, 2026

Diagnosing Generalization Failures from Representational Geometry Markers

Generalization, the ability to perform well beyond the training context, is a hallmark of biological and artificial intelligence, yet anticipating unseen failures remains a central challenge. Conventional approaches often take a

bottom-up'' mechanistic route by reverse-engineering interpretable features or circuits to build explanatory models. While insightful, these methods often struggle to provide the high-level, predictive signals for anticipating failure in real-world deployment. Here, we propose using a

top-down'' approach to studying generalization failures inspired by medical biomarkers: identifying system-level measurements that serve as robust indicators of a model's future performance. Rather than mapping out detailed internal mechanisms, we systematically design and test network markers to probe structure, function links, identify prognostic indicators, and validate predictions in real-world settings. In image classification, we find that task-relevant geometric properties of in-distribution (ID) object manifolds consistently forecast poor out-of-distribution (OOD) generalization. In particular, reductions in two geometric measures, effective manifold dimensionality and utility, predict weaker OOD performance across diverse architectures, optimizers, and datasets. We apply this finding to transfer learning with ImageNet-pretrained models. We consistently find that the same geometric patterns predict OOD transfer performance more reliably than ID accuracy. This work demonstrates that representational geometry can expose hidden vulnerabilities, offering more robust guidance for model selection and AI interpretability.

Chi-Ning Chou, Artem Kirsanov, Yao-Yuan Yang +1

Flatiron Institute · Harvard University · Google DeepMind

Neural Network Generalization OOD Generalization Neural Representation Geometry

Mar 1, 2026

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 28, 2026

A Gauge Theory of Superposition: Toward a Sheaf-Theoretic Atlas of Neural Representations

We develop a discrete gauge-theoretic framework for superposition in large language models (LLMs) that replaces the single-global-dictionary premise with a sheaf-theoretic atlas of local semantic charts. Contexts are clustered into a stratified context complex; each chart carries a local feature space and a local information-geometric metric (Fisher/Gauss-Newton) identifying predictively consequential feature interactions. This yields a Fisher-weighted interference energy and three measurable obstructions to global interpretability: (O1) local jamming (active load exceeds Fisher bandwidth), (O2) proxy shearing (mismatch between geometric transport and a fixed correspondence proxy), and (O3) nontrivial holonomy (path-dependent transport around loops). We prove and instantiate four results on a frozen open LLM (Llama-3.2-3B Instruct) using WikiText-103, a C4-derived English web-text subset, and the-stack-smol. (A) After constructive gauge fixing on a spanning tree, each chord residual equals the holonomy of its fundamental cycle, making holonomy computable and gauge-invariant. (B) Shearing lower-bounds a data-dependent transfer mismatch energy, turning

D_{\mathrm{shear}}

into an unavoidable failure bound. (C) We obtain non-vacuous certified jamming/interference bounds with high coverage and zero violations across seeds and hyperparameters. (D) Bootstrap and sample-size experiments show stable estimation of

D_{\mathrm{shear}}

and

D_{\mathrm{hol}}

, with improved concentration on well-conditioned subsystems.

Hossein Javidnia

School of Computing, Dublin City University, Ireland

LLM Interpretability Representation Geometry in Language Models Neural Representation Geometry

Feb 28, 2026

Neural Functional Alignment Space: Brain-Referenced Representation of Artificial Neural Networks

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

Ruiyu Yan, Hanqi Jiang, Yi Pan +4

Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA · School of Computing, University of Georgia, Athens, GA 30602, USA · Department of Biomedical Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA +1

Neural Representation Geometry Cross-Modal Alignment Representational Similarity Analysis

Feb 27, 2026

NeuroHex: A Brain-Inspired Hex Coordinate System to Enable Highly Computationally-Efficient World Models for Continuous Online-Adaptive Learning

NeuroHex is a brain-inspired hexagonal coordinate system designed to support highly efficient world models and reference frames for online adaptive AI systems. Inspired by the hexadirectional firing structure of grid cells in the human brain, NeuroHex adopts a cubic isometric hexagonal coordinate formulation that provides full 60° rotational symmetry and low-cost translation, rotation and distance computation. We develop a mathematical framework that incorporates ring indexing, quantized angular encoding, and a hierarchical library of foundational, simple, and complex geometric shape primitives. These constructs allow low-overhead point-in-shape tests and spatial matching operations that are expensive in Cartesian coordinate systems. To support realistic settings, we also develop a novel tool (OSM2Hex) that can process OpenStreetMap (OSM) data sets and convert them into the NeuroHex coordinate system. The OSM2Hex spatial abstraction processing pipeline can achieve a reduction of 90-99% in geometric complexity while maintaining the relevant spatial structure map for navigation. Our initial results, based on actual city and neighborhood scale data sets, demonstrate that NeuroHex offers a highly efficient substrate for building dynamic world models to enable adaptive spatial reasoning in autonomous energy-efficient AI systems with continuous online-adaptive learning (COAL) capability.

Quinn Jacobson, Joe Luo, Jingfei Xu +4

NeuroAI Computer Architecture Lab (NCAL), Carnegie Mellon University

Spatial Reasoning Neural Representation Geometry World Model Learning

Feb 27, 2026

From Prerequisites to Predictions: Validating a Geometric Hallucination Taxonomy Through Controlled Induction

We test whether a geometric hallucination taxonomy -- classifying failures as center-drift (Type~~1), wrong-well convergence (Type~~2), or coverage gaps (Type3) -- can distinguish hallucination types through controlled induction in GPT-2. Using a two-level statistical design with prompts ( $N = 15$ /group) as the unit of inference, we run each experiment 20 times with different generation seeds to quantify result stability. In static embeddings, Type3 norm separation is robust (significant in 18/20 runs, Holm-corrected in 14/20, median

r = +0.61

). In contextual hidden states, the Type~~3 norm effect direction is stable (19/20 runs) but underpowered at $N = 15$ (significant in 4/20, median $r = -0.28$ ). Types~~1 and2 do not separate in either space ( ${\leq}\,3/20$ runs). Token-level tests inflate significance by 4--16 $\times$ through pseudoreplication -- a finding replicated across all 20 runs. The results establish coverage-gap hallucinations as the most geometrically distinctive failure mode, carried by magnitude rather than direction, and confirm the Type1/2 non-separation as genuine at 124M parameters.

Matic Korun

Independent Researcher Ljubljana, Slovenia

Hallucination in Language Models Neural Representation Geometry

Feb 27, 2026

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 27, 2026

Leveraging Geometric Prior Uncertainty and Complementary Constraints for High-Fidelity Neural Indoor Surface Reconstruction

Neural implicit surface reconstruction with signed distance function has made significant progress, but recovering fine details such as thin structures and complex geometries remains challenging due to unreliable or noisy geometric priors. Existing approaches rely on implicit uncertainty that arises during optimization to filter these priors, which is indirect and inefficient, and masking supervision in high-uncertainty regions further leads to under-constrained optimization. To address these issues, we propose GPU-SDF, a neural implicit framework for indoor surface reconstruction that leverages geometric prior uncertainty and complementary constraints. We introduce a self-supervised module that explicitly estimates prior uncertainty without auxiliary networks. Based on this estimation, we design an uncertainty-guided loss that modulates prior influence rather than discarding it, thereby retaining weak but informative cues. To address regions with high prior uncertainty, GPU-SDF further incorporates two complementary constraints: an edge distance field that strengthens boundary supervision and a multi-view consistency regularization that enforces geometric coherence. Extensive experiments confirm that GPU-SDF improves the reconstruction of fine details and serves as a plug-and-play enhancement for existing frameworks. Source code will be available at https://github.com/IRMVLab/GPU-SDF

Qiyu Feng, Jiwei Shan, Shing Shin Cheng +1

Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China. · Department of Mechanical and Automation Engineering and T Stone Robotics Institute, The Chinese University of Hong Kong, Hong Kong.

UQ for 3D Reconstruction 3D Reconstruction Indoor Scene Reconstruction Neural Representation Geometry Signed Distance Functions

Feb 25, 2026

What Topological and Geometric Structure Do Biological Foundation Models Learn? Evidence from 141 Hypotheses

When biological foundation models such as scGPT and Geneformer process single-cell gene expression, what geometric and topological structure forms in their internal representations? Is that structure biologically meaningful or a training artifact, and how confident should we be in such claims? We address these questions through autonomous large-scale hypothesis screening: an AI-driven executor-brainstormer loop that proposed, tested, and refined 141 geometric and topological hypotheses across 52 iterations, covering persistent homology, manifold distances, cross-model alignment, community structure, and directed topology, all with explicit null controls and disjoint gene-pool splits. Three principal findings emerge. First, the models learn genuine geometric structure. Gene embedding neighborhoods exhibit non-trivial topology, with persistent homology significant in 11 of 12 transformer layers at p < 0.05 in the weakest domain and 12 of 12 in the other two. A multi-level distance hierarchy shows that manifold-aware metrics outperform Euclidean distance for identifying regulatory gene pairs, and graph community partitions track known transcription factor target relationships. Second, this structure is shared across independently trained models. CCA alignment between scGPT and Geneformer yields canonical correlation of 0.80 and gene retrieval accuracy of 72 percent, yet none of 19 tested methods reliably recover gene-level correspondences. The models agree on the global shape of gene space but not on precise gene placement. Third, the structure is more localized than it first appears. Under stringent null controls applied across all null families, robust signal concentrates in immune tissue, while lung and external lung signals weaken substantially.

Ihor Kendiukhov

Department of Computer Science, University of Tübingen, Tübingen, Germany

Neural Representation Geometry Genomic Foundation Models Geometric Representation Learning Topological Representation Learning Single-Cell Transcriptomics