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Sparse Prototype Code Underlies Classification and Prediction Across Modalities

Yehonatan Avidan, Daniel D. Lee, Haim Sompolinsky

arXiv:2608.15632v1cs.LGcond-mat.dis-nncs.AIstat.ML

TL;DR

The paper asks how complex high-dimensional representations can be understood geometrically for multiclass prediction. It analyzes prototype classification and derives a sparse mean-field theory based on centroid-aligned variability and radius renormalization. Across modalities and scales, the theory predicts accuracy while identifying geometric reorganization associated with improved performance.

  • Problem

    Interpreting high-dimensional neural representations remains difficult, and a predictive geometric theory of multiclass accuracy with class competition and rich within-class statistics is missing.

    Source: §1 · pp. 1–2

  • Method

    The paper uses prototype-based classification and derives a sparse mean-field theory from variability along true- and rival-class centroid directions plus a global radius renormalization.

    Source: §1 · p. 2

  • Results

    Across vision, audio, and language models, the theory’s geometric ingredients improve with model scale and mirror gains in classification accuracy.

    Source: §1 · p. 2, §7 · p. 10

  • Takeaways & Limitations

    Classification accuracy is governed by a sparse, centroid-aligned code embedded within the full representation space.

    Source: §7 · p. 10

    The framework assumes high-dimensional isotropic geometry and jointly Gaussian projection coefficients in parts of its analysis.

    Source: §3 · p. 4, §5 · p. 7

Abstract

from arXiv · show

Neural representations have become a central tool for studying the internal mechanisms of modern AI models, yet their complex high-dimensional structure makes them difficult to interpret. We show that classification tasks give rise to a universal representational geometry, shared across state-of-the-art models in vision, audio, and language processing. The key structure is that within-class variability is not random in representation space. Instead, its classifier-relevant component has strong and structured correlations with the class's own centroid and with the centroids of its competing classes. Building on this observation, we derive an analytical mean-field theory governed mainly by the variability along true-class and rival-class centroid coordinates, together with a global renormalization of the class radius that compensates for the non-Gaussian statistics of real representations. The theory accurately predicts classification accuracy across architectures and modalities. The relevant geometric quantities improve systematically with model scale, mirroring the observed gains in accuracy. A striking feature of the theory is its sparsity: accurate prediction requires only a small set of centroid coordinates associated with the true class and its strongest rivals - connecting our framework to sparse-feature extraction approaches such as sparse autoencoders. Together, these results provide a parsimonious predictive theory of neural representations and suggest that classification in deep networks is governed by a sparse, centroid-aligned structure embedded within the full high-dimensional representation space.

1 Introduction

The paper develops a geometric account of multiclass classification that links interpretable centroid-aligned variability to accuracy across modalities and model scales. Its key simplification is that a sparse set of rival-centroid coordinates captures the relevant structure.

Motivation and approach

  • A predictive geometric theory of multiclass accuracy is developed using prototype-based classification, relating accuracy directly to representation geometry.The approach addresses competition among multiple classes and rich within-class statistics.Source: §1 · p. 2

Universal geometry

  • Within-class variability is strongly correlated with the true-class centroid and competing-class centroids across vision, language, and audio backbones.Rotations that remove these correlations while preserving other geometric properties drive per-class accuracy to essentially 100%.Source: §1 · p. 2

Sparsity of competition

  • For each class, centroid-aligned correlations concentrate on a small set of active rivals sufficient to recover the geometric picture and reproduce predicted per-class accuracy.These rivals also carry semantically meaningful relations to the true class.Source: §1 · p. 2

Predictive theory

  • The mean-field theory uses variability along true- and rival-class centroid directions plus a global radius renormalization for heavy-tailed representation statistics.This sparse centroid basis predicts per-class accuracy across classes, architectures, and modalities.Source: §1 · p. 2

Scaling

  • The theory’s geometric ingredients improve systematically with model scale, mirroring gains in accuracy without requiring the smallest overall within-class variance.The identified measures characterize how larger models reorganize representational geometry.Source: §1 · p. 2

2 Setup and Notation

The setup represents each input as a hidden-layer vector and evaluates classes through normalized empirical centroids. Experiments use frozen representations from vision, language, and audio models across architectures and training objectives.

Representation and task

  • Each input is represented by a vector from a neural-network layer and assigned to one of P classes.The paper studies how representation geometry determines classification accuracy.Source: §2 · p. 3

Prototype classification

  • Prototype classification uses each class centroid—the empirical mean of its representations—as a normalized readout vector.Centroids are estimated from labeled data, and inference compares an example against all class prototypes using leave-one-out estimation.Source: §2 · pp. 3–4

Experimental setup

  • Frozen representations span vision, language, and audio backbones, including transformer and convolutional architectures and supervised, self-supervised, and contrastive objectives.Language examples are organized by lemma, retaining lemmas with at least 1000 examples.Source: §2 · p. 3

Manifold geometry

  • A class manifold is the point cloud for one class, with each representation decomposed into its centroid plus a zero-mean residual vector.The class radius measures overall within-class variance, while σ²_µ measures residual variance along a centroid direction.Source: §2 · p. 4

3 Isotropic Uncorrelated Model

The isotropic uncorrelated model provides a baseline in which centroids and residuals are independent Gaussian variables in a high-dimensional regime. Its theory predicts how class radius controls accuracy, from near-perfect separation to chance-level classification.

Model assumptions

  • The baseline model draws independent isotropic Gaussian centroids and class residuals, with examples given by x = c_k + δx.Residuals have covariance set by the class radius R²I_N.Source: §3 · p. 4
  • The analysis assumes N →∞, P = N^O(1) classes, and a radius R that may scale with dimensionality.These assumptions define the high-dimensional isotropic model.Source: §3 · p. 4

Accuracy theory

  • Theorem 3.3 derives isotropic-model accuracy using Gaussian-limit results for centroid overlaps and prototype logits.The proof relies on a centroid-overlap CLT, self-averaging, and a joint Gaussian logit distribution.Source: §3 · p. 4

Predictions

  • For R = O(1), accuracy approaches 1, whereas as R/√N →∞, accuracy collapses to 1/P.Large isotropic residuals overwhelm class information, making all classes equally likely.Source: §3 · p. 4
  • Non-perfect accuracy requires R = Ω(√(N/ln P)), and accuracy variance across centroid realizations is O((PR²)^−1).The high-dimensional accuracy is self-averaging.Source: §3 · p. 4

4 Empirical Results

Empirical representations deviate sharply from isotropic predictions: accuracy depends on class-specific, centroid-aligned variability rather than radius alone. Rotational manipulations show that residual–centroid correlations, especially with the true centroid, dominate classification geometry, while centroid–centroid correlations contribute little.

Accuracy changes on an O(1) radius scale

  • Accuracy falls from 100% to 61% across the narrow empirical radius range R ∈[0.4, 4.3], contradicting isotropic-model predictions.The isotropic model predicts decline only at R = Ω(N/ ln P) ∼20, far beyond the observed range.Source: §4 · p. 5

Large class-to-class variability at fixed radius

  • At R ∈[1.2, 1.3], 89 classes span 61.8%–99.6% accuracy despite similar radii, showing that radius alone cannot explain performance.The mean accuracy in this radius range is 90.4%.Source: §4 · p. 5

Strong residual–true-centroid correlations

  • Variability inflation leaves accuracy at 39.2% as R⋆→∞, far above chance at 0.1%, because residuals remain strongly correlated with the true centroid.The true-class projection variance exceeds every rival's in 93% of classes; those classes average 87.4% accuracy versus 60.1% otherwise.Source: §4.1 · p. 5

Geometric manipulations

  • Randomly rotating residuals preserves radii, centroid relations, and example-example structure while breaking residual–centroid correlations.This manipulation isolates the correlations most directly implicated in classification errors.Source: §4.1 · pp. 5–6

Breaking residual–centroid correlations recovers the isotropic model

  • After residual rotation, accuracy follows the isotropic model with MSE = 1.26 × 10−3 and reaches chance at R⋆→∞, while per-class variability shrinks by an order of magnitude.The per-class standard deviation falls from 1.7 × 10−2 to 2.3 × 10−3.Source: §4.1 · p. 6

Centroid-centroid correlations play a minor role

  • Breaking centroid–centroid correlations adds little, with the full-rotation curve differing from residual rotation by only MSE = 2.0×10−5.The results identify a sparse centroid-aligned geometry dominated by the true-class centroid.Source: §4.1 · p. 6

5 Centroid-Aligned Variability Model

The paper models classification using centroid-aligned residual variability concentrated on a sparse set of rival directions. A global radius renormalization accounts for heavy-tailed projections, enabling accurate per-class predictions across modalities.

  • Removing only residual–centroid correlations drives per-class accuracy to approximately 100%, while breaking centroid–centroid correlations has only a secondary effect.Source: §5 · p. 6
  • The model represents residuals using true- and rival-centroid directions within a restricted rival support whose centroid Gram matrix is assumed invertible.Source: §5 · pp. 6–7
  • The accuracy theory uses class radius R, rival projected variances {σµ}, centroid overlaps {gµ}, and residual correlation terms ρµ measured directly from representations.Source: §5 · p. 7

5.1 Sparsity

  • 6 rivals come within 3% of full P-way accuracy, 12 within 2%, and 37 within 1%, showing that prediction depends on a sparse competitor set.Source: §5.1 · p. 7
  • 14.4 rivals cover 90% of examples on average, while one rival accounts for more than half; selected rivals often have semantic or visual relations to the true class.Source: Figure 3 · p. 7, §5.1 · p. 8

5.2 Non-Gaussian tails and a global radius renormalization

  • A global transformation R → λR compensates for heavy-tailed projection extremes; λ⋆ correlates with inverse tail exponent 1/ν across 36 models at r = 0.90.Source: §5.2 · pp. 8–9
  • The λ-rescaled theory predicts per-class accuracy accurately in vision and language, with MSE=1.83 × 10^-3 and 4.43 × 10^-3, respectively.Source: Figure 4 · p. 9

5.3 Shared geometry across architectures and modalities

  • Across 36 vision, audio, and language backbones, the centroid-aligned geometry persists, including sparse rivals, heavy tails, and near-perfect accuracy after δ-rotation.Source: §5.3 · p. 9

6 Geometry and Accuracy Across Model Scales

Across model families, accuracy gains track systematic reorganizations of centroid-aligned geometry rather than uniformly reduced within-class variability. The theory isolates these geometric measures across scales and modalities.

  • Larger models improve the theory’s geometric ingredients alongside accuracy, identifying reorganizations associated with improved classification performance.Source: §6 · p. 9
  • Qwen3 accuracy rises from 41% to 57% while mean radius grows from 1.98 to 2.20 between 0.6B and 32B models.Source: §6 · p. 9
  • In DINOv3, ViT-7B has a mean radius 0.21 larger than ViT-H+ yet matches its accuracy, showing that reduced variability does not explain the scale trend.Source: §6 · p. 9
  • Across scale, variability aligns more with the true centroid, less with the hardest rival, while centroid and residual-projection correlations decrease.Source: §6 · p. 10
  • Representations become more heavy-tailed with scale, with ν decreasing from 19.3 to 4.5 in vision and from 21.9 to 11.4 in language.Source: §6 · p. 10
  • In autoregressive models, the best prototype classification is consistently found 1–3 layers before the final layer.Source: §6 · p. 10

7 Discussion

Across modalities and architectures, classification accuracy is governed by a sparse, centroid-aligned code rather than undifferentiated high-dimensional variability. The theory quantifies this structure, tracks its changes with model scale, and complements existing geometric frameworks.

  • Prototype accuracy is controlled by variability along the true-class centroid and a small set of rival-centroid coordinates.Source: §7 · p. 10
  • As models scale, accuracy improves through geometric reorganization rather than reduced class variance.Variability shifts toward the true-class coordinate, strongest-rival correlations decrease, and centroid directions become more decorrelated.Source: §7 · p. 11
  • The theory tracks standard linear classification as well as prototype classification, with close agreement except for a larger gap in bidirectional language models.Source: §7 · p. 11
  • The framework complements manifold-capacity theory by treating centroid-aligned within-class variability as dominant in this task, while random intercentroid-overlap fluctuations have little effect.Source: §7 · p. 11
  • The sparse class-supervised directions connect the framework to interpretability methods such as sparse autoencoders and dictionary learning.Source: §7 · p. 11
  • The compact geometric ingredients may support representation-quality diagnostics, model selection for fine-tuning, and geometry-aware training or distillation objectives.Source: §7 · p. 11

A.1 Prototype classification and linear readout comparison

The authors test whether prototype classification is a useful proxy for standard linear readouts using a held-out comparison. The methods align closely overall, but the correspondence is weaker for bidirectional language models.

  • A 90%/10% train-validation split compares a max-margin Crammer–Singer SVM with prototype scores computed from training centroids.Source: §A.1 · p. 16
  • The mean accuracy gap is 0.047 in vision and 0.063 in autoregressive models, indicating close agreement between the two classifiers.Source: §A.1 · p. 16
  • Bidirectional language models show a larger mean accuracy gap of 0.121 and lower per-class correlation, r = 0.58.The corresponding correlations are r = 0.87 in vision and r = 0.76 in autoregressive models.Source: §A.1 · p. 16
  • Figure 5 reports that the gap between centroid and trained linear-readout accuracy is small.Source: Figure 5 · p. 16
  • The comparison uses frozen pretrained representations spanning modalities and architecture families without fine-tuning or modifying the backbones.Source: B Experimental Setup · p. 16

B.1 Datasets

The study evaluates frozen representations from a broad roster of pretrained vision, language, and audio backbones on classification tasks. The datasets and preprocessing define modality-specific class labels and extracted features.

Datasets

  • The evaluation uses ImageNet-1K for vision, TinyStories token prediction for language, and VGGSound for audio.ImageNet-1K contains 1000 classes and approximately 1.28M images; VGGSound uses 310 classes and 183,630 clips.Source: §B.1 · pp. 16–17

Language preprocessing

  • Language classes are restricted to content-word lemmas, including NOUN, PROPN, VERB, ADJ, and ADV.Tokens sharing a lemma are unified into one class.Source: §B.1 · p. 17
  • Masked-word models use the final hidden state at a masked content-word position, while autoregressive models use the preceding hidden state for the next predicted word.Source: §B.1 · p. 17

Model roster

  • The canonical roster contains 36 pretrained backbones across the four modality groups, including transformer and convolutional architectures.Source: §B.2 · p. 18, Table 6 · p. 18

Feature extraction

  • Representations are taken from final-layer or pre-classifier pooled features, depending on the backbone family.Source: §B.3 · p. 19

Preprocessing

  • Each representation dimension is z-scored before analysis using its empirical mean and standard deviation.Source: §B.3 · p. 19

C.1 Isotropic model

The isotropic model treats centroids and within-class fluctuations as uncorrelated Gaussian variables, yielding analytical accuracy predictions and self-averaging results. The centroid-aligned model instead retains classifier-relevant rival coordinates and evaluates accuracy over a sparse rival set.

Isotropic baseline

  • The isotropic baseline assumes iid Gaussian centroids and isotropic Gaussian within-class fluctuations, retaining only N, P, and the noise-to-signal radius R.It discards correlations between centroids and fluctuations.Source: §C.1 · p. 19

Prototype classification

  • The prototype classifier predicts the class whose unit centroid has the largest dot product with the input representation.Centroids are empirical class means normalized to unit vectors; normalized logits preserve the argmax.Source: §C.1 · p. 19, §C.2 · p. 22

Analytical consequences

  • For R = O(1), isotropic classes are asymptotically separable and accuracy approaches 1; for large residual radius, accuracy collapses to chance, 1/P.The isotropic model also identifies a critical-radius scaling for non-perfect accuracy.Source: §C.1 · p. 21

Centroid-aligned model

  • The centroid-aligned model represents each rival logit as a centroid overlap plus radius-scaled projected variability, then computes correctness as an orthant probability.The conditional-independence assumption factorizes this probability into rival-wise Gaussian terms.Source: §C.2 · pp. 22–23

Sparse rivals

  • The centroid-aligned theory evaluates only a small rival set because its error saturates around K ≈10–20 rather than requiring all P classes.This sparse restriction is motivated by the observation that only a small number of classes are genuine competitors.Source: E Rival Selection · p. 23

E.2 Convergence of the K-way classification accuracy

The theory converges rapidly as the rival set grows: a small number of strongest competitors reproduces full classification accuracy, while heavy-tailed projections require global radius rescaling. Across modalities, the rescaled theory substantially improves prediction fidelity.

Rival selection

  • Margin and variance selectors reach full P-way accuracy with the fewest rivals, and projection variance is adopted because it is cheaper to measure and interpret.The conclusion is consistent across modalities; projection variance and conditional margin perform comparably within five rivals at every band.Source: §E.2 · p. 24

Convergence

  • At K = 20, the theory’s mean error is within 0.003 of its large-K floor for 31 of 36 backbones.The median number of rivals needed to come within 0.001 of the floor is 20.Source: §E.2 · p. 25

Heavy-tailed projections

  • The standardized centroid projections have pronounced positive tails, which increase extreme rival scores and make a Gaussian theory over-optimistic.A smaller Student-t exponent ν indicates a heavier tail.Source: F Heavy tail statistics and… · p. 26
  • The fitted global rescaling R → λR compensates for heavy-tailed projections by enlarging the effective within-class radius.The rescaling is fitted per model.Source: F Heavy tail statistics and… · p. 26

Rescaling validation

  • Without rescaling, predicted accuracy is high by 6% on average; with fitted λ, the error falls to 0.5%.The fitted rescaling is therefore retained in the main theory.Source: F Heavy tail statistics and… · p. 26
  • Across backbones, heavier tails correlate strongly with larger rescaling factors, with global Pearson r = 0.90 between λ and 1/ν.The reported within-family correlations are 0.89, 0.95, 0.81, and 0.89.Source: F Heavy tail statistics and… · p. 26, Table 11 · p. 27

G Depth analysis

Prototype classification depends on representation depth in a modality-specific way. Language models peak before the final block, whereas vision models generally peak at the final block.

Language

  • For autoregressive language models, content-word prototype accuracy is consistently highest a few blocks before the final layer and drops at the last block.The analysis identifies the best layer before measuring the geometry of the largest language models.Source: G Depth analysis · p. 29

Vision

  • For vision models, the final block gives the best prototype accuracy, with a plateau in DINOv3 7B.The comparison spans six Qwen3 autoregressive models and six DINOv3 vision transformers.Source: G Depth analysis · p. 29

Geometric measures

  • The class radius R does not predict the overall best accuracy across depth.Depth analysis therefore tracks additional geometric measures, including hardest-rival projection statistics and centroid overlap.Source: G Depth analysis · p. 29, Table 13 · p. 29

H Theoretical Predictions Across Models and Modalities

The centroid-aligned theory is tested unchanged across 36 state-of-the-art models spanning vision, language, and audio. It predicts per-class prototype accuracy strongly across modalities, while richer covariance corrections provide only a modest improvement at substantial parameter cost.

Cross-modal evaluation

Theory variants

  • The main theory uses a rank-one true–rival correlation structure, while the cluster expansion adds off-diagonal covariance corrections.Both variants are evaluated with K = 20 rivals and separately fitted global rescaling factors.Source: I Variants of the theory · pp. 30, 32, Table 14 · p. 33
  • The cluster expansion fits slightly better but adds K(K −1)/2 covariance parameters and is less transparent, so it is not used in the main text.It is presented as an additional theoretical result on the 36 tested backbones.Source: I Variants of the theory · p. 32

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