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On the Relationship Between Representation Geometry and Generalization in Deep Neural Networks

Sumit Yadav

arXiv:2602.00130v2cs.LG

TL;DR

The paper investigates whether representation geometry explains why some neural networks generalize better than others. It measures effective dimension and compression across vision and language models, finding domain-spanning predictive and intervention-based relationships with accuracy, while noting important scope and mechanistic limitations.

  • Problem

    Why some learned representations support better generalization remains incompletely explained, especially because classical capacity-based bounds can be vacuous for overparameterized networks.

  • Method

    The study measures unsupervised representation geometry across pretrained vision models, language encoders, and decoder-only LLMs, combining cross-domain evaluation with controlled noise and PCA interventions.

  • Results

    Across architectures and domains, effective-dimension metrics predict accuracy: output effective dimension reaches partial r = 0.75 and total compression partial r = -0.72, while decoder-only LLM model size has r = 0.07.

  • Takeaways & Limitations

    Effective dimension provides domain-agnostic predictive and causal information about neural-network performance and can be computed without labels.

  • Takeaways & Limitations

    The encoder–decoder divergence is empirical rather than theoretically proven, and the intervention evidence is limited to ResNet18 with strong baselines.

Abstract

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We investigate the relationship between representation geometry and neural network performance. Analyzing 52 pretrained ImageNet models across 13 architecture families, we show that effective dimension -- an unsupervised geometric metric -- strongly predicts accuracy. Output effective dimension achieves partial r=0.75 ($p < 10^(-10)$) after controlling for model capacity, while total compression achieves partial r=-0.72. These findings replicate across ImageNet and CIFAR-10, and generalize to NLP: effective dimension predicts performance for 8 encoder models on SST-2/MNLI and 15 decoder-only LLMs on AG News (r=0.69, p=0.004), while model size does not (r=0.07). We establish bidirectional causality: degrading geometry via noise causes accuracy loss (r=-0.94, $p < 10^(-9)$), while improving geometry via PCA maintains accuracy across architectures (-0.03pp at 95% variance). This relationship is noise-type agnostic -- Gaussian, Uniform, Dropout, and Salt-and-pepper noise all show $|r| > 0.90$. These results establish that effective dimension provides domain-agnostic predictive and causal information about neural network performance, computed entirely without labels.

1 Introduction

The paper asks which properties of learned representations support generalization, arguing that representation geometry may provide architecture- and domain-independent performance signals. It introduces geometric metrics and reports strong predictive and causal relationships with accuracy across models, datasets, and domains.

  • Classical VC-dimension and Rademacher-complexity bounds are often vacuous for overparameterized networks, motivating alternative characterizations of generalization.
  • Geometric properties may be fundamental across architectures because vision and language models increasingly measure datapoint distances similarly as they scale.
  • Total compression, defined as the log-ratio of output to input effective dimensionality, predicts accuracy across 52 pretrained models from 13 architecture families (r = -0.65; partial r = -0.71 controlling for model size).
  • Output effective dimension is the strongest individual accuracy predictor after controlling for model size (partial r = 0.75), with higher final-layer dimensionality associated with better performance.
  • Geometric relationships replicate across ImageNet and transferred CIFAR-10, extend to encoder and decoder-only language models, and support bidirectional intervention results involving noise and PCA.The reported interventions associate noise-based geometry degradation with accuracy loss and PCA-based geometry improvement with maintained accuracy.

2 Related Work

Related work connects representation geometry to information theory, convergence across architectures and modalities, intrinsic dimension, and generalization theory. These perspectives motivate measuring geometric structure directly while leaving its theoretical explanation unresolved.

  • The information bottleneck frames learning as compressing input information while preserving task-relevant information, but prior work questions whether compression universally explains generalization.
  • Prior studies report convergence toward shared representation geometry across vision and language models and show that embeddings can transfer across model families without paired data.
  • Representation-analysis methods link intrinsic dimensionality, layerwise expansion and compression, cross-network comparison, and neural collapse to structured representation geometry.
  • Generalization research has also examined flatness, dynamical isometry, singular-value distributions, compression-based bounds, double descent, and neural tangent kernels.

3 Problem Formulation

The paper formulates a network as a representation map followed by a classifier and characterizes representation geometry using effective dimension and total compression. It presents theoretical connections as intuition rather than proof, while identifying output effective dimension as the strongest empirical predictor of accuracy.

  • 3.1 Setup and Notation: A classifier is decomposed as f_θ = g ∘ h, where h maps inputs to representations and g maps representations to predictions.
  • 3.1 Setup and Notation: The representation matrix Z stacks sampled representations and is analyzed through effective dimensionality and class structure.
  • 3.2 Effective Dimensionality: Effective dimension, or participation ratio, measures how many dimensions contribute meaningfully to variance; it equals d under uniform variance and approaches 1 when variance concentrates in one direction.
  • 3.3 Total Compression: Total compression compares final- and initial-layer effective dimensions through a log-ratio across an L-layer network.
  • 3.3 Total Compression: Negative total compression indicates reduced effective dimensionality, whereas positive values indicate expansion.
  • 3.3 Total Compression: The log-ratio is dimensionless, supports a multiplicative interpretation, and enables comparison of equivalent relative transformations across differently scaled networks.
  • 3.4 Output Effective Dimension: Output effective dimension measures final-representation richness and is the strongest accuracy predictor after controlling for model size.
  • 3.5 Theoretical Connections (Informal): Connections to Rademacher complexity, Fisher information, and intrinsic dimension are presented as suggestive intuition, not rigorous explanations of generalization.

4 Methods

The methods combine pretrained-model analysis, cross-distribution evaluation, training-time checkpoints, geometric-signature extraction, and statistical prediction analyses. Effective-dimension features are computed from layer activations and related to accuracy using correlations, partial correlations, and feature importance.

  • The study analyzes 52 ImageNet-pretrained models across 13 architecture families, spanning 1.2M–306M parameters and 58.1%–85.8% top-1 accuracy.
  • Geometry is evaluated on native ImageNet validation images and transferred CIFAR-10 test images resized to 224×224.
  • Training-time analysis follows 11 models from six architecture families trained from scratch on CIFAR-10 for 50 epochs with checkpoints across eight epochs.
  • Algorithm 1 samples evaluation inputs, extracts activations at every layer, computes effective dimensions, and returns compression, bottleneck, input, and output geometric statistics.
  • The extraction pipeline uses 2,000 samples and randomized SVD when dimensionality exceeds 1,000 for computational efficiency.
  • Pearson correlations relate each geometric metric to accuracy across models, while partial correlations remove linear effects of log-parameter count.
  • Random Forest regressors predict accuracy from geometric features and provide Gini importance scores.

5 Results

Across 52 pretrained ImageNet models, total compression and output effective dimension consistently predict accuracy on both CIFAR-10 transfer and native ImageNet evaluations. These geometric metrics remain informative beyond model size, with output effective dimension showing the strongest partial relationship.

  • r = −0.64 on CIFAR-10 and r = −0.65 on ImageNet, with p < 10^-6 for both, for total compression versus accuracy.More negative compression corresponds to greater dimensionality reduction and higher accuracy.
  • Output effective dimension shows consistent accuracy correlations in both CIFAR-10 transfer and ImageNet native evaluations across architecture families.
  • After controlling for log(parameters), output effective dimension reaches partial r = 0.746, exceeding total compression at partial r = −0.720.Both are stronger after controlling for model size, indicating geometric information beyond capacity.
  • Geometric signatures strengthen after controlling for model size, showing they capture performance information not explained by capacity alone.

5.4 Feature Importance Analysis

Feature-importance and training-time analyses show that total compression carries substantial predictive information, while geometric signatures emerge early during training. Output effective dimension also acts as an early indicator of eventual performance across architectures.

  • > 50% of predictive power is accounted for by total compression in the accuracy regressor.Feature importance is measured by Gini impurity reduction.
  • Both output effective dimension and total compression become predictive of final accuracy early in training across 11 models and 6 architecture families.
  • Output effective dimension becomes significant earlier than accuracy as a predictor of final accuracy.Figure 6 compares R² progression for early accuracy and output effective dimension.
  • Accuracy ranges from 84.0% to 95.3% across 11 CIFAR-10-trained models, providing variance for correlation analysis.

5.6 Cross-Domain Generalization: NLP Results

Representation geometry predicts NLP performance across transformer encoders and both binary and three-way classification tasks. Output effective dimension remains strongly associated with accuracy after controlling for model size, supporting domain-agnostic applicability.

  • SST-2: r = −0.96 and R^2 = 0.92 for output effective dimension on SST-2, with partial r = −0.90 after controlling for model size.Lower output effective dimension predicts better accuracy in these encoder models.
  • SST-2: Compression shows the expected negative correlation on SST-2, with r = −0.60, as models that compress more achieve better performance.
  • MNLI: r = −0.94 for output effective dimension on MNLI, consistent with SST-2 despite MNLI being a more challenging three-way classification task.
  • Cross-domain conclusion: Geometric signatures predict performance across binary and multi-class NLP tasks, indicating they are not specific to vision.

5.7 Decoder-Only LLMs: Extending to Autoregressive Models

The study extends representation-geometry analysis to decoder-only language models and finds that geometric quality tracks performance independently of model scale. Decoder representations expand rather than compress, but transformation magnitude remains predictive across encoder and decoder regimes.

  • Evaluation: 15 decoder-only models across 5 architecture families are evaluated on AG News using pretrained representations and last-token pooling.Representations are analyzed without fine-tuning, using 2,000 balanced samples and layer-wise effective-dimension and compression measurements.
  • Geometric results: r=0.69, p=0.004: compression correlates with output effective dimension in decoder-only LLMs.Higher compression corresponds to richer final representations in the reported AG News analysis.
  • Geometric results: r=0.07, p=0.82: model size does not predict geometric quality, while architecture family matters more than scale.SmolLM and Qwen show higher compression than GPT-2 at equivalent sizes.
  • Encoder-decoder divergence: Decoder-only models exhibit positive compression because later layers expand representations toward the large vocabulary space.This contrasts with the negative compression pattern described for discriminative encoders.
  • Unified principle: r=0.72 for encoders and r=0.69 for decoders: transformation magnitude |𝒞| correlates with quality regardless of direction.The sign reflects different computational objectives, whereas magnitude provides a direction-agnostic geometric metric.
  • Caveat: The encoder-decoder sign reversal is an empirical pattern with a plausible explanation, not a formal theoretical result that must hold universally.The authors note that alternative explanations and exceptions across architectures or tasks remain possible.

6 Discussion

The discussion presents output effective dimension and total compression as complementary geometric indicators, then tests their causal relevance through controlled degradation and reduction of representation dimensions. The evidence supports a bidirectional geometry-performance link, while the authors identify limits in intervention scope, domain coverage, statistical power, and mechanistic understanding.

  • Geometric signatures: Output effective dimension captures representation richness, while total compression captures information refinement across the network.Their reported partial correlations with accuracy are 0.75 and -0.72, respectively.
  • Theoretical connections: Effective dimension is an unsupervised geometric proxy related to intrinsic dimensionality, flatness-generalization ideas, and information geometry.The paper connects it to the participation ratio and discusses total compression as an approximation of changes in Fisher information geometry.
  • Causal degradation: ΔEffDim=+666 and Δacc=-75.7pp for DenseNet121 at σ=0.6, compared with smaller accuracy losses for ResNet18 and ResNet34.The authors attribute DenseNet121’s sensitivity to perturbation propagation through dense skip connections.
  • Causal degradation: r=-0.94, p<10^-9: across 21 measurements, increases in effective dimension under noise track accuracy loss.The intervention uses ResNet18, ResNet34, and DenseNet121 trained on CIFAR-10 with strong 86-88% baselines.
  • Noise robustness: All four tested noise types show strong negative geometry-accuracy correlations, supporting noise-type agnosticism.Salt-and-pepper noise is strongest at r=-0.99, while dropout is weakest at r=-0.91.
  • Causal improvement: 14-16 principal components out of 512 preserve full accuracy across three architectures, with a mean loss of -0.03pp at 95% variance.PCA reduces effective dimension while maintaining accuracy, and DenseNet121 shows slight improvement in the reported table.
  • Bidirectional causality: The intervention evidence supports a bidirectional pathway: noise increases effective dimension and lowers accuracy, whereas PCA reduces effective dimension while maintaining accuracy.The reported bidirectional analysis gives r=-0.92 for degradation and mean Δacc=-0.05pp for PCA at ≥90% variance.
  • Limitations: The causal claims remain bounded because interventions were demonstrated on a limited architecture set and additional methods would strengthen the evidence.Suggested extensions include whitening and geometric regularization during training.

7 Conclusion

The paper concludes that output effective dimension and total compression provide label-free geometric information about model performance across architectures, datasets, and domains. Controlled interventions further support a causal geometry-performance relationship, while model size alone does not determine geometric quality in decoder-only LLMs.

  • Generalization: Geometric signatures generalize across vision models, NLP encoder models, and decoder-only LLMs, and become predictive before final accuracy stabilizes.The metrics are computed without labels, supporting use in representation-learning settings.
  • Main findings: Output effective dimension is the strongest predictor of accuracy, with partial r=0.75 and p<10^-10.It is interpreted as capturing representation richness.
  • Main findings: Total compression provides complementary information, with partial r=-0.72, by capturing information refinement across the network.Both metrics remain predictive after controlling for model capacity and replicate across ImageNet and CIFAR-10.
  • Scale and geometry: r=0.69, p=0.004 for compression versus output effective dimension, while hidden size shows r=0.07, p=0.82 in decoder-only LLMs.This supports the conclusion that model size does not determine geometric quality in that setting.
  • Causal evidence: r=-0.94, p<10^-9 under noise and mean Δacc=-0.03pp at 95% variance under PCA provide bidirectional intervention evidence.Degrading geometry causes accuracy loss, while reducing dimensions through PCA maintains accuracy across multiple architectures.

8 Appendix

The appendix documents implementation details, effective-dimension properties, and supplementary analyses of model scale, training dynamics, causal interventions, noise robustness, and cross-architecture PCA results.

  • Implementation: Experiments used 2× NVIDIA Tesla T4 GPUs, PyTorch 2.0, torchvision, Transformers, and scikit-learn, with geometry extraction taking 2 minutes per model.CIFAR-10 training took 30 minutes per model, while NLP fine-tuning took 15–45 minutes and LLM geometry extraction 5–10 minutes.
  • Effective-dimension properties: Effective dimension is bounded by 1 ≤ EffDim(Z) ≤ rank(Z) ≤ min(m, d), invariant under orthogonal transformations, approximately additive across independent subspaces, and continuous under small eigenvalue perturbations.These properties support comparing representations across architectures with different nominal dimensions.
  • Decoder-only LLM analysis: For decoder-only LLMs, compression and output effective dimension correlate at r = 0.69, whereas output effective dimension and hidden dimension correlate at r = 0.07.Effective dimension typically rises through early layers before stabilizing or decreasing, and compression signatures differ across model families.
  • Training dynamics: By epochs 20–50, output effective dimension and geometric metrics become strongly predictive of final accuracy, with R^2 > 0.7 by epochs 30–50 across six architecture families.At epoch 10, correlations remain weak with R^2 < 0.3, indicating that geometry can provide an earlier performance signal than accuracy itself.
  • Bidirectional intervention: Gaussian-noise degradation correlates with accuracy loss at r = −0.92, while PCA retaining at least 90% variance changes accuracy by only −0.03pp.The intervention comparison indicates that noisy dimension inflation harms accuracy more than variance-preserving dimension reduction.
  • Noise robustness and cross-architecture PCA: Across Gaussian, Uniform, Dropout, and Salt-pepper noise, correlations range from r = −0.91 to r = −0.99, with pooled r = −0.91.Across ResNet18, ResNet34, and DenseNet121, PCA at 95% variance uses 14–16 components and changes accuracy by −0.06pp to +0.02pp.
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