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On the impact of activation and normalization in obtaining isometric embeddings at initialization
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply layer normalization to representations | MLP representations · Arbitrary nonzero sample vectors | isometry: 1+\frac{\frac1n \sum_i^n(a_i-\bar{a})^2}{\bar{a}^2} | Helped |
| Apply batch normalization to representations | Batch-normalized representations · Mini-batch covariance | isometry: 1+\frac{\frac1d \sum_i^d(a_i-\bar{a})^2}{\bar{a}^2} | Helped |
| Use a nonlinear activation with βσ > 1 rather than a linear activation | Layer-normalized mean-field MLP · Non-degenerate input Gram matrices | isometry gap: -\log\Iso(G^\ell_*) \le \exp(-\ell\log\beta_\sigma - n\log\Iso(G^0_*) + \log(4n)) | Helped |
| Use a linear activation (βσ = 1) instead of a nonlinear activation | Layer-normalized mean-field MLP · Non-degenerate input Gram matrices | isometry: 1 | Hurt |
| Remove the centering offset by replacing activation σ(x) with σ(x)-c₀ (ablation) | MLP with activation · CIFAR10 | isometry | Helped |
| Replace layer normalization with constant scaling 1/√barσ(1) (ablation) | MLP with tanh, sigmoid, or ReLU · CIFAR10 | isometry | No clear effect |
| Replace layer normalization with constant scaling 1/√barσ(1) (ablation) | MLP with exponential activation · CIFAR10 | isometry | Hurt |
| Remove normalization while retaining activation (ablation) | MLP with nonlinear activation · CIFAR10 | isometry gap | Mixed |
| Use an activation with larger isometry strength | Shallow MLP with layer normalization · CIFAR10, up to 10 layers | SGD convergence | Helped |
| Use an activation with larger isometry strength | Deep MLP with layer normalization · CIFAR10, deeper than 10 layers | SGD convergence | No clear effect |
| Use mean-field-based activation gains instead of Xavier gain | MLP · Isometry across depth | isometry | Helped |
| Change the ordering of activation and normalization layers (ablation) | MLP · Isometry across depth | isometry | Hurt |
Closest papers by meaning
- Towards Training Without Depth Limits: Batch Normalization Without Gradient Explosion (ICLR 2024 · similarity 0.89)
- The Normalization Method for Alleviating Pathological Sharpness in Wide Neural Networks (NeurIPS 2019 · similarity 0.89)
- On the Nonlinearity of Layer Normalization (ICML 2024 · similarity 0.88)
- Beyond BatchNorm: Towards a Unified Understanding of Normalization in Deep Learning (NeurIPS 2021 · similarity 0.88)
- Normalizing the Normalizers: Comparing and Extending Network Normalization Schemes (ICLR 2017 · similarity 0.88)
- On the Importance of Gaussianizing Representations (ICML 2025 · similarity 0.86)
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