Spectral Normalization: Divide W by Its Top Singular Value for a 1-Lipschitz Layer
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A linear layer y = Wx stretches its input, and the worst-case stretch — the largest ‖Wx‖/‖x‖ over every x — is exactly the spectral norm σ(W) , the largest singular value of W . That number is also the layer's Lipschitz constant. In a GAN the discriminator is trained adversarially and its weights keep growing, so σ blows up, gradients explode, and training collapses. WGAN patched this with crude weight clipping (which kills capacity) and then a gradient penalty (an extra backward pass).…
1Key Takeaways
- A linear layer y = Wx stretches its input, and the worst-case stretch — the largest ‖Wx‖/‖x‖ over every x — is exactly the spectral norm σ(W) , the largest singular value of W .
- That number is also the layer's Lipschitz constant.
- In a GAN the discriminator is trained adversarially and its weights keep growing, so σ blows up, gradients explode, and training collapses.
- WGAN patched this with crude weight clipping (which kills capacity) and then a gradient penalty (an extra backward pass).….
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3Why it matters
Coding AI shifts how fast software ships and how much human review each change needs. DEV — ML reports that a linear layer y = Wx stretches its input, and the worst-case stretch — the largest ‖Wx‖/‖x‖ over every x — is exactly the spectral norm σ(W) , the largest singular value of W .
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