## Path Length Regularization

Introduced by Karras et al. in Analyzing and Improving the Image Quality of StyleGAN

Path Length Regularization is a type of regularization for generative adversarial networks that encourages good conditioning in the mapping from latent codes to images. The idea is to encourage that a fixed-size step in the latent space $\mathcal{W}$ results in a non-zero, fixed-magnitude change in the image.

We can measure the deviation from this ideal empirically by stepping into random directions in the image space and observing the corresponding $\mathbf{w}$ gradients. These gradients should have close to an equal length regardless of $\mathbf{w}$ or the image-space direction, indicating that the mapping from the latent space to image space is well-conditioned.

At a single $\mathbf{w} \in \mathcal{W}$ the local metric scaling properties of the generator mapping $g\left(\mathbf{w}\right) : \mathcal{W} \rightarrow \mathcal{Y}$ are captured by the Jacobian matrix $\mathbf{J_{w}} = \delta{g}\left(\mathbf{w}\right)/\delta{\mathbf{w}}$. Motivated by the desire to preserve the expected lengths of vectors regardless of the direction, we formulate the regularizer as:

$$\mathbb{E}_{\mathbf{w},\mathbf{y} \sim \mathcal{N}\left(0, \mathbf{I}\right)} \left(||\mathbf{J}^{\mathbf{T}}_{\mathbf{w}}\mathbf{y}||_{2} - a\right)^{2}$$

where $y$ are random images with normally distributed pixel intensities, and $w \sim f\left(z\right)$, where $z$ are normally distributed.

To avoid explicit computation of the Jacobian matrix, we use the identity $\mathbf{J}^{\mathbf{T}}_{\mathbf{w}}\mathbf{y} = \nabla_{\mathbf{w}}\left(g\left(\mathbf{w}\right)·y\right)$, which is efficiently computable using standard backpropagation. The constant $a$ is set dynamically during optimization as the long-running exponential moving average of the lengths $||\mathbf{J}^{\mathbf{T}}_{\mathbf{w}}\mathbf{y}||_{2}$, allowing the optimization to find a suitable global scale by itself.

The authors note that they find that path length regularization leads to more reliable and consistently behaving models, making architecture exploration easier. They also observe that the smoother generator is significantly easier to invert.

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