In Week 1, in the optional notebook of PPL, we have used **Linear Interpolation** to define PPL in w-space, and **spherical interpolation** to define PPL in z-space.

Also, in the notebook, it is mentioned that “**Because you sample points in z from a Gaussian, we use spherical interpolation instead of linear interpolation to interpolate in z-space.**”

Can anyone tell me how does the distribution help to decide which type of interpolation to use in which space? Additionally, I also wanted to ask if the 8-layer perception network that is used as a mapping from z-w space in StyleGAN, has anything to do with it?

Thanks in advance!

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Slerp is based on the fact that any point on the curve must be a [linear combination]of the ends. While *Slerp* interpolates along a great arc between two quaternions, it is also possible to interpolate along a straight line (in four-dimensional quaternion space) between those two quaternions.

Hey @cvetko.tim, thanks a lot for your reply. But do you mind breaking down your answer into a simpler version? I am not this well-rehearsed with Mathematics

Think of a sphere, like the Earth. You want to interpolate from one point (e.g. New York) to another (e.g. Tokyo).

You could either do it by going **through** the Earth (i.e. through its mantle and core), or by **walking** across the surface. ~~(you couldn’t actually walk, but you get the idea)~~

The former would be **linear interpolation**, the latter would be **spherical interpolation**.

The StyleGAN paper [1] says

(…) spherical interpolation (…) is the most appropriate way of interpolating in our normalized input latent space.

citing [2], which is where spherical interpolation for latent space is introduced. [2] argues that linear interpolation might go through locations that are unlikely given the latent space’s high dimensionality and Gaussian prior.

Using our previous analogy, you are likely to find a person on the Earth’s surface but very unlikely to find anyone deep inside its core.

Check out section 2 of [2] for more info.

[1] Karras, Tero, Samuli Laine, and Timo Aila. “A style-based generator architecture for generative adversarial networks.” *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition* (2019)

[2] White, Tom. “Sampling generative networks.” *arXiv preprint arXiv:1609.04468* (2016)

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Thanks a lot @pedrorohde. An amazing explanation indeed, and a fun one too. Now, I will keep in mind that **Journey to the center of the Earth** is nothing but trying to implement Linear Interpolation while traveling from New York to Tokyo

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