{%hackmd SybccZ6XD %}

How to control?
> PCA (Principal Component Analysis)

## PCA

Convariance matrix
> $$
\begin{gathered}
\begin{bmatrix} Var(a) & Convar(a, b) \\ Convar(a, b) & Var(b) \end{bmatrix}
\end{gathered}
$$

Eigen value and vactor
> There exists a vector that makes the covariance equal to 0.
$$
\begin{gathered}
\begin{bmatrix} Var(a) & Convar(a, b) \\ Convar(a, b) & Var(b) \end{bmatrix}
\begin{bmatrix} v_1 \\ v_2 \end{bmatrix} =
\begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix}
\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}
\end{gathered}
$$

## PCA in the styleGAN

Architecture
> 

Steps
- sample N random vectors $z_{1:N}$
- compute the corresponding $w_i = M(z_i)$
- compute PCA of these $w_{1:N}$ and find new basis V for W
- $w' = w + Vx$

## Result

$E(v_i, j-k)$ to denote edit directions; for example, $E(v_1, 0-3)$ means moving along component v1 at the first four layers only.
The first few components control large-scale variations, including apparent gender expression and head rotation.

## Problem

Rotating a dog often causes its mouth to open, perhaps a product of correlations in portraits of dogs.