We are training a variational auto$-$encoder $($VAE$).$ It consists of the following parts: the input vector x$,$

the latent vector z$,$ the encoder that models the probability q$($z$|$x$),$ and the decoder that models p $($x$|$z$).$

Based on these notations, let$$s look at several problems:

$($a$)$ We assume the latent vector z in R$2$ follows a multi$-$variate Gaussian distribution N $.$ Please compute

the output dimension of the encoder q$()$ under the following cases and briefly explain why. $($If

$$output dimension$$ is not clear enough for you, think of it as $$how many real numbers r in R are

needed to output for the sampling of latent vectors.$)$

$$ We assume N follows a multi$-$variate Gaussian distribution with an identity matrix as the

covariance matrix.

$$ We assume N follows a multi$-$variate Gaussian distribution with an diagonal matrix as the

covariance matrix.