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I am trying to wrap my head around VAE's and have trouble understanding what is being visualized when people make scatter plots of the latent space. I think I understand the bottleneck concept; we go from $N$ input dimensions to $H$ hidden dimensions to a $Z$ dimensional Gaussian with $Z$ mean values, and $Z$ variance values. For example here (which is based off the official PyTorch VAE example), $N=784, H=400$ and $Z=20$.

When people make 2D scatter plots what do they actually plot? In the above example the bottleneck layer is 20 dimensional, which means there are 40 features (counting both $mu$ and $sigma$). Do people do PCA or tSNE or something on this? Even if $Z=2$ there is still four features so I don't understand how the scatter plot showing clustering, say in MNIST, is being made.

When people make 2D scatter plots what do they actually plot?

First case: when we want to get an embedding for specific inputs:

We either

1. Feed a hand-written character "9" to VAE, receive a 20 dimensional "mean" vector, then embed it into 2D dimension using t-SNE, and finally plot it with label "9" or the actual image next to the point, or




2. We use 2D mean vectors and plot directly without using t-SNE.

Note that "variance" vector is not used for embedding. However, its size can be used to show the degree of uncertainty. For example a clear "9" would have less variance than a hastily written "9" which is close to "0".

Second case: when we want to plot a random sample of z space:

1. 
   

   We select random values of z, which effectively bypasses sampling from mean and variance vectors,

   
   
   

   sample = Variable(torch.randn(64, ZDIMS))
   

   
   


2. 
   

   Then, we feed those z's to decoder, and receive images,

   
   
   

   sample = model.decode(sample).cpu()
   

   
   



3. Finally, we embed z's into 2D dimension using t-SNE, or use 2D dimension for z and plot directly.

Here is an illustration for the second case (drawn by the one and only paint):

As you see, the mean and variances are completely bypassed, we directly give the random z's to decoder.

The referenced article says the same thing, but less obvious:

Below you see 64 random samples of a two-dimensional latent space of MNIST digits that I made with the example below, with ZDIMS=2

and

VAE has learned a 20-dimensional normal distribution for any input digit

ZDIMS = 20
...
self.fc21 = nn.Linear(400, ZDIMS) # mu layer
self.fc22 = nn.Linear(400, ZDIMS) # logvariance layer

which means it only refers to the z vector, bypassing mean and variance vectors.