In here, I learned that the learned MNIST manifold can be visualized as the image below (on page 10, figure 4(b)).

My understanding for this visualization is:

We start from probability integral transform which allows us to sample from a continuous distribution by sampling from a uniform distribution on $[0, 1]^d$.

In this example, we want to sample from the prior $p_{\theta}(z)$ which is a bivariate normal distribution, so we can sample from a uniform distribution on a unit square $[0, 1] \times [0, 1] $, and then transform the sampled points to $z$ by the inverse CDF of $p_{\theta}(z)$.

So, in the picture, the square is actually a unit square, and those points that the digits sit are the points that we sampled from this uniform distribution. Since each point can be converted to a $z$, which is a point in the manifold $p_{\theta}(x)$. And each such $z$ can be further decoded by the trained $p_{\theta}(x|z)$. So, we have those reconstructed digits sitting on those points (those digits are means of $p_{\theta}(x|z)$).

But I don't know whether this understanding is correct or not.

Moreover, what can we learn from this visualization?

Learned MNIST manifold

  • $\begingroup$ It's very well explained in the picture caption: "Visualisations of learned data manifold for generative models with two-dimensional latent space, learned with AEVB. Since the prior of the latent space is Gaussian, linearly spaced coordinates on the unit square were transformed through the inverse CDF of the Gaussian to produce values of the latent variables $\mathbf{z}$ . For each of these values $\mathbf{z}$, we plotted the corresponding generative $p_{\boldsymbol{\theta}}(\mathbf{x}\vert\mathbf{z})$ with the learned parameters $\boldsymbol{\theta}$". Which part you don't understand? $\endgroup$
    – DeltaIV
    Commented May 9, 2018 at 22:27
  • $\begingroup$ @ DeltaIV Thank you for pointing this out, I was stuck with the definition of a manifold. I have tried to give an answer, will you check whether my answer is correct or not? $\endgroup$ Commented May 10, 2018 at 3:38
  • $\begingroup$ Better to write comments, than to self-answer your question, especially when you end your self-answer with a different question: "What can we learn from this visualization?". Either edit this question or delete it, so that the new question is "What can we learn from this visualization?". Then I will answer it. $\endgroup$
    – DeltaIV
    Commented May 10, 2018 at 8:15
  • 1
    $\begingroup$ @ DeltaIV I have deleted my answer and edited my post, thank you for your advice :-) $\endgroup$ Commented May 10, 2018 at 8:28
  • $\begingroup$ you're welcome. I'm writing your answer, but in meantime please satisfy my curiosity: how did you understand that the digits are means of $p_{\theta}(x|z)$? That's a little dirty secret well-known to VAE practitioners, i.e., that usually you don't really sample from $p_{\theta}(x|z)$ when visualizing! But only show the mean. However, it's rarely (if ever at all) described in the papers, and even though I read the Kingma & Welling paper some time ago, I'm pretty sure it didn't mention it. So, how did you guess? $\endgroup$
    – DeltaIV
    Commented May 10, 2018 at 12:18

1 Answer 1


Your understanding is very good, save for a couple minor details:

  • to create the image we don't sample from a uniform distribution on $I=[0,1]×[0,1]$. We actually define a uniform $20\times20$ grid in $I$, and we compute the values of $\mathbf{z}=(z_1,z_2)$ using the inverse of the Gaussian CDF (note that $z_1$ and $z_2$ are assumed independent, zero-mean and unit-variance here). If you use the term sampling, one would expect the initial values to be drawn randomly from the bivariate uniform distribution on $I$, and we wouldn't have the nice-looking grid.

  • the "learned data manifold" Kingma & Welling are talking of, is the the decoder $p_{\boldsymbol{\theta}}(\mathbf{x}\vert\mathbf{z})$. To be more precise, it's the two vector functions $\boldsymbol{\mu}_{\boldsymbol{\theta}}(\mathbf{z}):\mathbb{R}^2\to\mathbb{R}^{784}$ and $\log{\boldsymbol{\sigma}}_{\boldsymbol{\theta}}^2(\mathbf{z}):\mathbb{R}^2\to\mathbb{R}^{784}$, whose expressions in the case of a Gaussian MLP are:

$$ \begin{align} \boldsymbol{\mu}(\mathbf{z}) & = \mathbf{W}_2(\tanh({\mathbf{W}_1}\mathbf{z} + \mathbf{b}_1))+\mathbf{b}_2 \\ \log{\boldsymbol{\sigma}^2}(\mathbf{z}) & = \mathbf{W}_3(\tanh({\mathbf{W}_1}\mathbf{z} + \mathbf{b}_1))+\mathbf{b}_3 \end{align} $$

However, as you correctly realized, the visualization most likely shows only the 400 values (corresponding to the grid of 400 $\mathbf{z}$ values) of the mean function $\boldsymbol{\mu}_{\boldsymbol{\theta}}(\mathbf{z})$. This is a dirty little VAE trick - when creating visualizations, people often don't really sample from $p_{\boldsymbol{\theta}}(\mathbf{x}\vert\mathbf{z}_i)$, but they just show the means of the distributions, to get better-looking images 1. Since $\boldsymbol{\mu}_{\boldsymbol{\theta}}(\mathbf{z})$ is the output of a neural network, it's a smooth function of $\mathbf{z}$, thus you get the nice-looking, smooth visualization in Figure 4.

What can we learn from this visualization?

We can see what features of the input space (the digits) have been coded into the latent variables, i.e., we can visualize the representation of the input space in terms of latent variables. Here the latent space has dimension 2: in a ideal world, we would like $z_1$ to encode some specific property of the images (for example, which digit we are generating, 0, 1, 2, etc.) and $z_2$ to encode a separate, independent property (for example, the slanting angle with which the digit has been drawn). This would be called a disentangled representation, and it's of interest because if we have one, then we can very easily generate images with some specific properties (for example, woman with long/short hair, man with/without sunglasses, etc.), by "fine-tuning" one or more of the latent variables.

If you use a vanilla VAE, it's very seldom the case that you learn a disentangled representation (you'd typically need $\beta$-VAE or InfoVAE/WassersteinVAE). In most of the cases, each latent variable affects all of the visual properties of the generated images, making it very difficult to perform the "fine-tuning" mentioned before. For example, in our case you can clearly see that neither $z_1$ or $z_2$ alone determine which digit is drawn. If you fix one variable and change only the other one, you can't generate all of the digits: to do so, you must vary both variables at the same time. The degree of slanting, instead, seems to be (very roughly) controlled by $z_2$: values of $z_2$ close to the center of the square seem to lead to more vertical digits, while values closer to the extremes seem to lead to more slanted digits, though the level of slanting is also affected (to a minor degree) by the current value of $z_1$.

Finally, one last (unsurprising) thing that we can learn, is that in the latent space two digits are close together if they're visually similar, not if they're consecutive in base 10. So 3 is close to 8, and also to 2 and 5, but it's not close to 4, 1 or 6, for example. As I said, this was to be expected, but still, it's nice to have a visual confirmation that our VAE trained well.

1Of course if you explicitly say that you're sampling from your generative model, then in that case you're really drawing a sample $\mathbf{x}\sim p_{\boldsymbol{\theta}}(\mathbf{x}\vert\mathbf{z}_i)$, i.e., you're using $\log{\boldsymbol{\sigma}}_{\boldsymbol{\theta}}^2(\mathbf{z}_i)$ too (see for example Figure 5 of the same paper).


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