Can you sample from an autoencoder? You can do gibbs sampling from a deep belief network (with RBMs as modules) however, can you sample from an autoencoder?
 A: Yes, it is possible. After you train it, you have $$z = f(X^Tw + b)$$ as your encoder.
You can now record the empirical distribution of $P(z_i)$ where $i \in d'$ is a feature index in the hidden layer space $d' \ll d$. 
You can do this by inputting your original data set and measuring for example the frequency of values one encoded feature $z_i$ (let's say independently of the others, for starters).
Once you have some ideas about the distrubtion of $P(z)$ then you could randomly add noise and select different combinations of values of $z_i$ to generate new data using the decoder. 
$$x' = f((z_k + \epsilon_k)^Tw + b)$$
Where $k$ is a combination of latent values with randomly injected noise $\epsilon$.
You might also want to look into Variational Auto-Encoders (VAE), check out this reddit thread.
A: The system proposed by Florin, which is the right way to try to sample from an autoencoder, only works if you have a good estimate of the distribution of features $P(z)$.  However, this doesn't scale well since in most cases the features have structural information (they are correlated or anti-correlated) and so you cannot just model them independently.  Also their distributions are usually far from Gaussian so it's not easy to model their variation from the mean.
If you want to sample from your model, you should just use a generative model.
