Evaluating an autoencoder: possible approaches? Literature suggests that Antoencoders can be effective in dimensionality reduction, like PCA. PCA can be evaluated based on the variance of each principal component generated. How to do the same for autoencoder? 
One way is to that we can reconstruct the input from the encoded representation from the autoencoder and can check the reconstruction error. But can we check the variance, like in PCA?
 A: Autoencoders are data-specific, which means that they will only be able to compress data similar to what they have been trained on. So, the usefulness of features that have been learned by hidden layers could be used for evaluating the efficacy of the method. 
For this reason, I think one way to evaluate an autoencoder efficacy in dimensionality reduction is cutting the output of the middle hidden layer and compare the accuracy/performance of your desired algorithm by this reduced data rather than using original data.
A: 
PCA can be evaluated based on the variance of each principal
  component generated.

Actually it measures the same thing as reconstruction error, but in a different way.
Let's fix $k$ and put this more precisely
$X$ - data matrix, $X'$ - best rank-$k$ approximation (rank-$k$ PCA). 
To calculate $X'$ you need to do SVD and then only take $k$ singular vectors with biggest singular values.
Eckhart-Young theorem then tells us that this $X'$ also minimizes Frobenius norm of $X-X'$. Frobenius norm is defined as 
$$\|X-X'\|_F = \sqrt{\sum_{n,m}(X_{n,m} - X'_{n,m})^2}$$
So
$$\|X-X'\|_F^2 =\sum_{n,m}(X_{n,m} - X'_{n,m})^2 = \sum_{n}\|X_n - X'_n\|^2$$
The last expression is the reconstruction error.
Back to evaluating PCA
The above fragment just says that for fixed rank $k$ we know how to find reconstruction error. I think you mentioned the fact that you can also easily evaluate how the reconstruction changes when you vary the $k$. 
This is where evaluating autoencoder and PCA diverges: the latent variables of autoencoder aren't guaranteed to be orthogonal. That means you can't decompose reconstruction error as in the case of PCA. Also since the coding/decoding in autoencoders is nonlinear, you don't know how the variance in the latent space translates to variance in input space.
