自动编码器是特定于数据的，这意味着它们只能压缩与训练过的数据相似的数据。因此，隐藏层学习到的特征的有用性可用于评估该方法的有效性。

出于这个原因，我认为评估自动编码器在降维方面的功效的一种方法是削减中间隐藏层的输出，并通过这些减少的数据而不是使用原始数据来比较所需算法的准确性/性能。

可以根据生成的每个主成分的方差来评估 PCA。

实际上，它测量与重建误差相同的东西，但以不同的方式。

让我们修正并更准确地说$k$

$X$ - 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.