If an autoencoder is trying to solve a regression problem, the most common choice as a loss function is the Mean Square Error
$$ L_{\textrm{MSE}} = \textrm{MSE} = \frac{1}{n}\sum_{i = 1}^{n}\left\vert\vert\mathbf{x}_{i}-\tilde{\mathbf{x}}_{i}\right\vert\vert^{2}_2. $$