The reconstruction error (RE) is a metric that gives you an indication of how good (or bad) the autoencoder was able to reconstruct the input observation \( \mathbf{x}_{i} \). The most typical RE used is the MSE
$$ \textrm{RE}\equiv \textrm{MSE} = \frac{1}{n}\sum_{i = 1}^{n}\left\vert\vert\mathbf{x}_{i}-\tilde{\mathbf{x}}_{i}\right\vert\vert^{2}_2. $$