Extracting and Composing Robust Features with Denoising Autoencoders

The Denoising Autoencoder is a modification to the basic autoencoder.
The denoising autoencoder partially corrupts the initial input vector $\mathbf{x}$ in a stochastic manner:
$\tilde{\mathbf{x}}^{(i)} \sim \mathcal{M}_\mathcal{D}(\tilde{\mathbf{x}}^{(i)} \vert \mathbf{x}^{(i)})$
The model is then trained to recover the original input by minimizing:
$L_\text{DAE}(\theta, \phi) = \frac{1}{n} \sum_{i=1}^n (\mathbf{x}^{(i)} - f_\theta(g_\phi(\tilde{\mathbf{x}}^{(i)})))^2$
where $\mathcal{M}_\mathcal{D}$ defines the mapping from the true data samples to the noisy or corrupted ones.