Note that we have \( g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{q} \) The decoder and the output of the network \( \tilde{\mathbf{x}}_{i} \) can be written then as a second generic function of the latent features
$$ \tilde{\mathbf{x}}_{i} = f\left(\mathbf{h}_{i}\right) = f\left(g\left(\mathbf{x}_{i}\right)\right), $$where \( \tilde{\mathbf{x}}_{i}\mathbf{\in }\mathbb{R}^{n} \).
Training an autoencoder simply means finding the functions \( g(\cdot) \) and \( f(\cdot) \) that satisfy
$$ \textrm{arg}\min_{f,g}< \left[\Delta (\mathbf{x}_{i}, f(g\left(\mathbf{x}_{i}\right))\right]>. $$