442 3.3. Deep Clustering Framework 3.3.1. Variational Autoencoders (VAE) A Variational Autoencoder (VAE) is a generative probabilistic model that extends the traditional autoencoder architecture by introducing a structured and regularized latent space. Unlike deterministic autoencoders, which learn a fixed latent representation, a VAE assumes that the latent variables follow a probability distribution. Given a input (, the encoder does not map it to a single talent vector <, but instead learns the parameters of a conditional probability distribution. S +(-|D)=ΝUV(D),X2(D)Y (2) Where 7(() and 8(() are the mean and standard deviation vectors learned by the encoder network, and © denotes the encoder parameters. A latent sample is obtained using the reparameterization trick: -=V(D)+X(D)∙[, [~Ν(0, I) (3) This formulation enables gradient-based optimization while maintaining stochastic sampling in the latent space. The key contribution of the variational component is the imposition of a prior distribution over the latent variables, typically: ](-)=Ν(0, I) (4) This constraint regularizes the latent space, forcing the encoded representations to follow a continuous and structured manifold. As a result, the latent space becomes smooth and continuous, similar inputs are mapped to nearby latent regions, overfitting is reduced, the representation becomes more robust to noise variability. This probabilistic regularization is particularly relevant in industrial process data, where nonlinear interactions and operational variability generate complex high-dimensional patterns. Figure 18. Variational Autoencoder estructure.
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