Variational auto encoders map input onto a distribution by replacing the bottleneck vector with separate vectors for mean and standard deviation. The training process involves a reconstruction loss and KL divergence to ensure the learned distribution is close to a standard Gaussian. To address the issue of gradients not passing through a sampling node, the reparameterization trick is used, where the latent vector is split into fixed parameters (mu and sigma) and a stochastic part (epsilon) for which gradients don't need to be computed, enabling end-to-end training.

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