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The encoder maps each input x to a distribution qϕ(z|x) in the latent space (often Gaussian), not to a single point.
In a variational autoencoder (VAE), the encoder and decoder form a probabilistic autoencoding pipeline around a latent space. The encoder takes an input data point x (e.g., an image) and maps it into a probability distribution over latent variables z, rather than a single latent vector. This latent distribution is typically parameterized (commonly as a Gaussian) by neural networks that output parameters such as the mean and (optionally) variance. The latent space is the probabilistic bottleneck: sampling from the encoder’s distribution produces a latent sample z. The decoder then maps from this latent space back to the input space, producing a reconstructed output x′ that is probabilistically modeled to be as similar as possible to the original x. Training is done jointly by optimizing an objective that balances (1) reconstruction quality (how close x′ is to x) and (2) regularization that aligns the encoder’s approximate posterior with a chosen prior over latents. This is commonly handled via the evidence lower bound (ELBO), and gradients through sampling are enabled using the reparameterization trick.
The encoder maps each input x to a distribution qϕ(z|x) in the latent space (often Gaussian), not to a single point.
The latent space is sampled from qϕ(z|x); the decoder maps sampled z to a distribution over reconstructions x′.
Training optimizes reconstruction error plus a divergence term (often expressed via the ELBO) to regularize the latent distribution toward a prior, using the reparameterization trick for backpropagation.
A neural network that maps input x to parameters of a latent distribution qϕ(z|x).
The probabilistic representation space of latent variables z from which the model samples during generation.
A neural network that maps sampled latent variables z back to the input space, producing a reconstruction distribution for x′.
The encoder’s approximate distribution over latents given x, used to represent uncertainty in the latent encoding.
A chosen distribution over latent variables that the learned latent distribution is regularized to match.
A method that rewrites sampling from qϕ(z|x) using an external noise variable so gradients can be backpropagated through the stochastic latent sampling.
The training objective that lower-bounds the log-likelihood of the data and combines reconstruction and latent regularization terms.
“Can you explain what "The encoder maps each input x to a distribution qϕ(z|x) in the latent space (often Gaussian), not to a single point." means in simple terms?”