Variational autoencoders are often associated with the autoencoder model because of its architectural affinity, but with significant differences in the goal and mathematical formulation. Variational autoencoders are probabilistic generative models that require neural networks as only a part of their overall structure, as e.g. in VQ-VAE. The neural network components are typically referred to as the encoder and decoder for the first and second component respectively. The first neural network maps the input variable to a latent space that corresponds to the parameters of a variational distribution. In this way, the encoder can produce multiple different samples that all come from the same distribution. The decoder has the opposite function, which is to map from the latent space to the input space, in order to produce or generate data points. Both networks are typically trained together with the usage of the reparameterization trick, although the variance of the noise model can be learned separately.
A variational autoencoder is a generative model with a prior and noise distribution respectively. Usually such models are trained using the Expectation-Maximization meta-algorithm (e.g. probabilistic PCA, (spike & slab) sparse coding). Such a scheme optimizes a lower bound of the data likelihood, which is usually intractable, and in doing so requires the discovery of q-distributions, or variational posteriors. These q distributions are normally parameterized for each individual data point in a separate optimization process. However, variational autoencoders use a neural network as an amortized approach to jointly optimize across data points. This neural network takes as input the data points themselves, and outputs parameters for the variational distribution. As it maps from a known input space to the low-dimensional latent space, it is called the encoder.
The decoder is the second neural network of this model. It is a function that maps from the latent space to the input space, e.g. as the means of the noise distribution. It is possible to use another neural network that maps to the variance, however this can be omitted for simplicity. In such a case, the variance can be optimized with gradient descent.
To optimize this model, one needs to know two terms: the "reconstruction error", and the Kullback–Leibler divergence. Both terms are derived from the free energy expression of the probabilistic model, and therefore differ depending on the noise distribution and the assumed prior of the data. The KL-D from the free energy expression maximizes the probability mass of the q distribution that overlaps with the p distribution, which unfortunately can result in mode-seeking behaviour. The "reconstruction" term is the remainder of the free energy expression, and requires a sampling approximation to compute its expectation value.
The basic scheme of a variational autoencoder. The model receives as input. The encoder compresses it into the latent space. The decoder receives as input the information sampled from the latent space and produces as similar as possible to .
From the point of view of probabilistic modelling, one wants to maximize the likelihood of the data by their chosen parameterized probability distribution . This distribution is usually chosen to be a Gaussian which is parameterized by and respectively, and as a member of the exponential family it is easy to work with as a noise distribution. Simple distributions are easy enough to maximize, however distributions where a prior is assumed over the latents results in intractable integrals. Let us find via marginalizing over .
where represents the joint distribution under of the observable data and its latent representation or encoding . According to the chain rule, the equation can be rewritten as
In the vanilla variational autoencoder, is usually taken to be a finite-dimensional vector of real numbers, and to be a Gaussian distribution. Then is a mixture of Gaussian distributions.
It is now possible to define the set of the relationships between the input data and its latent representation as
Unfortunately, the computation of is expensive and in most cases intractable. To speed up the calculus to make it feasible, it is necessary to introduce a further function to approximate the posterior distribution as
with defined as the set of real values that parametrize . This is sometimes called amortized inference, since by "investing" in finding a good , one can later infer from quickly without doing any integrals.
In this way, the problem is of finding a good probabilistic autoencoder, in which the conditional likelihood distribution is computed by the probabilistic decoder, and the approximated posterior distribution is computed by the probabilistic encoder.
As in every deep learning problem, it is necessary to define a differentiable loss function in order to update the network weights through backpropagation.
For variational autoencoders, the idea is to jointly optimize the generative model parameters to reduce the reconstruction error between the input and the output, and to make as close as possible to . As reconstruction loss, mean squared error and cross entropy are often used.
As distance loss between the two distributions the reverse Kullback–Leibler divergence is a good choice to squeeze under .
is equivalent to simultaneously maximizing and minimizing . That is, maximizing the log-likelihood of the observed data, and minimizing the divergence of the approximate posterior from the exact posterior .
For a more detailed derivation and more interpretations of ELBO and its maximization, see its main page.
The scheme of the reparameterization trick. The randomness variable is injected into the latent space as external input. In this way, it is possible to backpropagate the gradient without involving stochastic variable during the update.
does not allow one to put the inside the expectation, since appears in the probability distribution itself. The reparameterization trick (also known as stochastic backpropagation) bypasses this difficulty.
The most important example is when is normally distributed, as .
The scheme of a variational autoencoder after the reparameterization trick.
Since we reparametrized , we need to find . Let by the probability density function for , then
where is the Jacobian matrix of with respect to . Since , this is
Many variational autoencoders applications and extensions have been used to adapt the architecture to other domains and improve its performance.
-VAE is an implementation with a weighted Kullback–Leibler divergence term to automatically discover and interpret factorised latent representations. With this implementation, it is possible to force manifold disentanglement for values greater than one. This architecture can discover disentangled latent factors without supervision.
The conditional VAE (CVAE), inserts label information in the latent space to force a deterministic constrained representation of the learned data.
Some structures directly deal with the quality of the generated samples or implement more than one latent space to further improve the representation learning.
^Dilokthanakul, Nat; Mediano, Pedro A. M.; Garnelo, Marta; Lee, Matthew C. H.; Salimbeni, Hugh; Arulkumaran, Kai; Shanahan, Murray (2017-01-13). "Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders". arXiv:1611.02648 [cs.LG].