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In this way, it is possible to backpropagate the gradient without involving stochastic variable during the update. The scheme of a variational autoencoder after the reparameterization trick. In Variational Autoencoders (VAEs), the VAE objective function, known as the Evidence Lower Bound (ELBO), is given by:
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Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as ...
Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the ...
Many variational autoencoders applications and extensions have been used to adapt the architecture to other domains and improve its performance. β {\displaystyle \beta } -VAE is an implementation with a weighted Kullback–Leibler divergence term to automatically discover and interpret factorised latent representations.
There are various equivalent formalisms, including Markov chains, denoising diffusion probabilistic models, noise conditioned score networks, and stochastic differential equations. [3] They are typically trained using variational inference. [4] The model responsible for denoising is typically called its "backbone".
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.
In probability theory, a McKean–Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients of the diffusion depend on the distribution of the solution itself. [1] [2] The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966. [3]