View source for Normalizing flow

A '''normalizing flow''' is a generative model that learns an invertible, differentiable transformation between a simple base distribution (typically a Gaussian) and a complex data distribution. Unlike a [[Variational Autoencoder|variational autoencoder]], which approximates the posterior with a fixed parametric family, a normalizing flow constructs the posterior through composition of simple, invertible transformations, each of which has a tractable Jacobian determinant.

The key property of a normalizing flow is exact likelihood evaluation. Because the transformation is invertible, the change-of-variables formula applies: the log-likelihood of the data equals the log-likelihood of the base distribution plus the sum of the log-determinants of the Jacobian matrices at each transformation step. This exactness comes at a cost: the transformations must be carefully designed to preserve invertibility and computational tractability. Early flows used affine couplings and autoregressive structures; later work introduced '''[[Residual flow|residual flows]]''', continuous normalizing flows, and '''[[Neural ordinary differential equation|neural ordinary differential equations]]'''.

Normalizing flows occupy a middle ground in the generative modeling landscape. They are more expressive than a simple Gaussian prior but less flexible than a [[Diffusion model|diffusion model]], which does not require invertibility. The tension between expressiveness and tractability is the central design problem of the field.

[[Category:Machine Learning]]
[[Category:Mathematics]]
[[Category:Computer Science]]