Normalizing Flows
Exact densities through invertible, differentiable transformations.
Motivation
Normalizing flows build a complex distribution by pushing a simple base density through a sequence of invertible maps. Unlike VAEs or GANs, they give an exact likelihood — the price is that every layer must be invertible with a tractable Jacobian determinant.
Method
The change-of-variables formula converts a base density into the model density. Architectures such as coupling layers and autoregressive flows are engineered so the Jacobian is triangular, making its determinant cheap to compute.
Mathematical core
In the continuous limit a flow becomes a Neural ODE: the transformation is the solution of an ordinary differential equation, and the log-density evolves by the instantaneous change-of-variables (an integral of the vector field's divergence). This is exactly the homeomorphism-interpolation perspective I care about — flows deform space continuously while preserving its topology.