Normalizing Flows: Chaining Invertible Maps

On the left, z zero is a standard Gaussian in the plane: a round green bell on the base plate. To its right stand two transparent flow plates.

The first map stretches and shears the plane, bending the green bell into the amber shape on the middle plate. The second map stretches the other coordinate, folding the density into a blue crescent.

Each layer is an invertible map with a triangular Jacobian, so its determinant is the product of the diagonal. The data density equals the base Gaussian density divided by both absolute determinants: where a layer stretches area, the density thins, and where it squeezes area, the density rises.

Inverting the flow evaluates likelihoods in reverse: take any data point x on the blue plate, pull it back layer by layer, and read its exact log density.

Your turn. Drag the orange sample point on the blue plate, and adjust the coupling slider. Find a spot where the flow squeezes area enough that the log density of x beats the base log density by at least one half.

By chaining simple invertible layers, normalizing flows transform a simple Gaussian into complex data distributions with exact density and sampling.