What is: Affine Coupling?

Affine Coupling is a method for implementing a normalizing flow (where we stack a sequence of invertible bijective transformation functions). Affine coupling is one of these bijective transformation functions. Specifically, it is an example of a reversible transformation where the forward function, the reverse function and the log-determinant are computationally efficient. For the forward function, we split the input dimension into two parts:

$\mathbf{x}\_{a}, \mathbf{x}\_{b} = \text{split}\left(\mathbf{x}\right)$

The second part stays the same $\mathbf{x}\_{b} = \mathbf{y}\_{b}$, while the first part $\mathbf{x}\_{a}$ undergoes an affine transformation, where the parameters for this transformation are learnt using the second part $\mathbf{x}\_{b}$ being put through a neural network. Together we have:

$\left(\log{\mathbf{s}, \mathbf{t}}\right) = \text{NN}\left(\mathbf{x}\_{b}\right)$

$\mathbf{s} = \exp\left(\log{\mathbf{s}}\right)$

$\mathbf{y}\_{a} = \mathbf{s} \odot \mathbf{x}\_{a} + \mathbf{t}$

$\mathbf{y}\_{b} = \mathbf{x}\_{b}$

$\mathbf{y} = \text{concat}\left(\mathbf{y}\_{a}, \mathbf{y}\_{b}\right)$

Image: GLOW