We will build a flexible and tractable bijective function by stacking a sequence of simple bijections.In each simple bijection, part of the input vector is updated using a function which is simple to invert,but which depends on the remainder of the input vector in a complex way. We refer to each of thesesimple bijections as an affine coupling layer. Given a D dimensional input x and d < D, the outputy of an affine coupling layer follows the equations

The algorithmic complexity of jointly sampling and com-puting the log-det-Jacobian terms of the inference modelscales as O(LN 2) + O(KD), where L is the number ofdeterministic layers used to map the data to the parame-ters of the flow, N is the average hidden layer size, K isthe flow-length and D is the dimension of the latent vari-ables. Thus the overall algorithm is at most quadratic mak-ing the overall approach competitive with other large-scalesystems used in practice.