Our main contribution and also our departure from the earlier RealNVP work is the addition of a reversible 1x1 convolution, as well as removing other components, simplifying the architecture overall.

The RealNVP architecture consists of sequences of two types of layers: layers with checkboard masking, and layers with channel-wise masking. We remove the layers with checkerboard masking, simplifying the architecture. The layers with channel-wise masking perform the equivalent of a repetition of the following steps:

1. Permute the inputs by reversing their ordering across the channel dimension.
2. Split the input into two parts, A and B, down the middle of the feature dimension.
3. Feed A into a shallow convolutional neural network. Linearly transform B according to the output of the neural network.
4. Concatenate A and B.

By chaining these layers, A updates B, then B updates A, then A updates B, etc. This bipartite flow of information is clearly quite rigid. We found that model performance improves by changing the reverse permutation of step (1) to a (fixed) shuffling permutation.

Taking this a step further, we can also learn the optimal permutation. Learning a permutation matrix is a discrete optimization that is not amendable to gradient ascent. But because the permutation operation is just a special case of a linear transformation with a square matrix, we can make this work with convolutional neural networks, as permuting the channels is equivalent to a 1x1 convolution operation with an equal number of input and output channels. So we replace the fixed permutation with learned 1x1 convolution operations. The weights of the 1x1 convolution are initialized as a random rotation matrix. As we show in the figure below, this operation leads to significant modeling improvements. We’ve also shown that the computations involved in optimizing the objective function can be done efficiently through a LU decomposition of the weights.