  Wavelet transformation stands as a cornerstone in modern data analysis and
signal processing. Its mathematical essence is an invertible transformation
that discerns slow patterns from fast patterns in the frequency domain, which
repeats at each level. Such an invertible transformation can be learned by a
designed normalizing flow model. With a factor-out scheme resembling the
wavelet downsampling mechanism, a mutually independent prior, and parameter
sharing along the depth of the network, one can train normalizing flow models
to factor-out variables corresponding to fast patterns at different levels,
thus extending linear wavelet transformations to non-linear learnable models.
In this paper, a concrete way of building such flows is given. Then, a
demonstration of the model's ability in lossless compression task, progressive
loading, and super-resolution (upsampling) task. Lastly, an analysis of the
learned model in terms of low-pass/high-pass filters is given.
