  Filter banks are a popular tool for the analysis of piecewise smooth signals
such as natural images. Motivated by the empirically observed properties of
scale and detail coefficients of images in the wavelet domain, we propose a
hierarchical deep generative model of piecewise smooth signals that is a
recursion across scales: the low pass scale coefficients at one layer are
obtained by filtering the scale coefficients at the next layer, and adding a
high pass detail innovation obtained by filtering a sparse vector. This
recursion describes a linear dynamic system that is a non-Gaussian Markov
process across scales and is closely related to multilayer-convolutional sparse
coding (ML-CSC) generative model for deep networks, except that our model
allows for deeper architectures, and combines sparse and non-sparse signal
representations. We propose an alternating minimization algorithm for learning
the filters in this hierarchical model given observations at layer zero, e.g.,
natural images. The algorithm alternates between a coefficient-estimation step
and a filter update step. The coefficient update step performs sparse (detail)
and smooth (scale) coding and, when unfolded, leads to a deep neural network.
We use MNIST to demonstrate the representation capabilities of the model, and
its derived features (coefficients) for classification.
