  Despite a lack of theoretical understanding, deep neural networks have
achieved unparalleled performance in a wide range of applications. On the other
hand, shallow representation learning with component analysis is associated
with rich intuition and theory, but smaller capacity often limits its
usefulness. To bridge this gap, we introduce Deep Component Analysis (DeepCA),
an expressive multilayer model formulation that enforces hierarchical structure
through constraints on latent variables in each layer. For inference, we
propose a differentiable optimization algorithm implemented using recurrent
Alternating Direction Neural Networks (ADNNs) that enable parameter learning
using standard backpropagation. By interpreting feed-forward networks as
single-iteration approximations of inference in our model, we provide both a
novel theoretical perspective for understanding them and a practical technique
for constraining predictions with prior knowledge. Experimentally, we
demonstrate performance improvements on a variety of tasks, including
single-image depth prediction with sparse output constraints.
