  The graph neural network (GNN) has demonstrated its superior performance in
various applications. The working mechanism behind it, however, remains
mysterious. GNN models are designed to learn effective representations for
graph-structured data, which intrinsically coincides with the principle of
graph signal denoising (GSD). Algorithm unrolling, a "learning to optimize"
technique, has gained increasing attention due to its prospects in building
efficient and interpretable neural network architectures. In this paper, we
introduce a class of unrolled networks built based on truncated optimization
algorithms (e.g., gradient descent and proximal gradient descent) for GSD
problems. They are shown to be tightly connected to many popular GNN models in
that the forward propagations in these GNNs are in fact unrolled networks
serving specific GSDs. Besides, the training process of a GNN model can be seen
as solving a bilevel optimization problem with a GSD problem at the lower
level. Such a connection brings a fresh view of GNNs, as we could try to
understand their practical capabilities from their GSD counterparts, and it can
also motivate designing new GNN models. Based on the algorithm unrolling
perspective, an expressive model named UGDGNN, i.e., unrolled gradient descent
GNN, is further proposed which inherits appealing theoretical properties.
Extensive numerical simulations on seven benchmark datasets demonstrate that
UGDGNN can achieve superior or competitive performance over the
state-of-the-art models.
