  Graph convolutional neural networks (GCNNs) are popular deep learning
architectures that, upon replacing regular convolutions with graph filters
(GFs), generalize CNNs to irregular domains. However, classical GFs are prone
to numerical errors since they consist of high-order polynomials. This problem
is aggravated when several filters are applied in cascade, limiting the
practical depth of GCNNs. To tackle this issue, we present the neighborhood
graph filters (NGFs), a family of GFs that replaces the powers of the graph
shift operator with $k$-hop neighborhood adjacency matrices. NGFs help to
alleviate the numerical issues of traditional GFs, allow for the design of
deeper GCNNs, and enhance the robustness to errors in the topology of the
graph. To illustrate the advantage over traditional GFs in practical
applications, we use NGFs in the design of deep neighborhood GCNNs to solve
graph signal denoising and node classification problems over both synthetic and
real-world data.
