  In graph signal processing, one of the most important subjects is the study
of filters, i.e., linear transformations that capture relations between graph
signals. One of the most important families of filters is the space of shift
invariant filters, defined as transformations commute with a preferred graph
shift operator. Shift invariant filters have a wide range of applications in
graph signal processing and graph neural networks. A shift invariant filter can
be interpreted geometrically as an information aggregation procedure (from
local neighborhood), and can be computed easily using matrix multiplication.
However, there are still drawbacks to using solely shift invariant filters in
applications, such as being restrictively homogeneous. In this paper, we
generalize shift invariant filters by introducing and studying semi shift
invariant filters. We give an application of semi shift invariant filters with
a new signal processing framework, the subgraph signal processing. Moreover, we
also demonstrate how semi shift invariant filters can be used in graph neural
networks.
