  A fundamental problem in signal processing is to denoise a signal. While
there are many well-performing methods for denoising signals defined on regular
supports, such as images defined on two-dimensional grids of pixels, many
important classes of signals are defined over irregular domains such as graphs.
This paper introduces two untrained graph neural network architectures for
graph signal denoising, provides theoretical guarantees for their denoising
capabilities in a simple setup, and numerically validates the theoretical
results in more general scenarios. The two architectures differ on how they
incorporate the information encoded in the graph, with one relying on graph
convolutions and the other employing graph upsampling operators based on
hierarchical clustering. Each architecture implements a different prior over
the targeted signals. To numerically illustrate the validity of the theoretical
results and to compare the performance of the proposed architectures with other
denoising alternatives, we present several experimental results with real and
synthetic datasets.
