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In the area of graph signal processing, a graph is a set of nodes arbitrarily connected by weighted links; a graph signal is a set of scalar values associated with each node; and sampling is the problem of selecting an optimal subset of nodes from which a graph signal can be reconstructed. This paper proposes the use of spatial dithering on the vertex domain of the graph, as a way to conveniently find statistically good sampling sets. This is done establishing that there is a family of good sampling sets characterized on the vertex domain by a maximization of the distance between sampling nodes; in the Fourier domain, these are characterized by spectrums that are dominated by high frequencies referred to as blue-noise. The theoretical connection between blue-noise sampling on graphs and previous results in graph signal processing is also established, explaining the advantages of the proposed approach. Restricting our analysis to undirected and connected graphs, numerical tests are performed in order to compare the effectiveness of blue-noise sampling against other approaches.

In this paper we provide stability results for algebraic neural networks (AlgNNs) based on non commutative algebras. AlgNNs are stacked layered structures with each layer associated to an algebraic signal model (ASM) determined by an algebra, a vector space, and a homomorphism. Signals are modeled as elements of the vector space, filters are elements in the algebra, while the homomorphism provides a realization of the filters as concrete operators. We study the stability of the algebraic filters in non commutative algebras to perturbations on the homomorphisms, and we provide conditions under which stability is guaranteed. We show that the commutativity between shift operators and between shifts and perturbations does not affect the property of an architecture of being stable. This provides an answer to the question of whether shift invariance was a necessary attribute of convolutional architectures to guarantee stability. Additionally, we show that although the frequency responses of filters in non commutative algebras exhibit substantial differences with respect to filters in commutative algebras, their derivatives for stable filters have a similar behavior.

Graph neural networks (GNNs) have been used effectively in different applications involving the processing of signals on irregular structures modeled by graphs. Relying on the use of shift-invariant graph filters, GNNs extend the operation of convolution to graphs. However, the operations of pooling and sampling are still not clearly defined and the approaches proposed in the literature either modify the graph structure in a way that does not preserve its spectral properties, or require defining a policy for selecting which nodes to keep. In this work, we propose a new strategy for pooling and sampling on GNNs using graphons which preserves the spectral properties of the graph. To do so, we consider the graph layers in a GNN as elements of a sequence of graphs that converge to a graphon. In this way we have no ambiguity in the node labeling when mapping signals from one layer to the other and a spectral representation that is consistent throughout the layers. We evaluate this strategy in a synthetic and a real-world numerical experiment where we show that graphon pooling GNNs are less prone to overfitting and improve upon other pooling techniques, especially when the dimensionality reduction ratios between layers is large.

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