Sampling and Uniqueness Sets in Graphon Signal Processing

In this work, we study the properties of sampling sets on families of large graphs by leveraging the theory of graphons and graph limits. To this end, we extend to graphon signals the notion of removable and uniqueness sets, which was developed originally for the analysis of signals on graphs. We state the formal definition of a $\Lambda-$removable set and conditions under which a bandlimited graphon signal can be represented in a unique way when its samples are obtained from the complement of a given $\Lambda-$removable set in the graphon. By leveraging such results we show that graphon representations of graphs and graph signals can be used as a common framework to compare sampling sets between graphs with different numbers of nodes and edges, and different node labelings. Additionally, given a sequence of graphs that converges to a graphon, we show that the sequences of sampling sets whose graphon representation is identical in $[0,1]$ are convergent as well. We exploit the convergence results to provide an algorithm that obtains approximately close to optimal sampling sets. Performing a set of numerical experiments, we evaluate the quality of these sampling sets. Our results open the door for the efficient computation of optimal sampling sets in graphs of large size

Lie Group Algebra Convolutional Filters

In this paper we propose a framework to leverage Lie group symmetries on arbitrary spaces exploiting \textit{algebraic signal processing} (ASP). We show that traditional group convolutions are one particular instantiation of a more general Lie group algebra homomorphism associated to an algebraic signal model rooted in the Lie group algebra $L^{1}(G)$ for given Lie group $G$. Exploiting this fact, we decouple the discretization of the Lie group convolution elucidating two separate sampling instances: the filter and the signal. To discretize the filters, we exploit the exponential map that links a Lie group with its associated Lie algebra. We show that the discrete Lie group filter learned from the data determines a unique filter in $L^{1}(G)$, and we show how this uniqueness of representation is defined by the bandwidth of the filter given a spectral representation. We also derive error bounds for the approximations of the filters in $L^{1}(G)$ with respect to its learned discrete representations. The proposed framework allows the processing of signals on spaces of arbitrary dimension and where the actions of some elements of the group are not necessarily well defined. Finally, we show that multigraph convolutional signal models come as the natural discrete realization of Lie group signal processing models, and we use this connection to establish stability results for Lie group algebra filters. To evaluate numerically our results, we build neural networks with these filters and we apply them in multiple datasets, including a knot classification problem

Convolutional Learning on Multigraphs

Graph convolutional learning has led to many exciting discoveries in diverse areas. However, in some applications, traditional graphs are insufficient to capture the structure and intricacies of the data. In such scenarios, multigraphs arise naturally as discrete structures in which complex dynamics can be embedded. In this paper, we develop convolutional information processing on multigraphs and introduce convolutional multigraph neural networks (MGNNs). To capture the complex dynamics of information diffusion within and across each of the multigraph's classes of edges, we formalize a convolutional signal processing model, defining the notions of signals, filtering, and frequency representations on multigraphs. Leveraging this model, we develop a multigraph learning architecture, including a sampling procedure to reduce computational complexity. The introduced architecture is applied towards optimal wireless resource allocation and a hate speech localization task, offering improved performance over traditional graph neural networks

Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations

In this paper we discuss the results recently published in~[1] about algebraic signal models (ASMs) based on non commutative algebras and their use in convolutional neural networks. Relying on the general tools from algebraic signal processing (ASP), we study the filtering and stability properties of non commutative convolutional filters. We show how non commutative filters can be stable to small perturbations on the space of operators. We also show that although the spectral components of the Fourier representation in a non commutative signal model are associated to spaces of dimension larger than one, there is a trade-off between stability and selectivity similar to that observed for commutative models. Our results have direct implications for group neural networks, multigraph neural networks and quaternion neural networks, among other non commutative architectures. We conclude by corroborating these results through numerical experiments

Mejoramiento de la resolución espectral de imágenes hiperespectrales, por medio de un sistema óptico compresivo de múltiple-apertura

El sistema de sensado de imágenes espectrales, basado en la apertura codificada y de única toma (CASSI), captura la información espacial y espectral de una escena; mediante mediciones codificadas aleatorias capturadas en un sensor 2D. Un algoritmo basado en la teoría de sensado compresivo (CS), es utilizado para recuperar la escena tridimensional original a partir de las mediciones aleatorias capturadas. La calidad de reconstrucción de la escena depende exclusivamente, de la matriz de sensado del CASSI, la cual es determinada por la estructura de las aperturas codificadas que son utilizadas.En este artículo, se propone una generalización del sistema CASSI por medio del desarrollo de un sistema óptico multi-apertura, que permite el mejoramiento de la resolución espectral. En el sistema propuesto, un par de aperturas codificadas de alta resolución es introducido en el sistema CASSI, permitiendo así, la codificación tanto espacial como espectral de la imagen hiperespectral. Este enfoque permite la reconstrucción de cubos de datos hiperespectrales, donde el número de las bandas espectrales se aumenta significativamente respecto al original, y la calidad espacial es mejorada en gran medida. Así mismo, los experimentos simulados muestran mejoramiento en la relación de pico-de-señal-a-ruido (PSNR), junto con un mejor ajuste en las firmas espectrales reconstrui-das sobre los datos espectrales originales.The Coded Aperture Snapshot Spectral Imaging (CASSI) system captures the three-dimensional (3D) spatio-spectral information of a scene using a set of two-dimensional (2D) random-coded Focal Plane Array (FPA) measurements. A compressive sensing reconstruc-tion algorithm is then used to recover the underlying spatio-spectral 3D data cube. The quality of the reconstructed spectral images depends exclusively on the CASSI sensing matrix, which is determined by the structure of a set of random coded apertures. In this paper, the CASSI system is generalized by developing a multiple-aperture optical imaging system such that spectral resolution en-hancement is attainable. In the proposed system, a pair of high-resolution coded apertures is introduced into the CASSI system, allow-ing it to encode both spatial and spectral characteristics of the hyperspectral image. This approach allows the reconstruction of super-resolved hyperspectral data cubes, where the number of spectral bands is significantly increased and the quality in the spatial domain is greatly improved. Extensively simulated experiments show a gain in the peak-signal-to-noise ratio (PSNR), along with a better fit of the reconstructed spectral signatures to the original spectral data

Stability of Aggregation Graph Neural Networks

In this paper we study the stability properties of aggregation graph neural networks (Agg-GNNs) considering perturbations of the underlying graph. An Agg-GNN is a hybrid architecture where information is defined on the nodes of a graph, but it is processed block-wise by Euclidean CNNs on the nodes after several diffusions on the graph shift operator. We derive stability bounds for the mapping operator associated to a generic Agg-GNN, and we specify conditions under which such operators can be stable to deformations. We prove that the stability bounds are defined by the properties of the filters in the first layer of the CNN that acts on each node. Additionally, we show that there is a close relationship between the number of aggregations, the filter's selectivity, and the size of the stability constants. We also conclude that in Agg-GNNs the selectivity of the mapping operators is tied to the properties of the filters only in the first layer of the CNN stage. This shows a substantial difference with respect to the stability properties of selection GNNs, where the selectivity of the filters in all layers is constrained by their stability. We provide numerical evidence corroborating the results derived, testing the behavior of Agg-GNNs in real life application scenarios considering perturbations of different magnitude