Stefania Sardellitti

Lorenzo Giusti, Claudio Battiloro, Lucia Testa et al.

Since their introduction, graph attention networks achieved outstanding results in graph representation learning tasks. However, these networks consider only pairwise relationships among nodes and then they are not able to fully exploit higher-order interactions present in many real world data-sets. In this paper, we introduce Cell Attention Networks (CANs), a neural architecture operating on data defined over the vertices of a graph, representing the graph as the 1-skeleton of a cell complex introduced to capture higher order interactions. In particular, we exploit the lower and upper neighborhoods, as encoded in the cell complex, to design two independent masked self-attention mechanisms, thus generalizing the conventional graph attention strategy. The approach used in CANs is hierarchical and it incorporates the following steps: i) a lifting algorithm that learns {\it edge features} from {\it node features}; ii) a cell attention mechanism to find the optimal combination of edge features over both lower and upper neighbors; iii) a hierarchical {\it edge pooling} mechanism to extract a compact meaningful set of features. The experimental results show that CAN is a low complexity strategy that compares favorably with state of the art results on graph-based learning tasks.

Claudio Battiloro, Lucia Testa, Lorenzo Giusti et al.

Graph machine learning methods excel at leveraging pairwise relations present in the data. However, graphs are unable to fully capture the multi-way interactions inherent in many complex systems. An effective way to incorporate them is to model the data on higher-order combinatorial topological spaces, such as Simplicial Complexes (SCs) or Cell Complexes. For this reason, we introduce Generalized Simplicial Attention Neural Networks (GSANs), novel neural network architectures designed to process data living on simplicial complexes using masked self-attentional layers. Hinging on topological signal processing principles, we devise a series of principled self-attention mechanisms able to process data associated with simplices of various order, such as nodes, edges, triangles, and beyond. These schemes learn how to combine data associated with neighbor simplices of consecutive order in a task-oriented fashion, leveraging on the simplicial Dirac operator and its Dirac decomposition. We also prove that GSAN satisfies two fundamental properties: permutation equivariance and simplicial-awareness. Finally, we illustrate how our approach compares favorably with other simplicial and graph models when applied to several (inductive and transductive) tasks such as trajectory prediction, missing data imputation, graph classification, and simplex prediction.

Gabriele D'Acunto, Paolo Di Lorenzo, Francesco Bonchi et al.

Despite the large research effort devoted to learning dependencies between time series, the state of the art still faces a major limitation: existing methods learn partial correlations but fail to discriminate across distinct frequency bands. Motivated by many applications in which this differentiation is pivotal, we overcome this limitation by learning a block-sparse, frequency-dependent, partial correlation graph, in which layers correspond to different frequency bands, and partial correlations can occur over just a few layers. To this aim, we formulate and solve two nonconvex learning problems: the first has a closed-form solution and is suitable when there is prior knowledge about the number of partial correlations; the second hinges on an iterative solution based on successive convex approximation, and is effective for the general case where no prior knowledge is available. Numerical results on synthetic data show that the proposed methods outperform the current state of the art. Finally, the analysis of financial time series confirms that partial correlations exist only within a few frequency bands, underscoring how our methods enable the gaining of valuable insights that would be undetected without discriminating along the frequency domain.

Lucia Testa, Claudio Battiloro, Stefania Sardellitti et al.

In this work, we study the problem of stability of Graph Convolutional Neural Networks (GCNs) under random small perturbations in the underlying graph topology, i.e. under a limited number of insertions or deletions of edges. We derive a novel bound on the expected difference between the outputs of unperturbed and perturbed GCNs. The proposed bound explicitly depends on the magnitude of the perturbation of the eigenpairs of the Laplacian matrix, and the perturbation explicitly depends on which edges are inserted or deleted. Then, we provide a quantitative characterization of the effect of perturbing specific edges on the stability of the network. We leverage tools from small perturbation analysis to express the bounds in closed, albeit approximate, form, in order to enhance interpretability of the results, without the need to compute any perturbed shift operator. Finally, we numerically evaluate the effectiveness of the proposed bound.

Paolo Di Lorenzo, Sergio Barbarossa, Paolo Banelli et al.

The aim of this paper is to propose a least mean squares (LMS) strategy for adaptive estimation of signals defined over graphs. Assuming the graph signal to be band-limited, over a known bandwidth, the method enables reconstruction, with guaranteed performance in terms of mean-square error, and tracking from a limited number of observations over a subset of vertices. A detailed mean square analysis provides the performance of the proposed method, and leads to several insights for designing useful sampling strategies for graph signals. Numerical results validate our theoretical findings, and illustrate the performance of the proposed method. Furthermore, to cope with the case where the bandwidth is not known beforehand, we propose a method that performs a sparse online estimation of the signal support in the (graph) frequency domain, which enables online adaptation of the graph sampling strategy. Finally, we apply the proposed method to build the power spatial density cartography of a given operational region in a cognitive network environment.