The aim of this paper is to propose a novel framework to infer the sheaf Laplacian, including the topology of a graph and the restriction maps, from a set of data observed over the nodes of a graph. The proposed method is based on sheaf theory, which represents an important generalization of graph signal processing. The learning problem aims to find the sheaf Laplacian that minimizes the total variation of the observed data, where the variation over each edge is also locally minimized by optimizing the associated restriction maps. Compared to alternative methods based on semidefinite programming, our solution is significantly more numerically efficient, as all its fundamental steps are resolved in closed form. The method is numerically tested on data consisting of vectors defined over subspaces of varying dimensions at each node. We demonstrate how the resulting graph is influenced by two key factors: the cross-correlation and the dimensionality difference of the data residing on the graph's nodes.

While deep generative models are showing exciting abilities in computer vision and natural language processing, their adoption in communication frameworks is still far underestimated. These methods are demonstrated to evolve solutions to classic communication problems such as denoising, restoration, or compression. Nevertheless, generative models can unveil their real potential in semantic communication frameworks, in which the receiver is not asked to recover the sequence of bits used to encode the transmitted (semantic) message, but only to regenerate content that is semantically consistent with the transmitted message. Disclosing generative models capabilities in semantic communication paves the way for a paradigm shift with respect to conventional communication systems, which has great potential to reduce the amount of data traffic and offers a revolutionary versatility to novel tasks and applications that were not even conceivable a few years ago. In this paper, we present a unified perspective of deep generative models in semantic communication and we unveil their revolutionary role in future communication frameworks, enabling emerging applications and tasks. Finally, we analyze the challenges and opportunities to face to develop generative models specifically tailored for communication systems.

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.

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.

The aim of this work is to introduce Generalized Simplicial Attention Neural Networks (GSANs), i.e., novel neural architectures designed to process data defined on simplicial complexes using masked self-attentional layers. Hinging on topological signal processing principles, we devise a series of self-attention schemes capable of processing data components defined at different simplicial orders, such as nodes, edges, triangles, and beyond. These schemes learn how to weight the neighborhoods of the given topological domain in a task-oriented fashion, leveraging the interplay among simplices of different orders through the Dirac operator and its Dirac decomposition. We also theoretically establish that GSANs are permutation equivariant and simplicial-aware. Finally, we illustrate how our approach compares favorably with other methods when applied to several (inductive and transductive) tasks such as trajectory prediction, missing data imputation, graph classification, and simplex prediction.

Semantic communication is expected to be one of the cores of next-generation AI-based communications. One of the possibilities offered by semantic communication is the capability to regenerate, at the destination side, images or videos semantically equivalent to the transmitted ones, without necessarily recovering the transmitted sequence of bits. The current solutions still lack the ability to build complex scenes from the received partial information. Clearly, there is an unmet need to balance the effectiveness of generation methods and the complexity of the transmitted information, possibly taking into account the goal of communication. In this paper, we aim to bridge this gap by proposing a novel generative diffusion-guided framework for semantic communication that leverages the strong abilities of diffusion models in synthesizing multimedia content while preserving semantic features. We reduce bandwidth usage by sending highly-compressed semantic information only. Then, the diffusion model learns to synthesize semantic-consistent scenes through spatially-adaptive normalizations from such denoised semantic information. We prove, through an in-depth assessment of multiple scenarios, that our method outperforms existing solutions in generating high-quality images with preserved semantic information even in cases where the received content is significantly degraded. More specifically, our results show that objects, locations, and depths are still recognizable even in the presence of extremely noisy conditions of the communication channel. The code is available at https://github.com/ispamm/GESCO.