Topological Signal Processing and Learning: Recent Advances and Future Challenges

Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges capturing their interactions, leading to the establishment of the fields of graph signal processing (GSP) and graph machine learning (GML). Key graph-aware methods include Fourier transform, filtering, sampling, as well as topology identification and spatiotemporal processing. Although versatile, graphs can model only pairwise dependencies in the data. To this end, topological structures such as simplicial and cell complexes have emerged as algebraic representations for more intricate structure modeling in data-driven systems, fueling the rapid development of novel topological-based processing and learning methods. This paper first presents the core principles of topological signal processing through the Hodge theory, a framework instrumental in propelling the field forward thanks to principled connections with GSP-GML. It then outlines advances in topological signal representation, filtering, and sampling, as well as inferring topological structures from data, processing spatiotemporal topological signals, and connections with topological machine learning. The impact of topological signal processing and learning is finally highlighted in applications dealing with flow data over networks, geometric processing, statistical ranking, biology, and semantic communication.

Language-Oriented Semantic Latent Representation for Image Transmission

In the new paradigm of semantic communication (SC), the focus is on delivering meanings behind bits by extracting semantic information from raw data. Recent advances in data-to-text models facilitate language-oriented SC, particularly for text-transformed image communication via image-to-text (I2T) encoding and text-to-image (T2I) decoding. However, although semantically aligned, the text is too coarse to precisely capture sophisticated visual features such as spatial locations, color, and texture, incurring a significant perceptual difference between intended and reconstructed images. To address this limitation, in this paper, we propose a novel language-oriented SC framework that communicates both text and a compressed image embedding and combines them using a latent diffusion model to reconstruct the intended image. Experimental results validate the potential of our approach, which transmits only 2.09\% of the original image size while achieving higher perceptual similarities in noisy communication channels compared to a baseline SC method that communicates only through text.The code is available at https://github.com/ispamm/Img2Img-SC/ .

Opportunistic Information-Bottleneck for Goal-oriented Feature Extraction and Communication

The Information Bottleneck (IB) method is an information theoretical framework to design a parsimonious and tunable feature-extraction mechanism, such that the extracted features are maximally relevant to a specific learning or inference task. Despite its theoretical value, the IB is based on a functional optimization problem that admits a closed form solution only on specific cases (e.g., Gaussian distributions), making it difficult to be applied in most applications, where it is necessary to resort to complex and approximated variational implementations. To overcome this limitation, we propose an approach to adapt the closed-form solution of the Gaussian IB to a general task. Whichever is the inference task to be performed by a (possibly deep) neural-network, the key idea is to opportunistically design a regression sub-task, embedded in the original problem, where we can safely assume a (joint) multivariate normality between the sub-task's inputs and outputs. In this way we can exploit a fixed and pre-trained neural network to process the input data, using a tunable number of features, to trade data-size and complexity for accuracy. This approach is particularly useful every time a device needs to transmit data (or features) to a server that has to fulfil an inference task, as it provides a principled way to extract the most relevant features for the task to be executed, while looking for the best trade-off between the size of the feature vector to be transmitted, inference accuracy, and complexity. Extensive simulation results testify the effectiveness of the proposed methodhttps://info.arxiv.org/help/prep#comments and encourage to further investigate this research line.

Generative AI Meets Semantic Communication: Evolution and Revolution of Communication Tasks

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.

Stability of Graph Convolutional Neural Networks through the lens of small perturbation analysis

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.

Learning Multi-Frequency Partial Correlation Graphs

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.

Goal-oriented Communications for the IoT: System Design and Adaptive Resource Optimization

Internet of Things (IoT) applications combine sensing, wireless communication, intelligence, and actuation, enabling the interaction among heterogeneous devices that collect and process considerable amounts of data. However, the effectiveness of IoT applications needs to face the limitation of available resources, including spectrum, energy, computing, learning and inference capabilities. This paper challenges the prevailing approach to IoT communication, which prioritizes the usage of resources in order to guarantee perfect recovery, at the bit level, of the data transmitted by the sensors to the central unit. We propose a novel approach, called goal-oriented (GO) IoT system design, that transcends traditional bit-related metrics and focuses directly on the fulfillment of the goal motivating the exchange of data. The improvement is then achieved through a comprehensive system optimization, integrating sensing, communication, computation, learning, and control. We provide numerical results demonstrating the practical applications of our methodology in compelling use cases such as edge inference, cooperative sensing, and federated learning. These examples highlight the effectiveness and real-world implications of our proposed approach, with the potential to revolutionize IoT systems.

Generalized Simplicial Attention Neural Networks

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.

Generative Semantic Communication: Diffusion Models Beyond Bit Recovery

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.

Multi-user Goal-oriented Communications with Energy-efficient Edge Resource Management

Edge Learning (EL) pushes the computational resources toward the edge of 5G/6G network to assist mobile users requesting delay-sensitive and energy-aware intelligent services. A common challenge in running inference tasks from remote is to extract and transmit only the features that are most significant for the inference task. From this perspective, EL can be effectively coupled with goal-oriented communications, whose aim is to transmit only the information {\it relevant} to perform the inference task, under prescribed accuracy, delay, and energy constraints. In this work, we consider a multi-user/single server wireless network, where the users can opportunistically decide whether to perform the inference task by themselves or, alternatively, to offload the data to the edge server for remote processing. The data to be transmitted undergoes a goal-oriented compression stage performed using a convolutional encoder, jointly trained with a convolutional decoder running at the edge-server side. Employing Lyapunov optimization, we propose a method to jointly and dynamically optimize the selection of the most suitable encoding/decoding scheme, together with the allocation of computational and transmission resources, across all the users and the edge server. Extensive simulations confirm the effectiveness of the proposed approaches and highlight the trade-offs between energy, latency, and learning accuracy.