Topological Deep Learning (TDL) has emerged as a paradigm to process and learn from signals defined on higher-order combinatorial topological spaces, such as simplicial or cell complexes. Although many complex systems have an asymmetric relational structure, most TDL models forcibly symmetrize these relationships. In this paper, we first introduce a novel notion of higher-order directionality and we then design Directed Simplicial Neural Networks (Dir-SNNs) based on it. Dir-SNNs are message-passing networks operating on directed simplicial complexes able to leverage directed and possibly asymmetric interactions among the simplices. To our knowledge, this is the first TDL model using a notion of higher-order directionality. We theoretically and empirically prove that Dir-SNNs are more expressive than their directed graph counterpart in distinguishing non-isomorphic directed graphs. Experiments on a synthetic source localization task demonstrate that Dir-SNNs outperform undirected SNNs when the underlying complex is directed, and perform comparably when the underlying complex is undirected.

Graph neural networks excel at modeling pairwise interactions, but they cannot flexibly accommodate higher-order interactions and features. Topological deep learning (TDL) has emerged recently as a promising tool for addressing this issue. TDL enables the principled modeling of arbitrary multi-way, hierarchical higher-order interactions by operating on combinatorial topological spaces, such as simplicial or cell complexes, instead of graphs. However, little is known about how to leverage geometric features such as positions and velocities for TDL. This paper introduces E(n)-Equivariant Topological Neural Networks (ETNNs), which are E(n)-equivariant message-passing networks operating on combinatorial complexes, formal objects unifying graphs, hypergraphs, simplicial, path, and cell complexes. ETNNs incorporate geometric node features while respecting rotation, reflection, and translation equivariance. Moreover, being TDL models, ETNNs are natively ready for settings with heterogeneous interactions. We provide a theoretical analysis to show the improved expressiveness of ETNNs over architectures for geometric graphs. We also show how E(n)-equivariant variants of TDL models can be directly derived from our framework. The broad applicability of ETNNs is demonstrated through two tasks of vastly different scales: i) molecular property prediction on the QM9 benchmark and ii) land-use regression for hyper-local estimation of air pollution with multi-resolution irregular geospatial data. The results indicate that ETNNs are an effective tool for learning from diverse types of richly structured data, as they match or surpass SotA equivariant TDL models with a significantly smaller computational burden, thus highlighting the benefits of a principled geometric inductive bias. Our implementation of ETNNs can be found at https://github.com/NSAPH-Projects/topological-equivariant-networks.

Graph Neural Networks (GNNs) effectively learn from relational data by leveraging graph symmetries. However, many real-world systems—such as biological or social networks—feature multi-way interactions that GNNs fail to capture. Topological Deep Learning (TDL) addresses this by modeling and leveraging higher-order structures, with Combinatorial Complex Neural Networks (CCNNs) offering a general and expressive approach that has been shown to outperform GNNs. However, TDL lacks the principled and standardized frameworks that underpin GNN development, restricting its accessibility and applicability. To address this issue, we introduce Generalized CCNNs (GCCNs), a simple yet powerful family of TDL models that can be used to systematically transform any (graph) neural network into its TDL counterpart. We prove that GCCNs generalize and subsume CCNNs, while extensive experiments on a diverse class of GCCNs show that these architectures consistently match or outperform CCNNs, often with less model complexity. In an effort to accelerate and democratize TDL, we introduce TopoTune, a lightweight software for defining, building, and training GCCNs with unprecedented flexibility and ease.

Recent advances in *artificial intelligence* reveal the limits of purely predictive systems and call for a shift toward causal *and* collaborative reasoning. Drawing inspiration from the revolution of Grothendieck in mathematics, we introduce the *relativity of causal knowledge*, which posits structural causal models (SCMs) are inherently imperfect, subjective representations embedded within networks of relationships. By leveraging category theory, we arrange SCMs into a functor category and show that their observational and interventional probability measures naturally form convex structures. This result allows us to encode non-intervened SCMs with convex spaces of probability measures. Next, using sheaf theory, we construct the *network sheaf and cosheaf of causal knowledge*. These structures enable the transfer of causal knowledge across the network while incorporating interventional consistency and the perspective of the subjects, ultimately leading to the formal, mathematical definition of *relative causal knowledge*.

Topological neural networks (TNNs) are information processing architectures that model representations from data lying over topological spaces (e.g., simplicial or cell complexes) and allow for decentralized implementation through localized communications over different neighborhoods. Existing TNN architectures have not yet been considered in realistic communication scenarios, where channel effects typically introduce disturbances such as fading and noise. This paper aims to propose a novel TNN design, operating on regular cell complexes, that performs over-the-air computation, incorporating the wireless communication model into its architecture. Specifically, during training and inference, the proposed method considers channel impairments such as fading and noise in the topological con-volutional filtering operation, which takes place over different signal orders and neighborhoods. Numerical results illustrate the architecture’s robustness to channel impairments during testing and the superior performance with respect to existing architectures, which are either communication-agnostic or graph-based.

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.

Latent Graph Inference (LGI) relaxed the reliance of Graph Neural Networks (GNNs) on a given graph topology by dynamically learning it. However, most of LGI methods assume to have a (noisy, incomplete, improvable, ...) input graph to rewire and can solely learn regular graph topologies. In the wake of the success of Topological Deep Learning (TDL), we study Latent Topology Inference (LTI) for learning higher-order cell complexes (with sparse and not regular topology) describing multi-way interactions between data points. To this aim, we introduce the Differentiable Cell Complex Module (DCM), a novel learnable function that computes cell probabilities in the complex to improve the downstream task. We show how to integrate DCM with cell complex message-passing networks layers and train it in an end-to-end fashion, thanks to a two-step inference procedure that avoids an exhaustive search across all possible cells in the input, thus maintaining scalability. Our model is tested on several homophilic and heterophilic graph datasets and it is shown to outperform other state-of-the-art techniques, offering significant improvements especially in cases where an input graph is not provided.

We introduce TopoX, a Python software suite that provides reliable and user-friendly building blocks for computing and machine learning on topological domains that extend graphs: hypergraphs, simplicial, cellular, path and combinatorial complexes. TopoX consists of three packages: TopoNetX facilitates constructing and computing on these domains, including working with nodes, edges and higher-order cells; TopoEmbedX provides methods to embed topological domains into vector spaces, akin to popular graph-based embedding algorithms such as node2vec; TopoModelX is built on top of PyTorch and offers a comprehensive toolbox of higher-order message passing functions for neural networks on topological domains. The extensively documented and unit-tested source code of TopoX is available under MIT license at https://pyt-team.github.io.

This paper introduces topological Slepians, i.e., a novel class of signals defined over topological spaces (e.g., simplicial complexes) that are maximally concentrated on the topological domain (e.g., over a set of nodes, edges, triangles, etc.) and perfectly localized on the dual domain (e.g., a set of frequencies). These signals are obtained as the principal eigenvectors of a matrix built from proper localization operators acting over topology and frequency domains. Then, we suggest a principled procedure to build dictionaries of topological Slepians, which theoretically provide non-degenerate frames. Finally, we evaluate the effectiveness of the proposed topological Slepian dictionary in two applications, i.e., sparse signal representation and denoising of edge flows.

Weighing the topological domain over which data can be represented and analysed is a key strategy in many signal processing and machine learning applications, enabling the extraction and exploitation of meaningful data features and their (higher order) relationships. Our goal in this paper is to present topological signal processing tools for weighted simplicial complexes. Specifically, relying on the weighted Hodge Laplacian theory, we propose efficient strategies to jointly learn the weights of the complex and the filters for the solenoidal, irrotational and harmonic components of the signals defined over the complex. We numerically assess the effectiveness of the proposed procedures.

In this work we introduce a convolution operation over the tangent bundle of Riemannian manifolds exploiting the Connection Laplacian operator. We use this convolution operation to define tangent bundle filters and tangent bundle neural networks (TNNs), novel continuous architectures operating on tangent bundle signals, i.e. vector fields over manifolds. We discretize TNNs both in space and time domains, showing that their discrete counterpart is a principled variant of the recently introduced Sheaf Neural Networks. We formally prove that this discrete architecture converges to the underlying continuous TNN. We numerically evaluate the effectiveness of the proposed architecture on a denoising task of a tangent vector field over the unit 2-sphere.

The goal of this paper is to introduce pooling strategies for simplicial convolutional neural networks. Inspired by graph pooling methods, we introduce a general formulation for a simplicial pooling layer that performs: i) local aggregation of simplicial signals; ii) principled selection of sampling sets; iii) downsampling and simplicial topology adaptation. The general layer is then customized to design four different pooling strategies (i.e., max, top-k, self-attention, and separated top-k) grounded in the theory of topological signal processing. Also, we leverage the proposed layers in a hierarchical architecture that reduce complexity while representing data at different resolutions. Numerical results on real data benchmarks (i.e., flow and graph classification) illustrate the advantage of the proposed methods with respect to the state of the art.

Since their introduction, graph attention networks achieved outstanding results in graph representation learning tasks. However, these networks consider only pairwise relations between features associated to the nodes and then are unable to fully exploit higher-order and long-range interactions present in many real world data-sets. In this paper, we introduce a neural architecture operating on data defined over the nodes and the edges of a graph, represented as the 1-skeleton of a regular cell complex, able to capture insightful higher-order and long-range 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 is hierarchical and it incorporates the following steps: i) a lifting algorithm that learns (additional) edge features from node features; ii) a cell attention mechanism to find the optimal combination of edge features over both lower and upper neighbors; iii) a hierarchical edge pooling mechanism to extract a compact meaningful set of features. The experimental results show that this method compares favorably with state of the art results on graph-based learning tasks while maintaining a low complexity.

The aim of this work is to propose a novel dynamic resource allocation strategy for adaptive Federated Learning (FL), in the context of beyond 5G networks endowed with Reconfigurable Intelligent Surfaces (RISs). Due to time-varying wireless channel conditions, communication resources (e.g., set of transmitting devices, transmit powers, bits), computation parameters (e.g., CPU cycles at devices and at server) and RISs reflectivity must be optimized in each communication round, in order to strike the best trade-off between power, latency, and performance of the FL task. Hinging on Lyapunov stochastic optimization, we devise an online strategy able to dynamically allocate these resources, while controlling learning performance in a fully data-driven fashion. Numerical simulations implement distributed training of deep convolutional neural networks, illustrating the effectiveness of the proposed FL strategy endowed with multiple reconfigurable intelligent surfaces.

This work aims to propose a novel architecture and training strategy for graph convolutional networks (GCN). The proposed architecture, named Autoencoder-Aided GCN (AA-GCN), compresses the convolutional features in an information-rich embedding at multiple hidden layers, exploiting the presence of autoencoders before the point-wise nonlinearities. Then, we propose a novel end-to-end training procedure that learns different graph representations per layer, jointly with the GCN weights and auto-encoder parameters. As a result, the proposed strategy improves the computational scalability of the GCN, learning the best graph representations at each layer in a data-driven fashion. Several numerical results on synthetic and real data illustrate how our architecture and training procedure compares favorably with other state-of-the-art solutions, both in terms of robustness and learning performance.

The aim of this paper is to propose a novel dynamic resource allocation strategy for energy-efficient federated learning at the wireless network edge, with latency and learning performance guarantees. We consider a set of devices collecting local data and uploading processed information to an edge server, which runs stochastic gradient descent (SGD) to perform distributed learning and adaptation. Hinging on Lyapunov stochastic optimization tools, we dynamically optimize radio parameters (i.e., set of transmitting devices, transmit powers) and computation resources (i.e., CPU cycles at devices and at server) in order to strike the best trade-off between energy, latency, and performance of the federated learning task. The general framework is then customized to the case of federated least mean squares (LMS) estimation. Numerical results illustrate the effectiveness of our strategy to perform energy-efficient, low-latency, federated machine learning at the wireless network edge.

We study decentralized estimation of time-varying signals at a fusion center, when energy harvesting sensors transmit sampled data over rate-constrained links. We propose dynamic strategies to select radio parameters, sampling set, and harvested energy at each node, with the aim of estimating a time-varying signal while ensuring: 1) the accuracy of the recovery procedure and 2) the stability of the batteries around a prescribed operating level. The approach is based on stochastic optimization tools, which enable adaptive optimization without the need of apriori knowledge of the statistics of radio channels and energy arrivals processes. Numerical results validate the proposed approach for decentralized signal estimation under communication and energy constraints typical of Internet-of-Things scenarios.

The aim of this paper is to propose a novel distributed strategy for tensor completion, where (partial) data are collected over a network of agents with sparse, but connected, topology. The method hinges on the canonical polyadic decomposition, also known as PARAFAC, to complete the low-rank tensor in a distributed fashion. To deal with the nonconvex and distributed nature of the learning problem, we exploit a convexification/decomposition technique based on successive convex approximations, while using dynamic consensus to diffuse information over the network and force asymptotic agreement among the agents. Asymptotic convergence to stationary solutions of the centralized problem is established under mild conditions. Finally, numerical results assess the performance of the proposed method over both synthetic and real data.

We study decentralized estimation of time-varying signals at a fusion center (FC), when energy harvesting sensors transmit sampled data over rate-constrained links. We propose a dynamic strategy based on stochastic optimization for selecting radio parameters, sampling set, and harvested energy at each node, with the aim of estimating a time-varying signal with guaranteed performance while ensuring stability of the batteries around a prescribed operating level. Numerical results validate the proposed approach for dynamic signal estimation under communication and energy constraints.