Machine Learning algorithms are ubiquitous in key decision-making contexts such as justice, healthcare and finance, which has spawned a great demand for fairness in these procedures. However, the theoretical properties of such models in relation with fairness are still poorly understood, and the intuition behind the relationship between group and individual fairness is still lacking. In this paper, we provide a theoretical framework based on Sheaf Diffusion to leverage tools based on dynamical systems and homology to model fairness. Concretely, the proposed method projects input data into a bias-free space that encodes fairness constrains, resulting in fair solutions. Furthermore, we present a collection of network topologies handling different fairness metrics, leading to a unified method capable of dealing with both individual and group bias. The resulting models have a layer of interpretability in the form of closed-form expressions for their SHAP values, consolidating their place in the responsible Artificial Intelligence landscape. Finally, these intuitions are tested on a simulation study and standard fairness benchmarks, where the proposed methods achieve satisfactory results. More concretely, the paper showcases the performance of the proposed models in terms of accuracy and fairness, studying available trade-offs on the Pareto frontier, checking the effects of changing the different hyper-parameters, and delving into the interpretation of its outputs.

Recent advances in AI call for a paradigm shift from bit-centric communication to goal- and semantics-oriented architectures, paving the way for AI-native 6G networks. In this context, we address a key open challenge: enabling heterogeneous AI agents to exchange compressed latent-space representations while mitigating semantic noise and preserving task-relevant meaning. We cast this challenge as learning both the communication topology and the alignment maps that govern information exchange among agents, yielding a learned network sheaf equipped with orthogonal maps. This learning process is further supported by a semantic denoising end compression module that constructs a shared global semantic space and derives sparse, structured representations of each agent's latent space. This corresponds to a nonconvex dictionary learning problem solved iteratively with closed-form updates. Experiments with mutiple AI agents pre-trained on real image data show that the semantic denoising and compression facilitates AI agents alignment and the extraction of semantic clusters, while preserving high accuracy in downstream task. The resulting communication network provides new insights about semantic heterogeneity across agents, highlighting the interpretability of our methodology.

Predictive coding (PC) replaces global backpropagation with local optimization over weights and activations. We show that linear PC networks admit a natural formulation as cellular sheaves: the sheaf coboundary maps activations to edge-wise prediction errors, and PC inference is diffusion under the sheaf Laplacian. Sheaf cohomology then characterizes irreducible error patterns that inference cannot remove. We analyze recurrent topologies where feedback loops create internal contradictions, introducing prediction errors unrelated to supervision. Using a Hodge decomposition, we determine when these contradictions cause learning to stall. The sheaf formalism provides both diagnostic tools for identifying problematic network configurations and design principles for effective weight initialization for recurrent PC networks.

The deep learning revolution has spurred a rise in advances of using AI in sciences. Within physical sciences the main focus has been on discovery of dynamical systems from observational data. Yet the reliability of learned surrogates and symbolic models is often undermined by the fundamental problem of non-uniqueness. The resulting models may fit the available data perfectly, but lack genuine predictive power. This raises the question: under what conditions can the systems governing equations be uniquely identified from a finite set of observations? We show, counter-intuitively, that chaos, typically associated with unpredictability, is crucial for ensuring a system is discoverable in the space of continuous or analytic functions. The prevalence of chaotic systems in benchmark datasets may have inadvertently obscured this fundamental limitation. More concretely, we show that systems chaotic on their entire domain are discoverable from a single trajectory within the space of continuous functions, and systems chaotic on a strange attractor are analytically discoverable under a geometric condition on the attractor. As a consequence, we demonstrate for the first time that the classical Lorenz system is analytically discoverable. Moreover, we establish that analytic discoverability is impossible in the presence of first integrals, common in real-world systems. These findings help explain the success of data-driven methods in inherently chaotic domains like weather forecasting, while revealing a significant challenge for engineering applications like digital twins, where stable, predictable behavior is desired. For these non-chaotic systems, we find that while trajectory data alone is insufficient, certain prior physical knowledge can help ensure discoverability. These findings warrant a critical re-evaluation of the fundamental assumptions underpinning purely data-driven discovery.

We describe a theory and implementation of an intuitionistic decentralized framework for causal discovery using judo calculus, which is formally defined as j-stable causal inference using j-do-calculus in a topos of sheaves. In real-world applications -- from biology to medicine and social science -- causal effects depend on regime (age, country, dose, genotype, or lab protocol). Our proposed judo calculus formalizes this context dependence formally as local truth: a causal claim is proven true on a cover of regimes, not everywhere at once. The Lawvere-Tierney modal operator j chooses which regimes are relevant; j-stability means the claim holds constructively and consistently across that family. We describe an algorithmic and implementation framework for judo calculus, combining it with standard score-based, constraint-based, and gradient-based causal discovery methods. We describe experimental results on a range of domains, from synthetic to real-world datasets from biology and economics. Our experimental results show the computational efficiency gained by the decentralized nature of sheaf-theoretic causal discovery, as well as improved performance over classical causal discovery methods.

Higher-order relations are widespread in nature, with numerous phenomena involving complex interactions that extend beyond simple pairwise connections.

Sheaf diffusion has recently emerged as a promising design pattern for graph representation learning due to its inherent ability to handle heterophilic data and avoid oversmoothing. Meanwhile, cooperative message passing has also been proposed as a way to enhance the flexibility of information diffusion by allowing nodes to independently choose whether to propagate/gather information from/to neighbors. A natural question ensues: is sheaf diffusion capable of exhibiting this cooperative behavior? Here, we provide a negative answer to this question. In particular, we show that existing sheaf diffusion methods fail to achieve cooperative behavior due to the lack of message directionality. To circumvent this limitation, we introduce the notion of cellular sheaves over directed graphs and characterize their in- and out-degree Laplacians. We leverage our construction to propose Cooperative Sheaf Neural Networks (CSNNs). Theoretically, we characterize the receptive field of CSNN and show it allows nodes to selectively attend (listen) to arbitrarily far nodes while ignoring all others in their path, potentially mitigating oversquashing. Our experiments show that CSNN presents overall better performance compared to prior art on sheaf diffusion as well as cooperative graph neural networks.