arXiv subjects

Ana Luiza Tenório

Publications and source records attributed to Ana Luiza Tenório.

7 recordsLinked to original sources

arXiv2026

Cooperative Sheaf Neural Networks

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.

cs.LG↗

arXiv2026

Mitigating Over-squashing without Rewiring: A Sheaf Effective Resistance Perspective

Graph Neural Networks (GNNs) often struggle to capture long-range dependencies due to over-squashing -- a phenomenon in which the repeated compression of node embeddings into finite-size messages causes representations to collapse. Over-squashing is most often diagnosed as a property of the graph topology, with effective resistance serving as a principled measure of the bottleneck. We provide a complementary view on the matter: building on cellular sheaves, we introduce sheaf effective resistance, a generalization of effective resistance that depends on the sheaf attached to the graph, and we prove that for flat vector bundles, the over-squashing sensitivity in the Jacobian sense is upper bounded by a quantity related to the sheaf effective resistance between the nodes. The bottleneck thus need not lie in the graph itself: it can be relocated, and reduced, by adjusting the sheaf. We instantiate this idea in FlatNSD, a simple message-passing variant of Neural Sheaf Diffusion, and show that it implicitly learns to modulate total sheaf effective resistance, performing well on benchmarks designed to stress over-squashing without altering the original graph topology.

stat.ML↗

arXiv2026

Gaussian Sheaf Neural Networks

Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better represented by probability distributions than real vectors. Concretely, when node features are Gaussians, characterized by a mean and a covariance matrix, naively concatenating their parameters into a single vector and applying standard message passing discards the geometric and algebraic structure that governs means and covariances. We propose Gaussian Sheaf Neural Networks (GSNNs), a principled framework that incorporates these inductive biases into graph-based learning. Building on the theory of cellular sheaves, we derive a new Laplacian operator that generalizes the sheaf Laplacian to this setting and preserves its key properties. We complement our theoretical contributions with experiments on synthetic and real-world data that illustrate the practical relevance of GSNNs.

cs.LG↗

arXiv2025

The Category of Hypergroups: seeking for a generalized version of Abelian Categories

In this paper we study categorical properties of the category of abelian hypergroups that leads to the notion of hyper (almost) preadditive and hyper (almost) abelian categories. Our goal is to create a path towards a general theory of homological algebra for hyper-algebras. This is a first attempt to achieve this goal. We hope to improve the definitions and results, and provide more examples soon.

math.CT↗

arXiv2024

Expanding Čech cohomology for quantales

We expand Čech cohomology of a topological space $X$ with values in a presheaf on $X$ to Čech cohomology of a commutative ring with unity $R$ with values in a presheaf on $R$. The strategy is to observe that both the set of open subsets of $X$ and the set of ideals of $R$ provide examples of a (semicartesian) quantale. We study a particular pair of (adjoint) functors $(θ, τ)$ between the quantale of open subsets of $X$ and the quantale of ideals of $C(X)$, the ring of real-valued continuous functions on $X$. This leads to the main result of this paper: the $q$th Čech cohomology groups of $X$ with values on the constant sheaf $F$ on $X$ is isomorphic to the $q$th Čech cohomology groups of the ring $C(X)$ with values on a sheaf $F \circ τ$ on $C(X)$.

math.CT↗

arXiv2024

Grothendieck prelopologies: towards a closed monoidal sheaf category

In this paper, we present a generalization of Grothendieck pretopologies -- suited for semicartesian categories with equalizers $C$ -- leading to a closed monoidal category of sheaves, instead of closed cartesian category. This is proved through a different sheafification process, which is the left adjoint functor of the suitable inclusion functor but does not preserve all finite limits. If the monoidal structure in $C$ is given by the categorical product, all constructions coincide with those for Grothendieck toposes. The motivation for such generalization stems from a certain notion of sheaves on quantales that does not form a topos.

math.CT↗

arXiv2023

On sheaves on semicartesian quantales and their truth values

In this paper, we introduce a new definition of sheaves on semicartesian quantales, providing first examples and categorical properties. We note that our sheaves are similar to the standard definition of a sheaf on a locale, however, we prove in that in general it is not an elementary topos - since the lattice of external truth values of $Sh(Q)$, $Sub(1)$, is canonically isomorphic to the quantale $Q$ - placing this paper as part of a greater project towards a monoidal (not necessarily cartesian) closed version of elementary topos. To start the study the logical aspects of the category of sheaves we are introducing, we explore the nature of the "internal truth value objects" in such sheaves categories. More precisely, we analyze two candidates for subobject classifier for different subclasses of commutative and semicartesian quantales.

math.CT↗