arXiv subjects

Ferran Hernandez Caralt

Publications and source records attributed to Ferran Hernandez Caralt.

5 recordsLinked to original sources

arXiv2026

Cup and Cap Topological Neural Network

Topological Deep Learning (TDL) is designed to learn features associated with higher-dimensional simplices including not only nodes but also edges, triangles and so on. However, most TDL approaches, being based on boundary operators and Hodge Laplacians, have the limitation that features and signals cannot be lifted or lowered across more than one dimension per layer. To overcome this limitation, in this work, we propose the adoption of the cup and cap products. Specifically, we formulate the Cup and Cap Topological Neural Network (CCNN), a topological deep learning architecture designed to learn node-based variables (or 0-cochains) by taking into account their many-body interactions (e.g. triangles) present in the data. We validate CCNN by assessing its performance on the TopoBench datasets, revealing its competitiveness with respect to other simplicial complex neural networks.

cond-mat.dis-nn↗

arXiv2026

Get a GRIP, this will be a long TRIP: A Quantifiable Long-Range Framework for Verifying Over-squashing

Empirical claims about the connection between over-squashing and long-range interactions in GNNs, can only be trusted if the benchmarks used to validate them genuinely require long-range interactions. The de-facto standard, the Long Range Graph Benchmark, has been repeatedly shown to be saturated by tuned short-range models, with existing synthetic alternatives being tied to specific topologies. As such, there is a lack of principled certificate of long-rangedness on arbitrary graphs. This state reflects the absence of a precise characterization of long-ranged benchmarks. We address this fundamental gap by introducing four verifiable axioms: Predictability, Tightness, Strictly $k$-Range, and Topology-Invariance, that any task claiming to test $k$-hop interactions must satisfy. We formally prove that violating any one of them admits failure modes that undermine conclusions drawn from the task. Based on these axioms, we introduce TRIP (Truly Ranged Interactions Problem) and its generalisation GRIP (Generally Ranged Interactions Problem), constructive procedures that turn any graph into a provably long-ranged task by drawing features from stable distributions. Moreover, by construction, GRIP admits a closed-form, per-range Maximum-Likelihood oracle that yields the first a priori per-range lower bound on test error available on any benchmark. Using our framework, we: (i) audit 4 common long-range benchmarks and identify their failures modes with respect to our axioms; (ii) on TRIP-instantiated topologies, we find a popular notion of curvature is uncorrelated with GNN performance, supporting topological-vs-computational bottleneck distinction; and (iii) we show that a novel benchmark's over-squashing measures factors beyond pure long-rangedness. Code to use the framework and reproduce experiments is released https://github.com/ferranhernandezc/graph-grip.

cs.LG↗

arXiv2026

Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification

In the transductive setting, where the full graph is observed but node labels are only partially available, progress in semi-supervised node classification has largely focused on architectural innovation. In this paper, we revisit an orthogonal axis: the training objective. We start from a simple observation: transductive models produce predictions for every node during training, including nodes without labels. These unlabeled-node predictions may contain useful training signal, but standard supervised objectives discard them because no ground-truth labels are available. Inspired by the decomposition of cross-entropy into a label-dependent alignment term and a label-independent entropy term, we propose prediction confidence as a natural way to extract this signal in the absence of labels. This motivates Transductive Sharpening (TS): a loss-level modification that minimizes prediction entropy on unlabeled nodes while counterbalancing this effect on labeled nodes. We evaluate Transductive Sharpening across a wide range of node-classification benchmarks and observe consistent performance improvements without requiring any changes to the backbone architecture. Code is available at https://github.com/transductive-sharpening/tunedGNN.

cs.LG↗

arXiv2026

On the Necessity of Learnable Sheaf Laplacians

Sheaf Neural Networks (SNNs) were introduced as an extension of Graph Convolutional Networks to address oversmoothing on heterophilous graphs by attaching a sheaf to the input graph and replacing the adjacency-based operator with a sheaf Laplacian defined by (learnable) restriction maps. Prior work motivates this design through theoretical properties of sheaf diffusion and the kernel of the sheaf Laplacian, suggesting that suitable non-identity restriction maps can avoid representations converging to constants across connected components. Since oversmoothing can also be mitigated through residual connections and normalization, we revisit a trivial sheaf construction to ask whether the additional complexity of learning restriction maps is necessary. We introduce an Identity Sheaf Network baseline, where all restriction maps are fixed to the identity, and use it to ablate the empirical improvements reported by sheaf-learning architectures. Across five popular heterophilic benchmarks, the identity baseline achieves comparable performance to a range of SNN variants. Finally, we introduce the Rayleigh quotient as a normalized measure for comparing oversmoothing across models and show that, in trained networks, the behavior predicted by the diffusion-based analysis of SNNs is not reflected empirically. In particular, Identity Sheaf Networks do not appear to suffer more significant oversmoothing than their SNN counterparts.

cs.LG↗

arXiv2024

Joint Diffusion Processes as an Inductive Bias in Sheaf Neural Networks

Sheaf Neural Networks (SNNs) naturally extend Graph Neural Networks (GNNs) by endowing a cellular sheaf over the graph, equipping nodes and edges with vector spaces and defining linear mappings between them. While the attached geometric structure has proven to be useful in analyzing heterophily and oversmoothing, so far the methods by which the sheaf is computed do not always guarantee a good performance in such settings. In this work, drawing inspiration from opinion dynamics concepts, we propose two novel sheaf learning approaches that (i) provide a more intuitive understanding of the involved structure maps, (ii) introduce a useful inductive bias for heterophily and oversmoothing, and (iii) infer the sheaf in a way that does not scale with the number of features, thus using fewer learnable parameters than existing methods. In our evaluation, we show the limitations of the real-world benchmarks used so far on SNNs, and design a new synthetic task -- leveraging the symmetries of n-dimensional ellipsoids -- that enables us to better assess the strengths and weaknesses of sheaf-based models. Our extensive experimentation on these novel datasets reveals valuable insights into the scenarios and contexts where SNNs in general -- and our proposed approaches in particular -- can be beneficial.

cs.LG↗