arXiv subjects

Pablo Hernández-García

Publications and source records attributed to Pablo Hernández-García.

3 recordsLinked to original sources

arXiv2026

From Persistence to Resilience: New Betti Numbers for Analyzing Robustness in Simplicial Complex Networks

We show how persistence can be used to measure the robustness of cohomological cycles in finite simplicial complexes. We introduce two families of invariants: thick Betti numbers, which measure whether connected components and holes are supported by simplices of sufficiently high dimension; and cohesive Betti numbers, which measure the strength of higher-order adjacencies supporting cohomology classes. We then study robustness under simplicial degradation processes by combining an attack parameter with either the thick or the cohesive parameter, and by analyzing the resulting ladder modules through their image, kernel, and cokernel persistence modules. This allows us to distinguish features that remain structurally robust from those that lose thickness or cohesion during the degradation process. Finally, we prove stability results using the second network distance, which provides a theoretical reliability guarantee for the proposed constructions.

math.AT↗

arXiv2026

Interval Decompositions for Multipersistence Modules over Finite Posets and Robustness of Sheaf Data on Simplicial Complexes

We prove structure theorems for multipersistence modules indexed by finite posets that are not totally ordered. Specifically, we consider pointwise finite-dimensional modules over the opposite of the poset of non-empty subsets of a finite set, and give sufficient conditions, expressed through transition morphisms, for such modules to split as direct sums of interval modules. In the general case, the interval summands and multiplicities are explicitly determined by dimensions at finitely many indices. Although the assumptions may look algebraically restrictive, we show that they have a natural geometric origin in a robustness theory of cellular sheaf data over simplicial complexes, where one studies how algebraic information, compatibility constraints, and cohomological obstructions persist under structural failures. We extend thickness and cohesion from simplicial cohomology to cellular sheaves: thickness detects the dependence of cohomology classes on high-dimensional support, while cohesion captures the influence of higher-order adjacencies on the cohomological features. We leverage our abstract structure theorems to obtain interval decompositions for the resulting geometric cohesion modules. Finally, we introduce biparameter persistence constructions for sheaf resilience, tracking whether global sections and cohomological obstructions remain detectable on thick or cohesive substructures during topological degradation.

math.AT↗

arXiv2024

Attending to Topological Spaces: The Cellular Transformer

Topological Deep Learning seeks to enhance the predictive performance of neural network models by harnessing topological structures in input data. Topological neural networks operate on spaces such as cell complexes and hypergraphs, that can be seen as generalizations of graphs. In this work, we introduce the Cellular Transformer (CT), a novel architecture that generalizes graph-based transformers to cell complexes. First, we propose a new formulation of the usual self- and cross-attention mechanisms, tailored to leverage incidence relations in cell complexes, e.g., edge-face and node-edge relations. Additionally, we propose a set of topological positional encodings specifically designed for cell complexes. By transforming three graph datasets into cell complex datasets, our experiments reveal that CT not only achieves state-of-the-art performance, but it does so without the need for more complex enhancements such as virtual nodes, in-domain structural encodings, or graph rewiring.

cs.LG↗