arXiv subjects

Vicente Bosca

Publications and source records attributed to Vicente Bosca.

4 recordsLinked to original sources

arXiv2026

The $\infty$-Oreo$^{^\circledR}$

What happens when a food product contains a version of itself? The Oreo Loaded -- a cookie whose filling contains real Oreo cookie crumbs -- can be viewed as the result of mixing a Mega Stuf Oreo into a Mega Stuf Oreo. Iterating this process yields a sequence of increasingly self-referential cookies; taking the limit gives the $\infty$-Oreo. We model the iteration as an affine recurrence on the creme fraction of the filling, prove convergence, and compute the limit exactly: the stuf of the $\infty$-Oreo is approximately $95.8\%$~creme and $4.2\%$~wafer. We then extend the framework to pairs of foods that reference each other, deriving a coupled recursion whose fixed point defines a \emph{bi-$\infty$ food}, and illustrate the construction with M\&M Cookies and Crunchy Cookie M\&M's. Finally, we classify $\infty$-foods by the number of foods in the recursion and introduce \emph{homological foods}, whose recursive structure is governed by cycles in a directed graph of commercially available products. We close with a conjecture. All products used in this paper can be purchased at a supermarket.

math.HO↗

arXiv2026

Neural Networks as Local-to-Global Computations

We construct a cellular sheaf from any feedforward ReLU neural network by placing one vertex for each intermediate quantity in the forward pass and encoding each computational step - affine transformation, activation, output - as a restriction map on an edge. The restricted coboundary operator on the free coordinates is unitriangular, so its determinant is $1$ and the restricted Laplacian is positive definite for every activation pattern. It follows that the relative cohomology vanishes and the forward pass output is the unique harmonic extension of the boundary data. The sheaf heat equation converges exponentially to this output despite the state-dependent switching introduced by piecewise linear activations. Unlike the forward pass, the heat equation propagates information bidirectionally across layers, enabling pinned neurons that impose constraints in both directions, training through local discrepancy minimization without a backward pass, and per-edge diagnostics that decompose network behavior by layer and operation type. We validate the framework experimentally on small synthetic tasks, confirming the convergence theorems and demonstrating that sheaf-based training, while not yet competitive with stochastic gradient descent, obeys quantitative scaling laws predicted by the theory.

math.AT↗

arXiv2026

Selective Adaptation of Beliefs and Communication on Cellular Sheaves

We extend opinion dynamics on discourse sheaves to incorporate "directional stubbornness": agents may hold fixed positions in specified directions of their opinion stalk while remaining flexible in others. This converts the equilibrium problem from harmonic extension to a forced sheaf equation: the free-opinion component satisfies a sheaf Poisson equation with forcing induced by the clamped directions. We develop a parallel theory for "selective learning" of expression policies. When only a designated subset of incidence maps may adapt, the resulting gradient flow is sheaf diffusion on an auxiliary structure sheaf whose global sections correspond to sheaf structures making a fixed opinion profile publicly consistent. For joint evolution of beliefs and expressions, we give conditions (and regularized variants) guaranteeing convergence to nondegenerate equilibria, excluding spurious agreement via vanishing opinions or trivialized communication maps. Finally, we derive stagnation bounds in terms of the rate ratio between opinion updating and structural adaptation, quantifying when rapid rhetorical accommodation masks nearly unchanged beliefs, and conversely when flexible beliefs conform to rigid communication norms.

math.OC↗

arXiv2025

Topological Signatures of ReLU Neural Network Activation Patterns

This paper explores the topological signatures of ReLU neural network activation patterns. We consider feedforward neural networks with ReLU activation functions and analyze the polytope decomposition of the feature space induced by the network. Mainly, we investigate how the Fiedler partition of the dual graph and show that it appears to correlate with the decision boundary -- in the case of binary classification. Additionally, we compute the homology of the cellular decomposition -- in a regression task -- to draw similar patterns in behavior between the training loss and polyhedral cell-count, as the model is trained.

cs.LG↗