arXiv subjects

Jose Antonio Lara Benitez

Publications and source records attributed to Jose Antonio Lara Benitez.

4 recordsLinked to original sources

arXiv2026

A Sharp Norm Inequality and Buzano's Inequality via Determinants

We give a short linear-algebraic proof of the inequality $$ \|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{n}}{2}\,\|x\|_2^2, $$ valid for every $x\in\mathbb{R}^n$. This inequality relates three fundamental norms on finite-dimensional spaces and has applications in optimization and numerical analysis. Our proof exploits the determinantal structure of a parametrized family of quadratic forms, and we show the constant $(1+\sqrt{n})/2$ is optimal. The inequality is a special case of Buzano's inequality, and the same method also proves Buzano's inequality for real vectors.

math.CA↗

arXiv2026

Mixtures of Neural Operators Reduce Active Complexity in Operator Learning

Operator-learning systems are not governed solely by total parameter count; for one query, the relevant bottleneck can be the model that must be loaded and evaluated. We study this distinction for classical neural operators on compact Sobolev subsets through a constructive comparison between routed mixtures of neural operators (MoNOs) and a fixed single-neural-operator construction. The comparison concerns expert-active complexity relative to that baseline, with total stored size and routing search accounted separately. A MoNO routes each input function through a tree to one expert. Our main theorem shows that every scalar uniformly continuous nonlinear operator with bounded output Sobolev radius on the approximation set admits a MoNO approximation whose active expert has smaller depth, width, and rank scaling than the analyzed single-neural-operator construction; for Lipschitz targets these expert quantities are bounded by $\mathcal{O}(\varepsilon^{-1})$. The theorem turns localization into an operator-level accounting of active expert size, routing depth, and number of experts. We also prove a quantitative universal approximation theorem for the underlying neural-operator architecture, with explicit dependence on compact-set diameter and modulus of continuity.

cs.LG↗

arXiv2026

Neural Operators Can Discover Functional Clusters

Operator learning is reshaping scientific computing by amortizing inference across infinite families of problems. While neural operators (NOs) are increasingly well understood for regression, far less is known for classification and its unsupervised analogue: clustering. We prove that sample-based neural operators can learn any finite collection of classes in an infinite-dimensional reproducing kernel Hilbert space, even when the classes are neither convex nor connected, under mild kernel sampling assumptions. Our universal clustering theorem shows that any $K$ closed classes can be approximated to arbitrary precision by NO-parameterized classes in the upper Kuratowski topology on closed sets, a notion that can be interpreted as disallowing false-positive misclassifications. Building on this, we develop an NO-powered clustering pipeline for functional data and apply it to unlabeled families of ordinary differential equation (ODE) trajectories. Discretized trajectories are lifted by a fixed pre-trained encoder into a continuous feature map and mapped to soft assignments by a lightweight trainable head. Experiments on diverse synthetic ODE benchmarks show that the resulting practical SNO recovers latent dynamical structure in regimes where classical methods fail, providing evidence consistent with our universal clustering theory.

cs.LG↗

arXiv2026

Hybrid operator learning of wave scattering maps in high-contrast media

Surrogate modeling of wave propagation and scattering (i.e. the wave speed and source to wave field map) in heterogeneous media has significant potential in applications such as seismic imaging and inversion. High-contrast settings, such as subsurface models with salt bodies, exhibit strong scattering and phase sensitivity that challenge existing neural operators. We propose a hybrid architecture that decomposes the scattering operator into two separate contributions: a smooth background propagation and a high-contrast scattering correction. The smooth component is learned with a Fourier Neural Operator (FNO), which produces globally coupled feature tokens encoding background wave propagation; these tokens are then passed to a vision transformer, where attention is used to model the high-contrast scattering correction dominated by strong, spatial interactions. Evaluated on high-frequency Helmholtz problems with strong contrasts, the hybrid model achieves substantially improved phase and amplitude accuracy compared to standalone FNOs or transformers, with favorable accuracy-parameter scaling.

eess.SP↗