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arXiv · 2608.20812

Resolution-Consistent Greedy Neural Approximation on Infinite-Dimensional Spaces

Pablo M. Berná·Antonio Falcó·Diego Mondéjar
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Abstract

We develop constructive approximation and learning guarantees for shallow neural models with infinite-dimensional inputs observed through finitely many coordinates. The analysis is based on a parameter-normalized neural dictionary and its associated weighted variation class. Within this class, the approximation error separates into a distribution-dependent coordinate-truncation term and a greedy finite-width term. For empirical regression, a fully-corrective greedy procedure yields population guarantees whose statistical complexity is uniform in the retained input resolution. The same framework extends to Hilbert-valued responses without an explicit dependence on the output dimension. The dimension-free statements are statistical, not computational: selecting a new neuron still requires solving a nonconvex parameter-search problem. The quasi-Polish construction underlying recent infinite-dimensional universal approximation results provides a motivating example, and synthetic experiments illustrate the predicted resolution, width, and sample-size regimes.

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Pablo M. Berná, Antonio Falcó, Diego Mondéjar. 2026-08-21. Resolution-Consistent Greedy Neural Approximation on Infinite-Dimensional Spaces. https://arxiv.org/abs/2608.20812

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