Resolution-Consistent Greedy Neural Approximation on Infinite-Dimensional Spaces

• URL: http://arxiv.org/abs/2608.20812v1
• Date: Fri, 21 Aug 2026 07:33:27 GMT
• Title: Resolution-Consistent Greedy Neural Approximation on Infinite-Dimensional Spaces
• Abstract summary: We develop constructive approximation and learning guarantees for shallow neural models with infinite-dimensional inputs.<n>For empirical regression, a fully-run greedy procedure yields responses whose statistical dependence on the output dimension is uniform.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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