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Greedy Neural Approximation on Infinite-Dimensional Spaces.

Pablo M. Bern\'a, Antonio Falc\'o, Diego Mond\'ejar· August 24, 2026 View original

Key takeaways

  • New approximation and learning guarantees for shallow neural models with infinite-dimensional inputs.
  • Error separates into coordinate-truncation and greedy finite-width terms.
  • A fully-corrective greedy procedure yields resolution-uniform statistical complexity.
  • The framework extends to Hilbert-valued responses without output dimension dependence.

Who benefits

Scientific ComputingSignal ProcessingQuantitative FinanceClimate ModelingMedical Imaging

Summary

This research develops constructive approximation and learning guarantees for shallow neural models with infinite-dimensional inputs observed through finitely many coordinates. It introduces a parameter-normalized neural dictionary and a fully-corrective greedy procedure, yielding population guarantees whose statistical complexity is uniform in the retained input resolution.

The challenge of approximating functions with neural networks when inputs reside in infinite-dimensional spaces, but are only observed through a finite number of coordinates, is a fundamental problem in machine learning. This paper addresses this by providing constructive approximation and learning guarantees for shallow neural models under these conditions. The analysis centers on a parameter-normalized neural dictionary and its corresponding weighted variation class. Within this framework, the total approximation error can be distinctly separated into two components: a distribution-dependent term related to coordinate truncation, and a greedy finite-width term. This separation offers a clearer understanding of how different factors contribute to the overall error. For empirical regression tasks, the researchers propose a fully-corrective greedy procedure. This method yields population guarantees where the statistical complexity remains uniform regardless of the input resolution retained. The framework also extends to Hilbert-valued responses without explicit dependence on the output dimension. While the dimension-free statements are statistical, the computational aspect of selecting new neurons still involves solving a non-convex parameter-search problem.

Why it matters

Researchers and advanced AI engineers can leverage these theoretical guarantees to design more robust and efficient neural networks for high-dimensional data, particularly in fields like scientific computing, signal processing, and functional data analysis.

How to implement this in your domain

  1. 1Explore the theoretical framework for designing shallow neural models with infinite-dimensional inputs.
  2. 2Apply the parameter-normalized neural dictionary concept to develop more robust approximation algorithms.
  3. 3Investigate the fully-corrective greedy procedure for empirical regression in high-dimensional settings.
  4. 4Consider the implications of resolution-consistent guarantees for developing models that generalize well across varying input resolutions.

Original post by Pablo M. Bern\'a, Antonio Falc\'o, Diego Mond\'ejar

"arXiv:2608.20812v1 Announce Type: new 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 a…"

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Originally posted by Pablo M. Bern\'a, Antonio Falc\'o, Diego Mond\'ejar on X · view source

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