Learning in function space: A continuous formulation and its limits

doi:10.1016/j.jfranklin.2026.109031

Learning in function space: A continuous formulation and its limits

2026-08-31

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Angshul Majumdar

DOI：10.1016/j.jfranklin.2026.109031

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Learning problems are frequently optimized from finitely many point evaluations although the unknown predictor belongs naturally to an infinite-dimensional space. A direct function-space formulation is therefore useful only if the population problem, perturbations of the data-generating primitives, and empirical discretizations can be connected under assumptions that are verifiable before the desired convergence conclusions are invoked.

Keywords:

Function-space learning

Compact observation operator

Empirical risk minimization

Γ-convergence

Concentration inequality

Strong convergence

Reproducing-kernel Hilbert space

Learning in function space: A continuous formulation and its limits

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