Generalization error of minimum weighted norm and kernel interpolation

Weilin Li

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arXiv · 2008.03365

Generalization error of minimum weighted norm and kernel interpolation

Weilin Li

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Abstract

We study the generalization error of functions that interpolate prescribed data points and are selected by minimizing a weighted norm. Under natural and general conditions, we prove that both the interpolants and their generalization errors converge as the number of parameters grow, and the limiting interpolant belongs to a reproducing kernel Hilbert space. This rigorously establishes an implicit bias of minimum weighted norm interpolation and explains why norm minimization may either benefit or suffer from over-parameterization. As special cases of this theory, we study interpolation by trigonometric polynomials and spherical harmonics. Our approach is from a deterministic and approximation theory viewpoint, as opposed to a statistical or random matrix one.

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Weilin Li. 2020-08-07. Generalization error of minimum weighted norm and kernel interpolation. https://arxiv.org/abs/2008.03365

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