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Minimum-Norm Interpolation in Weighted Variation Spaces: Direct and Inverse Problems

  • Weizhao Li
  • Lei Shi
  • Hanfei Zhou
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Abstract

We study minimum-norm interpolation in weighted variation spaces generated by ReLUs ridge functions on the unit ball, where s is a positive integer. For a target f in the weighted variation space and a node set with sufficiently small fill distance h, every minimum-norm interpolant satisfies the direct bound |fβˆ’β„Xhf|L2(𝔹d)≲d,sh(2s+1)/4|f|𝒱s. Under a boundary separation condition on the nodes, we construct a matching lower bound for the worst-case interpolation error. For the inverse problem, we introduce dual certificates and derive an estimate analogous to a Bernstein inequality. Under suitable upper bounds on dictionary size and spectral lower bounds for the atomic Gram matrices, sufficiently fast L2 convergence of minimum-norm interpolants along geometrically decreasing fill distances implies that the target belongs to 𝒱s. As a byproduct, the best L2 approximation error En(f)L2(𝔹d) for shallow ReLUs networks of width at most n satisfies En(f)L2(𝔹d)≲d,snβˆ’1/2βˆ’(2s+1)/(2d)|f|𝒱s, with the weighted atomic cost of an approximating network bounded by a constant multiple of |f|𝒱s, independently of n.

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Cite

BibTeXRIS
BibTeX
@misc{li2026minimumnorm,
  title = {{Minimum-Norm Interpolation in Weighted Variation Spaces: Direct and Inverse Problems}},
  author = {Weizhao Li and Lei Shi and Hanfei Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01105},
  eprint = {2611.01105},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01105},
}
RIS
TY  - UNPB
TI  - Minimum-Norm Interpolation in Weighted Variation Spaces: Direct and Inverse Problems
AU  - Weizhao Li
AU  - Lei Shi
AU  - Hanfei Zhou
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01105
UR  - https://arxiv.org/abs/2611.01105
AB  - We study minimum-norm interpolation in weighted variation spaces generated by ReLU^s ridge functions on the unit ball, where s is a positive integer. For a target f in the weighted variation space and a node set with sufficiently small fill distance h, every minimum-norm interpolant satisfies the direct bound |fβˆ’β„_X_hf|_L_2(𝔹^d)≲_d,sh^(2s+1)/4|f|_𝒱_s. Under a boundary separation condition on the nodes, we construct a matching lower bound for the worst-case interpolation error. For the inverse problem, we introduce dual certificates and derive an estimate analogous to a Bernstein inequality. Under suitable upper bounds on dictionary size and spectral lower bounds for the atomic Gram matrices, sufficiently fast L_2 convergence of minimum-norm interpolants along geometrically decreasing fill distances implies that the target belongs to 𝒱_s. As a byproduct, the best L_2 approximation error E_n(f)_L_2(𝔹^d) for shallow ReLU^s networks of width at most n satisfies E_n(f)_L_2(𝔹^d)≲_d,sn^βˆ’1/2βˆ’(2s+1)/(2d)|f|_𝒱_s, with the weighted atomic cost of an approximating network bounded by a constant multiple of |f|_𝒱_s, independently of n.
ER  -