2020/08/07 by Weilin Li, Li, Weilin
Computer Science · Engineering · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques #cs.IT #cs.NA #math.IT #math.NA
paper · pdf · doi:10.48550/arxiv.2008.03365
31 pages, 2 figures. To appear in SIAM Journal on Mathematics of Data Science
openalex publication_date 2020/08/07 · arxiv created 2021/02/09 · arxiv updated 2021/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.