Weighted nonlocal operators and their applications in semi-supervised learning

2024/12/20 by Qiang Du, Du, Qiang, James Scott +1
Decision Sciences · #Analysis of PDEs (math.AP) #FOS: Mathematics #Multi-Criteria Decision Making

paper · pdf · doi:10.48550/arxiv.2412.16109

openalex publication_date 2024/12/20 · openalex created_date 2024/12/24 · openalex updated_date 2026/07/28

Abstract

Motivated by problems in machine learning, we study a class of variational problems characterized by nonlocal operators. These operators are characterized by power-type weights, which are singular at a portion of the boundary. We identify a range of exponents on these weights for which the variational Dirichlet problem is well-posed. This range is determined by the ambient dimension of the problem, the growth rate of the nonlocal functional, and the dimension of the boundary portion on which the Dirichlet data is prescribed. We show the variational convergence of solutions to solutions of local weighted Sobolev functionals in the event of vanishing nonlocality.

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