Weighted nonlocal operators and their applications in semi-supervised learning

TL;DR

This paper investigates weighted nonlocal operators with singular boundary behavior, establishing conditions for well-posedness and demonstrating convergence to local solutions as nonlocal effects diminish.

Contribution

It introduces a class of weighted nonlocal operators with power-type weights and characterizes the parameter ranges for well-posed variational problems, including convergence results.

Findings

Identified exponent ranges for well-posedness based on dimension and boundary conditions

Proved variational convergence of nonlocal solutions to local weighted Sobolev solutions

Analyzed the impact of boundary singularities on nonlocal operator behavior

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.