Continuum Limits of Nonlocal $p$-Laplacian Variational Problems on Graphs

Abstract

In this paper, we study a nonlocal variational problem which consists of minimizing in L 2 the sum of a quadratic data fidelity and a regularization term corresponding to the L p -norm of the nonlocal gradient. In particular, we study convergence of the numerical solution to a discrete version of this nonlocal variational problem to the unique solution of the continuum one. To do so, we derive an error bound and highlight the role of the initial data and the kernel governing the nonlocal interactions. When applied to variational problem on graphs, this error bound allows us to show the consistency of the discretized variational problem as the number of vertices goes to infinity. More precisely, for networks in convergent graph sequences (simple and weighted deterministic dense graphs as well as random inhomogeneous graphs), we prove convergence and provide rate of convergence of solutions for the discrete models to the solution of the continuum problem as the number of vertices grows.

Torsional rigidity in random walk spaces

“…In the case of the random walk space associated with a locally finite weighted discrete graph G = (V, E) (as defined in Example 2.9), the m G -Laplace operator coincides with the graph Laplacian (also called the normalized graph Laplacian) studied by many authors (see, for example, [4], [5], [16], [18], [24]):…”

Discrete p-bilaplacian Operators on Graphs

“…where A : H(V ) → H(V ) is a linear operator, f ∈ H(V ), λ > 0 is the regularization parameter, and F d (•; p) is given by (1). Problems of the form (6) can be of great interest for graph-based regularization in machine learning and inverse problems in imaging; see [7] and references therein. Problem ( 6) is well-posed under standard assumptions.…”

“…where τ, σ > 0, proj Hg(V ) is the orthogonal projector on the subspace H g (V ) (which has a trivial closed form), 1/p + 1/q = 1, and prox σ q • q q is the proximal mapping of the proper lsc convex function σ q • q q . The latter can be computed easily, see [7] for details. Combining [3,Theorem 1], Proposition 1 and Theorem 1, the convergence guarantees of ( 8) are summarized in the following proposition.…”

Partial differential equations and variational methods for geometric processing of images

“…The techniques developed to associate point cloud based functions with continuum functions have recently also been applied to prove consistency of other statistical methods [40,41] and to show that certain artificial neural networks have continuum limits that take the form of ODEconstrained variational models [79]. For discrete-to-continuum limit results, also graphon methods have been considered [45,46,58] 1.6. Further applications…”