2019 # 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 app…

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“…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]):…”

confidence: 99%

“…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]):…”

confidence: 99%

“…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.…”

confidence: 99%

“…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.…”

confidence: 99%

“…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…”

confidence: 99%