The first studies concerning the limit behavior of difference operators on random graphs were carried out in [
33] and the follow-up work [
32], which considers the consistency of spectral clustering on graphs. The main motivation of our paper is the works [
22,
35] where the
\(\varGamma \)-convergence of the discrete functionals
$$\begin{aligned} E^{(p)}_n(\mathbf {u}) = \frac{1}{s_n^p n^2}\sum _{x,y\in \varOmega _n} \eta _{s_n}(\left| x-y\right| ) \left| \mathbf {u}(x)-\mathbf {u}(y)\right| ^p \end{aligned}$$
(13)
toward a continuum functional of the form
$$\begin{aligned} {\mathcal {E}}^{(p)}(u) = \sigma _\eta \int _{\varOmega } \left| \nabla {u}\right| ^p \rho ^2(x)~\mathrm {d}x, \end{aligned}$$
for
\(1\le p<\infty \) and
u smooth enough, is shown. Here,
\(\rho \) is the density of the measure
\(\mu \) according to which the points
\(x_i\) are distributed. The
\(\varGamma \)-convergence is considered w.r.t. the
\(TL^{p}\) topology, which allows to compare discrete functions
\(\mathbf {u}\in L^{p}(\nu _n)\) with continuum functions
\(u\in L^{p}(\mu )\) via an optimal transport ansatz. Here,
\(\nu _n\) denotes the empirical measure for the set
\(\varOmega _n\), see (
21). In particular, the problem they study is connected to the extension task (
1) by considering the constrained functional
$$\begin{aligned} E^{(p)}_{n,\mathrm {cons}}(\mathbf {u}):= {\left\{ \begin{array}{ll} E^{(p)}_n(\mathbf {u})&{}\text { if } \mathbf {u}(x_i) = g(x_i) \text { for }x_i\in {\mathcal {O}},\\ \infty &{}\text { else}, \end{array}\right. } \end{aligned}$$
and the associated minimization problem
$$\begin{aligned} \inf _{\mathbf {u}\in L^{p}(\nu _n)}E^{(p)}_{n,\mathrm {cons}}(\mathbf {u}). \end{aligned}$$
(14)
Here, the constraint set
\({\mathcal {O}}\) is assumed to be a fixed collection of finitely many points. The main result they show ensures that minimizers of the functionals
\(E^{(p)}_{n,\mathrm {cons}}\) converge uniformly toward minimizers of a respective constrained version of
\({\mathcal {E}}^{(p)}\) under appropriate assumptions on the scaling
\(s_n\). The motivation for considering the limit
\(p\rightarrow \infty \) for above problems is given in [
2], where the graph
p-Laplacian for
\(p<d\) is noticed to have undesirable properties, namely the solution of problem (
14) tends to equal a constant at the majority of the vertices with sharp spikes at the labeled data. Using a suitable scaling, this problem does not occur in the case
\(p>d\) (which is intuitively connected to the Sobolev embedding
\(W^{1,p}(\varOmega ) \hookrightarrow C^{0,1-d/p}({{\overline{\varOmega }}})\) for
\(p>d\) see [
1, Ch. 4]) and, in particular, as pointed out in [
30, Sec. 3] one generally hopes for better interpolation properties in the case
\(p\rightarrow \infty \). However, in this limit the interpretation of the measure
\(\mu \) and its density changes, namely only its support enters the problem. The functional
\({\mathcal {E}}^{(p)}\) incorporates information about the distribution of the data in the form of the density
\(\rho \). Using standard properties of
\(L^p\)-norms, the limit
\(p\rightarrow \infty \) on the other hand reads
$$\begin{aligned} \mu {\text {-}}{{\,\mathrm{ess\,sup}\,}}_{x\in \varOmega } \left| \nabla {u}\right| = \lim _{p\rightarrow \infty } \big ({\mathcal {E}}^{(p)}(u)\big )^{1/p} \end{aligned}$$
and thus only the support of the measure
\(\mu \) is relevant. This phenomenon was already observed in [
2,
14] which the authors informally described as the limit ‘forgetting’ the distribution of the data. Furthermore, this observation is consistent with our results, since the limit
\(n\rightarrow \infty \) is independent of the sequence
\(\left( x_{i}\right) _{i\in \mathbb {N}}\subset \varOmega \) as long as it fills out the domain
\(\varOmega \) sufficiently fast, see Theorem
1 for the precise condition. In the case
\(p<\infty \) the minimization task (
1) is equivalent to the graph
p-Laplace equation, for which the formal limit
\(p\rightarrow \infty \) leads to the graph infinity Laplace equation
$$\begin{aligned} \begin{aligned} \varDelta _\infty \mathbf {u}&= 0 \text { in } V{\setminus }{\mathcal {O}},\\ \mathbf {u}&= g\text { in }{\mathcal {O}}, \end{aligned} \end{aligned}$$
(15)
where the operator
\(\varDelta _\infty \) on
\(\varOmega _n\) is defined as
$$\begin{aligned} \varDelta _\infty \mathbf {u}(x):=\max _{y\in \varOmega _n}\omega (x,y)(\mathbf {u}(y) - \mathbf {u}(x)) + \min _{y\in \varOmega _n}\omega (x,y)(\mathbf {u}(y) - \mathbf {u}(x)). \end{aligned}$$
One should note that the unique solution of (
15) also solves the Lipschitz learning task (
2), see, e.g., [
14,
30]. A first study concerning the continuum limit of this problem is carried out in [
14]. The main result therein states that the solutions of the discrete problems converge uniformly to the viscosity solution of the continuum equation,
$$\begin{aligned} \begin{aligned} \varDelta _\infty u&=0 \text { in }\varOmega ,\\ u&=g\text { on }{\mathcal {O}}, \end{aligned} \end{aligned}$$
(16)
where the continuum infinity Laplacian for a smooth function
u is defined as
$$\begin{aligned} \varDelta _\infty u := \langle \nabla u, D^2 u\nabla u\rangle = \sum _{i,j=1}^d \partial _i u\,\partial _j u\, \partial ^2_{ij}u. \end{aligned}$$
(17)
The considered domain is the flat torus, i.e.,
\(\varOmega =\mathbb {R}^{d}{\setminus }\mathbb {Z}^d\) and again the constraint set
\({\mathcal {O}}\) is assumed to be fixed and finite. Furthermore, the only requirement on the sequence of points
\(\varOmega _n\subset \varOmega \) is characterized by the value
\(r_n\) defined in (
7).