Minimal Lipschitz and ∞-harmonic extensions of vector-valued functions on finite graphs

@article{Bavcak2020MinimalLA,
  title={Minimal Lipschitz and ∞-harmonic extensions of vector-valued functions on finite graphs},
  author={Miroslav Bavc'ak and Johannes Hertrich and Sebastian Neumayer and Gabriele Steidl},
  journal={Information and Inference: A Journal of the IMA},
  year={2020}
}
This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called $\mathrm{lex}$ and $L\mbox{-}\mathrm{lex}$ minimal extensions are actually the same and call them minimal Lipschitz extensions. Then, we prove that the solution of the graph $p$-Laplacians converge to these extensions as $p\to \infty$. Furthermore, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters… 
1 Citations

Figures from this paper

On $$\alpha $$-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces
<jats:p>We introduce the class of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\alpha $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">

References

SHOWING 1-10 OF 46 REFERENCES
Minimal Lipschitz Extensions for Vector-Valued Functions on Finite Graphs
TLDR
It is proved that the minimizers of functionals involving grouped \(\ell _p\)-norms converge to these extensions as \(p\rightarrow \infty \).
Vector‐valued optimal Lipschitz extensions
Consider a bounded open set $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}U \subset \R^n$ and a Lipschitz function $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}g: \partial U \to \R^m$. Does this
Algorithms for Lipschitz Learning on Graphs
TLDR
This work develops fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices using the absolutely minimal Lipschitz extension.
Infinity-Laplacians on Scalar- and Vector-Valued Functions and Optimal Lipschitz Extensions on Graphs
TLDR
The thesis gives an overview of the existing theory and provides some novel results on the approximation of tight Lipschitz extensions for vector-valued functions.
On the p-Laplacian and ∞-Laplacian on Graphs with Applications in Image and Data Processing
TLDR
A new family of partial difference operators on graphs and study equations involving these operators are introduced which enables to interpolate adaptively between Laplacian diffusion-based filtering and morphological filtering, i.e., erosion and dilation.
Minimal lipschitz extension
The thesis is concerned to some mathematical problems on minimal Lipschitz extensions. Chapter 1: We introduce some basic background about minimal Lipschitz extension (MLE) problems. Chapter 2: We
On the numerical approximation of $$\infty $$∞-harmonic mappings
A map $$u : \Omega \subseteq \mathbb {R}^n \longrightarrow \mathbb {R}^N$$u:Ω⊆Rn⟶RN, is said to be $$\infty $$∞-harmonic if it satisfies 1The system (1) is the model of vector-valued Calculus of
Local Lipschitz property for the Chebyshev center mapping over N-nets
We prove local Lipschitz property of the map which puts in correspondence to each exact N-net its Chebyshev center. If dimension of Euclidean or Lobachevsky space is greater than 1 and the net
A Graph Framework for Manifold-Valued Data
TLDR
The basic calculus needed to formulate variational models and partial differential equations for manifold-valued functions are introduced and the proposed graph framework for two particular families of operators, namely, the isotropic and anisotropic graph $p$-Laplacian operators, $p\geq1$.
Nonlocal Inpainting of Manifold-Valued Data on Finite Weighted Graphs
TLDR
This paper introduces a new graph infinity-Laplace operator based on the idea of discrete minimizing Lipschitz extensions, which is used to formulate the inpainting problem as PDE on the graph.
...
...