[Paper Review] Nonlocal $p$-Laplacian Variational problems on graphs

Yosra Hafiene, Jalal Fadili+1 more|arXiv (Cornell University)|Oct 30, 2018

Advanced Mathematical Modeling in Engineering44 references4 citations

TL;DR

This paper establishes the consistency of discrete nonlocal $p$-Laplacian variational problems on graphs by deriving explicit error bounds between discrete solutions and their continuum limit. It proves convergence and provides convergence rates for dense and random graph sequences, showing that solutions to the discrete problem converge to the unique solution of the continuum variational problem as the number of vertices increases, with rates dependent on data smoothness and kernel discretization.

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.

Motivation & Objective

• To rigorously analyze the consistency of discrete nonlocal $p$-Laplacian variational problems on graphs as the number of vertices tends to infinity.
• To derive explicit error bounds in $L^2$ between the discrete solution and the continuum solution, quantifying the influence of data and kernel discretization.
• To establish convergence rates for the discrete solution to the continuum solution across various graph sequences, including deterministic dense graphs and random inhomogeneous graphs.
• To provide theoretical justification for the use of nonlocal regularization on discrete data structures such as point clouds and images, linking discrete models to continuum PDEs.
• To offer insights into improving discrete algorithms by understanding their asymptotic behavior and convergence properties.

Proposed method

• Formulates a nonlocal variational problem on a continuum domain $\Omega = [0,1]$ with a regularization term based on the $L^p$-norm of the nonlocal gradient operator $\nabla_K u(x,y) = K(x,y)^{1/p}(u(y) - u(x))$.
• Proposes a discrete counterpart on graphs with $n$ vertices, minimizing a functional combining data fidelity $\|u_n - g_n\|_{L^2}^2$ and a discrete nonlocal $p$-Dirichlet energy $R_{n,p}(u_n, K_n)$.
• Derives a general $L^2$ error bound between the continuum solution $u^*$ and the discrete solution $u_n^*$, explicitly depending on the discretization error of the kernel $K$ and the initial data $g$.
• Applies the error bound to graph sequences converging to a limit graphon, proving convergence of discrete solutions to the continuum solution under mild regularity assumptions on $K$ and $g$.
• Uses graphon theory and random graph models (e.g., $G_{q_n}(n,K)$) to analyze convergence in random inhomogeneous graphs, establishing rates involving $n$ and logarithmic terms.
• Employs numerical experiments on point clouds and signals to validate theoretical convergence rates, using $p=1$ and $p=2$ with primal convergence and error tracking.

Experimental results

Research questions

• RQ1 Does the minimizer of the discrete nonlocal $p$-Laplacian problem on graphs converge to a unique solution of the continuum variational problem as the number of vertices $n \to \infty$?
• RQ2 What is the rate of convergence of the discrete solution to the continuum solution, and how does it depend on the smoothness of the data $g$, the kernel $K$, and the graph structure?
• RQ3 How do discretization errors in the kernel $K$ and the initial data $g$ affect the overall error between the discrete and continuum solutions?
• RQ4 Can the continuum limit provide theoretical justification and improvement strategies for discrete algorithms used in image, signal, and point cloud processing?
• RQ5 How do convergence rates behave in random graph models, such as inhomogeneous Erdős–Rényi graphs, and do they match theoretical predictions?

Key findings

• The discrete solution $u_n^*$ to the nonlocal $p$-Laplacian variational problem converges in $L^2$ to the unique minimizer $u^*$ of the continuum problem as $n \to \infty$ for convergent graph sequences.
• An explicit error bound is derived, showing that the $L^2$ error between the discrete solution and the continuum solution is controlled by the $L^2$-norm of the kernel discretization error and the data discretization error.
• For Lipschitz-continuous data and kernels, the convergence rate is $O(n^{-1/2})$ in the case of dense graph sequences, consistent with numerical experiments on point clouds.
• For random inhomogeneous graphs, the convergence rate is $O\left(\left(\frac{\log n}{n}\right)^{1/2}\right)$, which matches the theoretical prediction from Theorem 6.2 and is validated numerically.
• Numerical experiments on point cloud denoising with $p=1$ confirm the primal convergence rate of $o(1/k)$, and the error between discrete solutions and a reference solution decreases at a rate consistent with $O(n^{-1/2})$.
• The consistency results are robust to the choice of graph structure and hold in higher dimensions, with applications to image and point cloud processing demonstrated via numerical examples.