arXiv subjects

Kehan Shi

Publications and source records attributed to Kehan Shi.

8 recordsLinked to original sources

arXiv2026

An operator-splitting algorithm for the hypergraph $p$-Laplacian with applications to missing data recovery

Hypergraph $p$-Laplacian regularization is a fundamental model in data analysis with successful applications in various tasks. It aims to minimize a nonsmooth and typically large-scale objective function defined as the sum of the $p$-th powers of the Lipschitz regularization over hyperedges. In this paper, we propose an operator-splitting algorithm for the hypergraph $p$-Laplacian that allows us to handle hyperedges separately in a Gauss-Seidel fashion. Each subproblem can be viewed as a generalized graph Lipschitz learning on a hyperedge, for which we introduce an auxiliary variable to overcome the nonsmoothness and solve it with one step of the alternating direction method of multipliers (ADMM). The resulting algorithm performs proximal ADMM updates sequentially over the hyperedges, and its convergence is proven. We test the algorithm on missing data recovery problems, including image sparse inpainting and semi-supervised learning, to demonstrate that it is faster than existing methods.

math.NA↗

arXiv2026

A Nonlocal $p$-Laplacian Interface Model with Sharp Interface

We propose an energy-based nonlocal $p$-Laplacian interface problem. Neumann interface conditions are naturally formulated via the energy, while Dirichlet conditions are enforced through a penalty term. A key feature is that the model retains a sharp interface, which facilitates extension to other interface problems; we illustrate this by developing a nonlocal approximation for the $p$-Laplacian interface problem with membrane conditions. By establishing $\Gamma$-convergence and compactness, we prove that as the nonlocal horizon vanishes, minimizers of the nonlocal functionals converge to those of the local counterparts. Numerical experiments using an efficient finite element method confirm the convergence.

math.AP↗

arXiv2026

Continuum limit of hypergraph $p$-Laplacian equations on point clouds

This paper studies a class of $p$-Laplacian equations on point clouds that arise from hypergraph learning in a semi-supervised setting. Under the assumption that the point clouds consist of independent random samples drawn from a bounded domain $\Omega\subset\mathbb{R}^d$, we investigate the asymptotic behavior of the solutions as the number of data points tends to infinity, with the number of labeled points remains fixed. We show, for any $p>d$ in the viscosity solution framework, that the continuum limit is a weighted $p$-Laplacian equation subject to mixed Dirichlet and Neumann boundary conditions. The result provides a new discretization of the $p$-Laplacian on point clouds. Numerical experiments on high-dimensional data interpolation are presented.

math.AP↗

arXiv2025

Coupling local and nonlocal total variation flow for image despeckling

Nonlocal equations effectively preserve textures but exhibit weak regularization effects in image denoising, whereas local equations offer strong denoising capabilities yet fail to protect textures. To integrate the advantages of both approaches, this paper investigates a coupled local-nonlocal total variation flow for image despeckling. We establish the existence and uniqueness of the weak solution for the proposed equation. Several properties, including the equivalent forms of the weak solution and its asymptotic behavior, are derived. Furthermore, we demonstrate that the weak solutions of the proposed equation converge to the weak solution of the classical total variation flow under kernel rescaling. The importance of coupling is highlighted through comparisons with local and nonlocal models for image despeckling.

math.AP↗

arXiv2025

Nonlocal-to-local convergence of the $p$-Biharmonic evolution equation with the Dirichlet boundary condition

This paper studies the nonlocal $p$-biharmonic evolution equation with the Dirichlet boundary condition that arises in image processing and data analysis. We prove the existence and uniqueness of solutions to the nonlocal equation and discuss the large time behavior of the solution. By appropriately rescaling the nonlocal kernel, we further show that the solution converges to the solution of the classical $p$-biharmonic equation with the Dirichlet boundary condition. Numerical experiments are presented to demonstrate the effectiveness of the nonlocal $p$-biharmonic equation for image inpainting.

math.AP↗

arXiv2024

Hypergraph $p$-Laplacian equations for data interpolation and semi-supervised learning

Hypergraph learning with $p$-Laplacian regularization has attracted a lot of attention due to its flexibility in modeling higher-order relationships in data. This paper focuses on its fast numerical implementation, which is challenging due to the non-differentiability of the objective function and the non-uniqueness of the minimizer. We derive a hypergraph $p$-Laplacian equation from the subdifferential of the $p$-Laplacian regularization. A simplified equation that is mathematically well-posed and computationally efficient is proposed as an alternative. Numerical experiments verify that the simplified $p$-Laplacian equation suppresses spiky solutions in data interpolation and improves classification accuracy in semi-supervised learning. The remarkably low computational cost enables further applications.

math.NA↗

arXiv2024

Hypergraph $p$-Laplacian regularization on point clouds for data interpolation

As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. This paper explores the benefit of the hypergraph structure for the interpolation of point cloud data that contain no explicit structural information. We define the $\varepsilon_n$-ball hypergraph and the $k_n$-nearest neighbor hypergraph on a point cloud and study the $p$-Laplacian regularization on the hypergraphs. We prove the variational consistency between the hypergraph $p$-Laplacian regularization and the continuum $p$-Laplacian regularization in a semisupervised setting when the number of points $n$ goes to infinity while the number of labeled points remains fixed. A key improvement compared to the graph case is that the results rely on weaker assumptions on the upper bound of $\varepsilon_n$ and $k_n$. To solve the convex but non-differentiable large-scale optimization problem, we utilize the stochastic primal-dual hybrid gradient algorithm. Numerical experiments on data interpolation verify that the hypergraph $p$-Laplacian regularization outperforms the graph $p$-Laplacian regularization in preventing the development of spikes at the labeled points.

math.NA↗

arXiv2024

Continuum limit of $p$-biharmonic equations on graphs

This paper studies the $p$-biharmonic equation on graphs, which arises in point cloud processing and can be interpreted as a natural extension of the graph $p$-Laplacian from the perspective of hypergraph. The asymptotic behavior of the solution is investigated when the random geometric graph is considered and the number of data points goes to infinity. We show that the continuum limit is an appropriately weighted $p$-biharmonic equation with homogeneous Neumann boundary conditions. The result relies on the uniform $L^p$ estimates for solutions and gradients of nonlocal and graph Poisson equations. The $L^\infty$ estimates of solutions are also obtained as a byproduct.

math.AP↗