Related papers: When Locally Linear Embedding Hits Boundary

High-dimensional data arises in numerous applications, and the rapidly developing field of geometric deep learning seeks to develop neural network architectures to analyze such data in non-Euclidean domains, such as graphs and manifolds.…

We consider Riemannian optimization problems with inequality and equality constraints and analyze a class of Riemannian interior point methods for solving them. The algorithm of interest consists of outer and inner iterations. We show that,…

Diffusion-based inverse algorithms have shown remarkable performance across various inverse problems, yet their reliance on numerous denoising steps incurs high computational costs. While recent developments of fast diffusion ODE solvers…

Spectral clustering is a key research topic in the field of machine learning and data mining. Most of the existing spectral clustering algorithms are built upon Gaussian Laplacian matrices, which are sensitive to parameters. We propose a…

While the exterior Helmholtz problem with Dirichlet boundary conditions is always well-posed, the associated standard boundary integral equations are not if the squared wavenumber agrees with an eigenvalue of the interior Dirichlet problem.…

This paper proposed a new explicit nonlinear dimensionality reduction using neural networks for image retrieval tasks. We first proposed a Quasi-curvature Locally Linear Embedding (QLLE) for training set. QLLE guarantees the linear…

Given only a collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to solve elliptic partial differential equations (PDEs) supplemented with boundary conditions. Notice…

In manifold learning, algorithms based on graph Laplacians constructed from data have received considerable attention both in practical applications and theoretical analysis. In particular, the convergence of graph Laplacians obtained from…

Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…

The Laplace-Beltrami operator has established itself in the field of non-rigid shape analysis due to its many useful properties such as being invariant under isometric transformation, having a countable eigensystem forming an orthornormal…

The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable properties. Recently, we proposed a novel numerical method, Point…

The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this…

We demonstrate that locally linear embedding (LLE) inherently admits some unwanted results when no regularization is used, even for cases in which regularization is not supposed to be needed in the original algorithm. The existence of one…

A mesh-free numerical method for solving linear elliptic PDE's using the local kernel theory that was developed for manifold learning is proposed. In particular, this novel approach exploits the local kernel theory which allows one to…

The localization is one of the active and fundamental research in topology physics. Based on a generalized Su-Schrieffer-Heeger model with the quasiperiodic non-Hermitian emerging at the off-diagonal location, we propose a novel systematic…

The increasing availability of geometric data has motivated the need for information processing over non-Euclidean domains modeled as manifolds. The building block for information processing architectures with desirable theoretical…

We present a lightweight network that infers grouping and boundaries, including curves, corners and junctions. It operates in a bottom-up fashion, analogous to classical methods for sub-pixel edge localization and edge-linking, but with a…

Modern foundational models are often compressed via a combination of structured pruning and re-training to meet the strict compute, memory, and connectivity constraints of edge deployments. While state-of-the-art pruning schemes target the…

We provide a new interpretation of Hessian locally linear embedding (HLLE), revealing that it is essentially a variant way to implement the same idea of locally linear embedding (LLE). Based on the new interpretation, a substantial…

Multiplex network embedding is an effective technique to jointly learn the low-dimensional representations of nodes across network layers. However, the number of edges among layers may vary significantly. This data imbalance will lead to…