The study classifies biharmonic submanifolds in a sphere using specific eigenmaps.
problem Characterizing biharmonic submanifolds in a sphere.
method Classification based on bi-eigenmaps and buckling eigenmaps.
result Generalizations of Takahashi's characterization of minimal submanifolds in a sphere.
Transformers interpreted as probabilistic Laplacian Eigenmaps steps.
problem Improving transformer performance through probabilistic interpretation.
method Probabilistic Laplacian Eigenmaps model derivation and graph diffusion step.
result Subtracting identity from attention matrix improves transformer performance.
Paper analyzes convergence of Laplacian eigenmaps on submanifolds with singularities.
problem Analyzing convergence of Laplacian eigenmaps on submanifolds with singularities.
method Using ε-neighborhood graphs constructed from random points on the submanifold, the paper provides a spectral approximation result for the Laplacian.
result The convergence rate for the eigenvalue of the Laplacian is \( O\left(\left(\log n/n
ight)^{1/(m+2)}
ight) \), where \( m \) and \( n \) are the dimension of the manifold and the sample size, respectively.
Study shows rates for Laplacian-eigenmap methods in nonparametric regression.
problem Minimizing error in nonparametric regression using Laplacian-eigenmap.
method Adaptive and non-adaptive minimax rates using Sobolev space constraints.
result Extends minimax rates to various weighted Laplacian matrices.
Survey of Laplacian-based methods for data dimensionality reduction and embedding.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving important features and structures.
method Laplacian-based methods including spectral clustering, Laplacian eigenmap, locality preserving projection, graph embedding, and diffusion map.
result Comprehensive overview of various optimization variants and applications of Laplacian-based techniques.
Manifold learning and dimensionality reduction techniques are ubiquitous in science and engineering, but can be computationally expensive procedures when applied to large data sets or when similarities are expensive to compute. To date, little work has been done to investigate the tradeoff between computational resourc…
Study improves regularity estimates for harmonic maps into ellipsoids.
problem Independence of regularity estimates on harmonic maps with varying target dimensions.
method Analyzes harmonic maps into ellipsoids, uses Palais-Smale sequences, and critical metrics.
result Enhanced regularity estimates for Laplace harmonic eigenmaps.
Develops a framework for generating harmonic maps from a unit ball to a sphere.
problem Creating families of harmonic maps from a unit ball to a sphere.
method Based on Toth's machinery for generating eigenmaps, combined with a generalized radial projection.
result Provides theoretical background for constructing solutions to variational problems.
Proves isometric embeddings in Euclidean spaces for RCD spaces.
problem Isometric immersions of RCD spaces in Euclidean spaces.
method Analyzes regular isometric immersions and eigenmaps of compact non-collapsed RCD spaces.
result Eigenmaps of compact non-collapsed RCD spaces are locally bi-Lipschitz embeddings to spheres.
Root Laplacian Eigenmaps help in spectral embedding of graphs.
problem Efficient spectral embedding of graphs.
method Square root of graph-Laplacian operator.
result Improved spectral embedding techniques.
Proposes EOT eigenmaps for aligning and embedding multiple datasets.
problem Aligning and embedding multiple datasets with shared structures but individual distortions.
method Entropic Optimal Transport (EOT) eigenmaps, leveraging leading singular vectors of EOT plan matrix.
result Proves theoretical guarantees and favorable properties for aligning and embedding datasets.
The study uses unsupervised machine learning to identify top European football teams.
problem Selecting teams for the new European football Super League.
method Used Laplacian eigenmaps clustering on performance data.
result Successfully identified four clusters of teams based on performance metrics.
Neumann eigenmaps improve landmark-based diffusion map embeddings.
problem Landmark-based diffusion map embeddings can be computationally inefficient and unstable.
method NeuMaps use a renormalized Neumann Laplacian for eigendecomposition, incorporating landmarks as a subgraph.
result NeuMaps offer a computationally efficient and stable embedding method.
New method for triharmonic maps to spheres in various dimensions.
problem Creating triharmonic maps to spheres in different dimensions.
method Construction method based on eigenmaps and suitable deformations.
result Existence of triharmonic maps from Rm∖{0} into spheres. LDLE embeds manifolds in lower dimensions with low distortion.
problem Embedding manifolds in lower dimensions with low distortion.
method Constructs local views using global eigenvectors of the graph Laplacian, registers them using Procrustes analysis, and tears manifolds apart for intrinsic dimension embedding.
result LDLE preserves distances up to a constant scale with low distortion.
PCR-LE achieves optimal rates for nonparametric regression over Sobolev spaces.
problem Nonparametric regression over Sobolev spaces with random design.
method PCR-LE using Laplacian Eigenmaps on neighborhood graphs.
result PCR-LE achieves minimax rates of convergence for both estimation and goodness-of-fit testing.
We analyze the performance of a class of manifold-learning algorithms that find their output by minimizing a quadratic form under some normalization constraints. This class consists of Locally Linear Embedding (LLE), Laplacian Eigenmap, Local Tangent Space Alignment (LTSA), Hessian Eigenmaps (HLLE), and Diffusion maps.…
Heat kernels map RCD spaces to Riemannian manifolds.
problem Mapping RCD spaces to Riemannian manifolds using heat kernels.
method Using heat kernels to map RCD spaces into L2 space and then normalizing to achieve isometric immersions. result Compact RCD spaces with isometrically heat kernel immersions are isometric to unweighted smooth Riemannian manifolds.
Optimizes metrics on surfaces for eigenvalues.
problem Finding optimal metrics for eigenvalues on surfaces.
method Combining constructions of Palais-Smale-like sequences and techniques from Karpukhin et al.
result Existence of optimal metrics for various eigenvalues on surfaces.
Study Hodge Laplacians for manifold data, improving error bounds.
problem Approximating Laplace-Beltrami operator on differential forms.
method Higher-order graph Laplacians (Hodge Laplacians) as approximations.
result High-probability error bound for Dirichlet forms.
The paper extends manifold learning to arbitrary norms, improving molecular motion mapping.
problem Improving manifold learning for non-Euclidean norms.
method Determines the limiting differential operator for graph Laplacians using any norm.
result A modified Laplacian eigenmaps algorithm using Earthmover's distance outperforms Euclidean methods in molecular motion mapping.
Study spectral properties of graph Laplacian for manifold data.
problem Understanding spectral properties of graph Laplacian for manifold data.
method Non-asymptotic error bounds on spectral properties of empirical graph Laplacian.
result Eigenvalues and eigenspaces of empirical graph Laplacian are close to Laplace-Beltrami operator of manifold.
A new method simplifies HLLE for better robustness.
problem Improving robustness of Hessian locally linear embedding.
method Replacing Hessian with arbitrary weights and modifying manifold dimension.
result Achieved a new LLE-type method called tangential LLE.
Study evaluates graph-based semi-supervised learning under noisy label conditions.
problem Evaluation of semi-supervised learning algorithms under noisy label conditions.
method Compared graph-based semi-supervised algorithms under varying labeled data and label noise conditions.
result Laplacian Eigenmaps performed better than label propagation under noisy conditions.
In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [TY13] and [TXY14]. Eells and…
Introduces a probabilistic framework for dimension reduction methods.
problem Lack of clear probabilistic foundations for popular DR methods.
method A unifying statistical framework based on the coupling of hidden graphs using cross entropy.
result Existing DR methods suffer from a statistical deficiency that affects performance.
In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherica…
A new spline method for manifold learning using Hessian-based curvature penalties.
problem Learning manifolds with curvature penalties in high dimensions.
method Generalizes thin-plate splines to flat manifolds using Hessian matrices, minimizing square error with curvature constraints.
result Existence and uniqueness of the spline solution, expressed as Green's functions and Hessian approximations.
Neighbor embeddings balance attraction and repulsion to visualize data.
problem Visualizing high-dimensional datasets with trade-offs between continuous and discrete structures.
method Neighbor embeddings combine attractive and repulsive forces to visualize data.
result Changing the exaggeration parameter in t-SNE yields a spectrum of embeddings with a trade-off between continuous and discrete structures.
We consider the problem of estimating a function defined over n locations on a d-dimensional grid (having all side lengths equal to n1/d). When the function is constrained to have discrete total variation bounded by Cn, we derive the minimax optimal (squared) ℓ2 estimation error rate, parametrized by …
Overview of geometric analysis for manifold learning.
problem Analyzing high-dimensional data via spectral embeddings.
method Heat kernel and eigenfunctions on Riemannian manifolds.
result Uniform control of spectral embeddings on key classes of manifolds.
We give several construction methods and use them to produce many examples of proper biharmonic maps including biharmonic tori of any dimension in Euclidean spheres (Theorem 2.2, Corollaries 2.3, 2.4, and 2.6), biharmonic maps between spheres (Theorem 2.9) and into spheres (Theorem 2.10) via orthogonal multiplications …
We use the octonionic multiplication ⋅ of S7 to associate, to each unit normal section η of a submanifold M of S7, an octonionic Gauss map γη:M→S6, γη(x)=x−1⋅η(x), x∈M, where S6 is the unit sphere of T1S7, 1 i…
In this paper we study the family of embeddings Φt of a compact RCD∗(K,N) space (X,d,m) into L2(X,m) via eigenmaps. Extending part of the classical results by Bérard, Bérard-Besson-Gallot, known for closed Riemannian manifolds, we prove convergence as t↓0 of the rescaled pull-back metrics $Φ_t^*g…
In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man ifold geometry using either local or global features of the data. Building on t…
We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Wei…
We present GraphTSNE, a novel visualization technique for graph-structured data based on t-SNE. The growing interest in graph-structured data increases the importance of gaining human insight into such datasets by means of visualization. Among the most popular visualization techniques, classical t-SNE is not suitable o…
We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian…
Paper proposes a new method for supervised manifold learning using random forest proximities.
problem Existing supervised manifold learning methods fail to uncover meaningful embeddings due to using class-conditional distances.
method Proposes a data-geometry-preserving variant of random forest proximities as an initialization for manifold learning methods.
result Local and global structure preservation is near universal across manifold learning approaches using diffusion-based algorithms.
Method reconstructs missing wind farm data using graph theory and nearest neighbors.
problem Missing data in wind farm records due to sensor failures.
method Combines spectral graph theory and k-Nearest Neighbors to estimate missing data.
result Significant improvement in data reconstruction over existing methods.
Develops nonparametric regression for non-smooth functions using fractional Laplacian.
problem Non-smooth regression functions in high dimensions.
method Fractional Laplacian eigenmaps for L2-fractional Sobolev spaces. result Upper bound on estimation error of $n^{- rac{2s}{2s+d}}$.
We introduce the concept of Hypoelliptic Diffusion Maps (HDM), a framework generalizing Diffusion Maps in the context of manifold learning and dimensionality reduction. Standard non-linear dimensionality reduction methods (e.g., LLE, ISOMAP, Laplacian Eigenmaps, Diffusion Maps) focus on mining massive data sets using w…
Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eige…
Quantum computing improves graph neural network aggregation.
problem Limitations of classical GNNs in processing global graph features.
method Quantum computer-generated aggregation weights for graph neural networks.
result Quantum-enhanced GNN performs similarly to classical models on standard datasets.
Paper provides a performance guarantee for spectral clustering.
problem Finding the global solution to the minimum ratio cut problem.
method Two-step spectral clustering method with a rounding step, analyzed using two-to-infinity norm perturbation bounds.
result Spectral clustering is guaranteed to output the global solution under certain conditions.
Method detects trajectory outliers using Hodge Laplacian embeddings.
problem Detecting outliers in trajectory data on simplicial complexes.
method Flow-embeddings using Hodge 1-Laplacian of simplicial complexes.
result Classifies trajectories based on topological behavior.
The paper develops a new Laplacian for manifold learning from data.
problem Learning manifold structures from point cloud data.
method Constructs deformed Hodge Laplacians and proves their spectral convergence.
result Empirical operators converge to the classical Hodge Laplacian.
A new Helmholtzian operator from point clouds for flow analysis.
problem Analyzing flows and vector fields on manifolds from point cloud data.
method Estimation of manifold Helmholtzian from point cloud data using weighted 1-Laplacian.
result The Helmholtzian operator L1 effectively smooths, predicts, and extracts features from flows on manifolds.