We obtain global extensions of the celebrated Nash-Kuiper theorem for isometric immersions of compact manifolds with optimal Hölder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent $θ<1…
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48 results for “Nash-Kuiper embedding”
The paper proves isometric embeddings for smooth manifolds.
problem Embedding smooth manifolds isometrically into Euclidean spaces.
method Nash-Kuiper's convex integration construction and gluing technique.
result Infinitely many global isometric embeddings into 2n-dimensional Euclidean space.
The Nash-Kuiper Theorem states that the collection of -isometric embeddings from a Riemannian manifold into is -dense within the collection of all smooth 1-Lipschitz embeddings provided that . This result is now known to be a consequence of Gromov's more general -principle. Ther…
Hyperbolic Flowersmath.HO
Crochet models show hyperbolic geometry properties.
problem Visualizing hyperbolic geometry concepts.
method Triangulated hyperbolic plane models.
result Demonstrates various hyperbolic geometry properties.
Extends Nash-Kuiper theorem to higher Hölder exponents.
problem Constructing isometric immersions beyond Borisov's exponent.
method Novel corrugation ansatz, integration by parts, and algebraic decomposition.
result Flexibility of isometric immersions beyond Borisov's exponent.
In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by isometric embeddings. This statement clearly cannot be true for embeddings in general, due to the classi…
In this note we would like to present "an analysts' point of view" on the Nash-Kuiper theorem and in particular highlight the very close connection to some aspects of turbulence -- a paradigm example of a high-dimensional phenomenon.
New method approximates short immersions as C^{1,θ} isometric immersions for n ≥ 3.
problem Constructing C^{1,θ} isometric immersions of Riemannian metrics.
method Convex integration scheme with iterative integration by parts procedure.
result Uniform approximation of any short immersion by C^{1,θ} isometric immersions for θ < 1/(1+2(n-1)).
The paper improves the approximation of isometric immersions in high codimension.
problem Isometric immersions in high codimension.
method Uniform approximation of short immersions by isometric immersions.
result Achieved regularity for isometric immersions in local settings.
This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration …
We prove that, given a Riemannian metric on the -dimensional disk , any short immersion of into can be uniformly approximated with isometric immersions for any . This statement improves previous results by Yu.F. Borisov and of a joint paper of th…
A theory for the evolution of a metric driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash-Kuiper theorem is then applied to th…
Paper proves optimal decomposition for matrix fields, reducing convex integration steps.
problem Optimizing decomposition of symmetric matrix fields for convex integration.
method Algebraic geometry and topology applications to prove optimality.
result Optimal decomposition with fewer rank-one terms, improving Hölder regularity.
Embeddings are ubiquitous in machine learning, appearing in recommender systems, NLP, and many other applications. Researchers and developers often need to explore the properties of a specific embedding, and one way to analyze embeddings is to visualize them. We present the Embedding Projector, a tool for interactive v…
Proposes QQE for transforming and embedding data distributions.
problem Transforming and embedding data distributions for better representation or visualization.
method Quantile-Quantile Embedding (QQE) using quantile-quantile plot concept.
result QQE allows for better discrimination of classes in some cases.
Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
problem Embedding maps in higher dimensions without self-intersections.
method Lifting maps to embeddings in product spaces.
result Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
New embeddings for manifolds using heat kernels.
problem Constructing canonical conformal embeddings for manifolds.
method Employing heat kernel embedding from Bérard-Besson-Gallot'94 to find canonical conformal embeddings.
result Intrinsic construction of canonical conformal embeddings with dimensions growing exponentially with .
Introduces PELP for graph-enhanced word embeddings.
problem Combining graph side-information into static word embeddings.
method Probabilistic embeddings using Laplacian priors.
result Unified and flexible approach to various embedding methods.
Curvature regularization prevents distortion in graph embeddings.
problem Graph topology patterns distort in Euclidean space, making detection difficult.
method Proposes curvature regularization to enforce flatness in embedding manifolds.
result Significant improvements in five embedding methods on open graph datasets.
Word embeddings are a powerful approach for unsupervised analysis of language. Recently, Rudolph et al. (2016) developed exponential family embeddings, which cast word embeddings in a probabilistic framework. Here, we develop dynamic embeddings, building on exponential family embeddings to capture how the meanings of w…
Proposes cone embedding for better graph hierarchical structure representation.
problem Lack of natural and interpretable hierarchical indicators in graph embeddings.
method Metric cone embedding method to capture hierarchical structure.
result Extracts hierarchical structure from other graph embedding outputs.
Classifies linear embeddings of grassmannians and ind-grassmannians.
problem Understanding linear embeddings of grassmannians and ind-grassmannians.
method Classification through isomorphism of Picard groups and direct limits.
result Most linear embeddings of grassmannians are equivariant.
BC-Aligner maintains backward compatibility of embeddings after frequent updates.
problem Updating embeddings without requiring consumer teams to retrain their models.
method Learning backward compatible embeddings through BC-Aligner.
result BC-Aligner maintains backward compatibility with existing unintended tasks after multiple model version updates.
Unified framework for word embedding models using noise examples.
problem Improving word embedding models with negative sampling.
method Formulated a Word-Context Classification (WCC) framework that generalizes SkipGram word embedding models.
result The best noise distribution is the data distribution, improving both performance and training speed.
Embedding calculus proves convergence for surfaces.
problem Proving convergence of embedding calculus for surfaces.
method Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two.
result Relates Johnson filtration of mapping class group to embedding calculus.
Models use embeddings and attention for better claim severity prediction.
problem Improving predictive modeling of claim severity with categorical data.
method Developed neural networks and attention-based architectures with embeddings.
result Attention-based models enhance predictive performance with contextual augmentation.
A fast graph embedding method for large graphs.
problem Efficiently embedding large graphs for various applications.
method One-hot graph encoder embedding with linear complexity.
result Graph encoder embedding is approximately normally distributed and converges to its mean.
Paper proves impossibility of three desirable properties in node embedding.
problem Understanding limitations of node embedding methods.
method Axiomatic approach to node embedding, proving impossibility of three properties.
result No node embedding method can satisfy all three desirable properties simultaneously.
The study characterizes and verifies equivariant embeddings of symmetric Kählerian manifolds.
problem Characterizing and verifying equivariant embeddings of symmetric Kählerian manifolds.
method Investigation motivated by Cartan and Wallach's theorem on symmetric spaces, focusing on and parallel plurimean curvature.
result If an equivariant embedding has parallel plurimean curvature, it is the extrinsically symmetric one.
Proves uniqueness of embedding complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifold into infinite-dimensional projective space.
method Fine estimates of asymptotics of a balanced embedding.
result Uniqueness of embedding proven.
The paper defines invariants for almost graph embeddings and explores their properties.
problem Understanding the properties and limitations of almost graph embeddings in the plane.
method Introducing and analyzing integer invariants (winding number, Wu numbers) for almost embeddings.
result Some values of invariants are realizable for almost embeddings but not for embeddings.
This paper proposes a new method for embedding sequences using Wasserstein distances.
problem Embedding sequences in a metric space for better pattern recognition.
method Develops a deep learning model that embeds sequences as distributions and uses Wasserstein distances for comparison.
result Distributional embeddings using Wasserstein distances outperform traditional vector embeddings.
Recently, click-through rate (CTR) prediction models have evolved from shallow methods to deep neural networks. Most deep CTR models follow an Embedding\&MLP paradigm, that is, first mapping discrete id features, e.g. user visited items, into low dimensional vectors with an embedding module, then learn a multi-layer pe…
This work analyzes PPR-based node embeddings and their topological information.
problem Understanding and interpreting PPR-based node embeddings.
method Unified framework and two methods for topology recovery.
result PPR-based embeddings maintain more topological information than random walk-based embeddings.
Network representation learning in low dimensional vector space has attracted considerable attention in both academic and industrial domains. Most real-world networks are dynamic with addition/deletion of nodes and edges. The existing graph embedding methods are designed for static networks and they cannot capture evol…
For leveled spatial graphs, we find a surface embedding that allows cellular embedding.
problem Finding a surface embedding for general spatial graphs is not always possible.
method Define leveled property, decompose graph into subgraphs, and construct surface.
result For leveled spatial graphs with a small number of levels, a surface can always be found.
Proposes spherical text embedding for better directional similarity.
problem Directional similarity is more effective but unsupervised text embeddings are typically learned in Euclidean space.
method Develops a spherical generative model and an efficient optimization algorithm for unsupervised word and paragraph embeddings.
result Achieves state-of-the-art performances on various text embedding tasks.
Long spacelike embeddings can be approximated by isometric ones.
problem Approximating long embeddings to isometric embeddings in Lorentzian spaces.
method Proving approximation by constructing isometric embeddings.
result Long spacelike embeddings can be -approximated by isometric embeddings.
Wrinkled Embeddingsmath.GT
A {\it wrinkled embedding} is a topological embedding which is a smooth embedding everywhere on except a set of -dimensional spheres, where has cuspidal corners. In this paper we prove that any rotation of the tangent plane field of a {\it smoothly embedded} submanifold $V\s…
Graph embeddings have become a key and widely used technique within the field of graph mining, proving to be successful across a broad range of domains including social, citation, transportation and biological. Graph embedding techniques aim to automatically create a low-dimensional representation of a given graph, whi…
Recent works reveal that network embedding techniques enable many machine learning models to handle diverse downstream tasks on graph structured data. However, as previous methods usually focus on learning embeddings for a single network, they can not learn representations transferable on multiple networks. Hence, it i…
The paper proves nonexistence and existence results for minimal surfaces in R^4.
problem Proving nonexistence and existence of minimal surfaces in R^4.
method Analyzing complete minimal immersions with finite total curvature and embedded planar ends.
result Existence of embedded minimal spheres in R^4 with 3 embedded planar ends.
Obtaining continuous representations of structural data such as directed acyclic graphs (DAGs) has gained attention in machine learning and artificial intelligence. However, embedding complex DAGs in which both ancestors and descendants of nodes are exponentially increasing is difficult. Tackling in this problem, we de…
Graphs embeddable on torus and linklessly in 3D can be embedded linklessly in standard torus.
problem Embedding linklessly in a standard torus for graphs embeddable on torus and in 3D.
method Analyzing graphs of order 9 and below, showing linkless embedding in standard torus.
result For graphs of order 9 and below, linkless embedding in standard torus is possible.
Geometrically transforms word embeddings into a common space for better comparison.
problem Comparing embeddings from different sources is challenging.
method Applies orthogonal rotations and Mahalanobis scaling to transform embeddings into a shared latent space.
result The method improves word similarity and analogy tasks.
MCE reduces embedding instability in nonlinear dimensionality reduction.
problem Embedding instability caused by random initialization.
method Median of multiple embeddings (MCE) based on large deviation theory.
result MCE achieves consistency at an exponential rate and effectively mitigates instability.
Natural language processing has improved tremendously after the success of word embedding techniques such as word2vec. Recently, the same idea has been applied on source code with encouraging results. In this survey, we aim to collect and discuss the usage of word embedding techniques on programs and source code. The a…
A new method for fast graph embedding using diffusion graphs.
problem Efficiently generating graph embeddings for large networks.
method Diffusion graphs for rapid vertex sequence generation.
result Improved accuracy and performance with higher edge density.