Local-to-global principle for Morse actions on symmetric spaces.
problem Recognizing Morse actions on symmetric spaces.
method Equivariant Morse quasiisometric embeddings of trees into symmetric spaces.
result Algorithmic recognizability of Morse actions and construction of Morse Schottky subgroups.
We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.
We analyze the asymptotic cones of Teichmüller space with the Teichmüller metric, (T(S),dT). We give a new proof of a theorem of Eskin-Masur-Rafi which bounds the dimension of quasiisometrically embedded flats in (T(S),dT). Our approach is an application of the ideas of Behrstock and Behrstock…
Stable cylinders found in hyperbolic groups and curve graphs.
problem Torsionfree hyperbolic groups and curve graphs of surfaces have globally stable cylinders.
method Generalised Sageev's construction to improve fine properties of hyperbolic spaces.
result Proved curve graphs of surfaces admit equivariant quasi-isometric embeddings in finite products of quasitrees.
Study compares hyperbolic and quasihyperbolic metrics in plane domains.
problem Comparing hyperbolic and quasihyperbolic metrics in plane domains.
method Analyzes metric spaces and boundaries of hyperbolic domains, proving equivalence and constructing counterexamples.
result Hyperbolic and quasihyperbolic metric spaces are quasiisometrically equivalent for finitely connected hyperbolic domains, but not in general.
We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex…
A sharp vanishing theorem for the Lp cohomology torsion of Riemannian manifolds with pinched negative curvature is given. It follows that certain negatively curved homogeneous spaces cannot be quasiisometric to better pinched manifolds.
The Lp-cohomology in degree 1 of Riemannian homogeneous spaces is computed. It turns out that reduced cohomology does not vanish exactly for spaces quasiisometric to negatively curved homogeneous spaces.
Lp-cohomology of rank one symmetric spaces of noncompact type is shown to be Hausdorff for values of p where this does not follow from curvature pinching. Using the multiplicative structure on Lp-cohomology, it is shown that no simply connected Riemannian manifold with strictly -1/4-pinched sectional curvature …
We build an augmentation of the Masur-Minsky marking complex by Groves-Manning combinatorial horoballs to obtain a graph we call the augmented marking complex, AM(S). Adapting work of Masur-Minsky, we prove that AM(S) is quasiisometric to Teichmüller space with the Teichmüller metric. A similar …
Study groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel.
problem Understanding PD3 groups and their properties. method Coarse generalization of Shapiro's lemma, homological isoperimetric inequalities, and Margolis's coarse homological algebra.
result Groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel are either torus or Klein-bottle bundles over S^1, or quasiisometric to Riemannian manifolds.
Surjectivity of Cannon-Thurston map proven for metric graph bundles.
problem Proving surjectivity of Cannon-Thurston map in metric graph bundles.
method Generalized Mj-Sardar's result to include more types of fibers.
result Continuous extension map between boundaries is surjective.
New groups with distinct Dehn functions and properties.
problem Finding groups with different Dehn functions.
method Created specific Lie groups and Carnot graded groups.
result Groups with uniform lattices have different asymptotic cones and Dehn functions.
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 t. 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.
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.
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.
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 CPn 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.
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.
Long spacelike embeddings can be approximated by isometric ones.
problem Approximating long embeddings to isometric embeddings in Lorentzian spaces.
method Proving approximation by constructing C1 isometric embeddings. result Long spacelike embeddings can be C0-approximated by C1 isometric embeddings. A {\it wrinkled embedding} f:Vn→Wm is a topological embedding which is a smooth embedding everywhere on V except a set of (n−1)-dimensional spheres, where f has cuspidal corners. In this paper we prove that any rotation of the tangent plane field TV⊂TW 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…
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.
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…
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.
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…
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…
Paper uses JIVE to decompose word embeddings, improving sentiment analysis performance.
problem Improving sentiment analysis performance on word embeddings.
method Joint and individual variance explained (JIVE) method for decomposition.
result Mapping word embeddings into joint components improves sentiment analysis performance.
Well-quasi-orders proved on embedded planar graphs.
problem Proving well-quasi-orders on embedded planar graphs.
method Careful analysis and extensions of classical methods for embedded minor relations.
result Embedded minor relations are well-quasi-orders on various classes of embedded planar graphs.
We seek to better understand the difference in quality of the several publicly released embeddings. We propose several tasks that help to distinguish the characteristics of different embeddings. Our evaluation of sentiment polarity and synonym/antonym relations shows that embeddings are able to capture surprisingly nua…