This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\rr^3$ and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every…
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48 results for “Positional embeddings”
We propose to formulate multi-label learning as a estimation of class distribution in a non-linear embedding space, where for each label, its positive data embeddings and negative data embeddings distribute compactly to form a positive component and negative component respectively, while the positive component and nega…
The paper proves the existence of at least 4 embedded minimal tori in a three-sphere with positive Ricci curvature.
problem Proving the existence of embedded minimal tori in three-spheres with positive Ricci curvature.
method The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory.
result There exist at least 4 distinct embedded minimal tori in the three-sphere with positive Ricci curvature.
This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.
problem Hexagonal grid firing patterns in grid cells.
method Learning a distance-preserving position embedding in neural space using a recurrent neural network.
result The conformal isometric embedding of 2D physical space into neural space explains hexagonal grid firing patterns.
Study estimates gaps in semigroup products, proving embedding properties.
problem Estimating singular value gaps in semigroup products.
method Lower estimates for singular value gaps of free products of semigroups in ping-pong position.
result Groups generated by semigroups in ping-pong position are quasi-isometrically embedded.
Totally geodesically embeddings of infinitely many closed 7-manifolds into 13-dimensional positively curved closed Riemannian manifolds are constructed. The problems of computing pinching constants and existence of other totally geodesical embeddings are discussed.
Paper shows graphs can be embedded in lower dimensions than expected.
problem Choosing the right embedding dimension for graph analysis.
method Utilizes hidden manifold structure to predict lower-dimensional embedding.
result Graphs can be embedded in much lower dimensions than previously thought.
Gaussian kernel fails on circle and related spaces.
problem Gaussian kernel's positive definiteness on non-Euclidean spaces.
method Analyzing the Gaussian kernel on the circle and related metric spaces.
result Gaussian kernel is not positive definite on the circle or spaces admitting circle embeddings.
Forecasting the future traffic flow distribution in an area is an important issue for traffic management in an intelligent transportation system. The key challenge of traffic prediction is to capture spatial and temporal relations between future traffic flows and historical traffic due to highly dynamical patterns of h…
We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral approximation.
The paper proves the existence of embedded hypersurfaces of constant mean curvature in manifolds with positive Ricci curvature.
problem Proving the existence of embedded hypersurfaces of constant mean curvature in manifolds with positive Ricci curvature.
method Using the Allen--Cahn min-max scheme with a non-zero constant prescribing function.
result The existence of embedded, closed λ-CMC hypersurfaces with Morse index 1 for any prescribed non-zero constant λ.
For graphs generated from stochastic blockmodels, adjacency spectral embedding is asymptotically consistent. Further, adjacency spectral embedding composed with universally consistent classifiers is universally consistent to achieve the Bayes error. However when the graph contains private or sensitive information, trea…
New method embeds dynamic networks with stability for node behavior.
problem Embed time-evolving node representations with stability.
method Unfolded adjacency spectral embedding for dynamic networks.
result Method satisfies cross-sectional and longitudinal stability.
In contrast with what happens for Legendrian embeddings, there always exist positive loops of Legendrian immersions.
Study classifies Halin graphs with positive curvature.
problem Classifying Halin graphs with specific curvature.
method Analyzing generalized Halin graphs formed by connecting tree leaves.
result Identified all generalized Halin graphs with positive Lin-Lu-Yau curvature.
This work provides the first unifying theoretical framework for node (positional) embeddings and structural graph representations, bridging methods like matrix factorization and graph neural networks. Using invariant theory, we show that the relationship between structural representations and node embeddings is analogo…
We consider the problem of vertex classification for graphs constructed from the latent position model. It was shown previously that the approach of embedding the graphs into some Euclidean space followed by classification in that space can yields a universally consistent vertex classifier. However, a major technical d…
A new method for federated learning with only positive labels.
problem Learning with only positive labels leads to poor classifier performance.
method Federated Averaging with Spreadout (FedAwS) framework.
result FedAwS can almost match the performance of conventional learning with negative labels.
If a class of finitely generated groups Curly(G) is closed under isometric amalgamations along free subgroups, then every G in Curly(G) can be quasi-isometrically embedded in a group Hat(G) in Curly(G) that has no proper subgroups of finite index. Every compact, connected, non-positively curved space X admits an isomet…
MEANTIME improves sequential recommendation by using multi-temporal embeddings and attention mechanisms.
problem Limited use of timestamp information and information bottleneck in sequential recommendation models.
method MEANTIME employs multiple types of temporal embeddings and attention mechanisms to capture diverse patterns from user behavior sequences.
result MEANTIME outperforms state-of-the-art sequential recommendation methods.
The study finds large Betti numbers in minimal hypersurfaces with positive Ricci curvature.
problem Minimal hypersurfaces with large Betti numbers in manifolds with positive Ricci curvature.
method Constructing sequences of manifolds with embedded minimal hypersurfaces.
result Minimal hypersurfaces have unbounded first Betti numbers.
The book is devoted to constructing embedding finite-dimensional maps into trivial bundles and investigating the corresponding general position properties.
Let M be a 3-manifold (possibly with boundary). We show that, for any positive integer g, there exists an open nonempty set of metrics on M for each of which there are stable compact embedded minimal surfaces of genus g with arbitrarily large area. This extends the result of Colding and Minicozzi for g=1.
Generalized Tanaka prolongation ensures convergence of formal embeddings of complex manifolds.
problem Ensuring convergence of formal embeddings of complex manifolds under weaker conditions.
method Formulated and proved generalized Tanaka prolongation for geometric structures.
result Convergence of formal embeddings holds under weaker semi-positive normal bundle conditions.
Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
Proposes a hierarchical clustering method for positive and negative dissimilarities.
problem Clustering dissimilarities, especially positive and negative.
method Hierarchical correlation clustering followed by tree preserving embedding.
result Performance on various datasets.
In this note it is shown that every 7-dimensional Eschenburg space can be totally geodesically embedded into infinitely many topologically distinct 13-dimensional Bazaikin spaces. Furthermore, examples are given which show that, under the known construction, it is not always possible to totally geodesically embed a pos…
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
In this article, we study a free boundary isometric embedding problem for abstract Riemannian two-manifolds with the topology of the disc. Under the assumption of positive Gauss curvature and geodesic curvature of the boundary being equal to one, we show that any such disc may be isometrically embedded into the Euclide…
Paper introduces a new distance measure for Gaussian Mixture Models.
problem Developing a new distance measure for Gaussian Mixture Models.
method Embedding K-component Gaussian Mixture Models into the manifold of symmetric positive definite matrices and calculating a lower bound for the Fisher-Rao metric.
result Demonstrated effectiveness through experiments on standard datasets.
Due to Čencov's theorem, there exists a unique family of invariant symmetric -tensor fields on the space of positive probability measures on a set of -points indexed by under Markov embeddings. We deform Markov embeddings keeping sufficiency, and prove existence and uniqueness of invariant f…
In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space . This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the sol…
We describe a construction of complete embedded self-translating surfaces under mean curvature flow by desingularizing the intersection of a finite family of grim reapers in general position.
Study of 3-manifolds in 5-sphere using bridge decompositions.
problem Understanding embeddings of 3-manifolds in 5-sphere.
method Introduce and study bridge decompositions, use multisections of 5-manifolds.
result Every embedded 3-manifold admits a bridge decomposition.
A large class of positive finite presentations of the braid groups is found and studied. It is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive relations.
A new condition for the genericity of ergodic measures on non-positively curved Riemannian manifoldsmath.DS
The study shows that ergodic measures are not generic on non-positively curved manifolds.
problem Determining the genericity of ergodic measures on non-positively curved Riemannian manifolds.
method Investigates the existence of an open isometric embedding of a product manifold with a factor isometric to .
result The closure of the set of ergodic measures does not encompass all invariant measures, indicating the failure of genericity.
Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…
Improved speech enhancement with larger neural networks using novel embeddings and biases.
problem Decreased robustness of speech enhancement models to real-world use cases.
method Frequency-positional embeddings, semi-supervised training, biased loss function.
result Better performance on real recordings with improved large neural network architecture.
Found a minimal surface in curved space.
problem Minimal surfaces in curved spaces.
method Constructed a metric with a specific surface.
result Found a minimal surface of index 1.
Surfaces in 4-Manifoldsmath.DG
In this paper we introduce a technique, called rim surgery, which can change a smooth embedding of an orientable surface of positive genus and nonnegative self-intersection in a smooth 4-manifold while leaving the topological embedding unchanged.
We break down transformer embeddings into interpretable components revealing hidden geometric structures.
problem Understanding the hidden geometry and interpretability of transformer models.
method Decomposed transformer embeddings into position, context, and residual components.
result Pervasive mathematical structure in transformer embeddings, including position and context vectors.
Estimates class prior for unlabeled data using kernel embedding.
problem Estimating class prior in PU learning scenario where only positive and full population samples are available.
method Direct estimator based on distribution matching and kernel embedding in Reproducing Kernel Hilbert Space.
result Asymptotic consistency and explicit deviation bound for the estimator.
Computes knot filtered ECH for torus knots on tight 3-sphere.
problem Computing knot filtered ECH for torus knots.
method Generalized knot filtered ECH definition, Morse-Bott methods, energy filtered Seiberg-Witten theory.
result Computed knot filtered ECH for T(2,q) knots (q odd, positive).
GraphReach improves GNN performance by incorporating node positions.
problem Existing GNNs fail to capture node positions, leading to inaccurate predictions.
method GraphReach uses reachability estimations from anchor nodes to capture global node positions.
result GraphReach achieves up to 40% relative improvement in accuracy compared to state-of-the-art GNNs.
DDMI generates high-quality INRs by adapting positional embeddings.
problem Existing INR generative models fail to produce high-quality representations.
method DDMI uses adaptive positional embeddings and a D2C-VAE to enhance expressive power.
result DDMI outperforms existing models across multiple modalities and datasets.
We study the embedded Calabi-Yau problem for complete embedded constant mean curvature surfaces of finite topology or of positive injectivity radius in a simply-connected three-dimensional Lie group X endowed with a left-invariant Riemannian metric. We first prove a half-space theorem for constant mean curvature surfac…
Uncovering Locally Low-dimensional Structure in Networks by Locally Optimal Spectral Embeddingstat.ML
LASE improves local network structure visualization by targeting locally low-dimensional regions.
problem Global spectral embedding fails to capture local geometric features in sparse, transitive networks.
method Local Adjacency Spectral Embedding (LASE) using weighted spectral decomposition.
result LASE reveals locally low-dimensional structure, improving local reconstruction and visualization.
Every lens space has a locally flat embedding in a connected sum of 8 copies of the complex projective plane and a smooth embedding in n copies of the complex projective plane for some positive integer n. We show that there is no n such that every lens space smoothly embeds in n copies of the complex projective plane.