arXiv research

Paper explores coarse embeddings between symmetric spaces and Euclidean buildings, answering open questions.

This paper tightens the generalization error bound for graph embedding in non-Euclidean spaces.

Improves hierarchical clustering in Euclidean space using autoencoders.

Paper constructs a new type of hypersurface in Euclidean spaces.

Topological manifolds can be embedded flatly in high-dimensional Euclidean space and are locally retracts.

Euclidean embeddings of data are fundamentally limited in their ability to capture latent semantic structures, which need not conform to Euclidean spatial assumptions. Here we consider an alternative, which embeds data as discrete probability distributions in a Wasserstein space, endowed with an optimal transport metri…

In this paper, we construct compact embedded $λ$-hypersurfaces with the topology of torus which are called $λ$-torus in Euclidean spaces $\mathbb {R}^{n+1}$.

FastMap-D embeds directed graphs using potential fields.

We prove that any compact almost complex manifold $(M, J)$ of real dimension $2m$ admits a pseudo-holomorphic embedding in a Euclidean space of dimension $4m + 2$, endowed with a suitable non-standard almost complex structure. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class…

New research shows hyperbolic embeddings are useful for global consistency tasks in graphs.

The paper provides bounds for embedding manifolds into Euclidean spaces with group actions.

Proves isometric embeddings in Euclidean spaces for RCD spaces.

Origami creates flat torus models of any size.

A lower bound for the number of 3-periodical billiard trajectories in a manifold embedded in Euclidean space is obtained.

Notes on embedding criteria for smooth manifolds.

In this paper two zero-dimensional compact sets with equal topological and fractal dimensions but embedded in Euclidean space by different ways are under study. Diffraction of plane electromagnetic wave propagated and reflected by fractal surfaces is considered for each of these compact sets placed in vacuum. It is obt…

Paper proposes a matrix optimization model for reliable Euclidean embedding from noisy data.

Study rational homotopy types of embedding spaces of manifolds.

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash $C^1$ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also sh…

Researchers describe a specific type of submanifolds in Euclidean space.

Survey simplifies embedding theorems for manifolds.

New proof for symmetric spaces with rectangular lattices.

Due to Janet-Cartan's theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M,g), it is in general hard to determine the least dimensional Euclidean space into which (M,g) can be locally isom…

New symplectic barriers found in ball embeddings.

We classify all rotational surfaces in Euclidean space whose principal curvatures $κ_1$ and $κ_2$ satisfy the linear relation $κ_1=aκ_2+b$, where $a$ and $b$ are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find clos…

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, Σ)$ among all possible embedded closed/bordered surfaces with given geometric/algebraic g…

New condition ensures submanifolds are skew in small areas.

New method constructs surfaces with constant mean curvature.

We consider the problem of embedding a relation, represented as a directed graph, into Euclidean space. For three types of embeddings motivated by the recent literature on knowledge graphs, we obtain characterizations of which relations they are able to capture, as well as bounds on the minimal dimensionality and preci…

We show that a pseudo-holomorphic embedding of an almost-complex $2n$-manifold into almost-complex $(2n + 2)$-Euclidean space exists if and only if there is a CR regular embedding of the $2n$-manifold into complex $(n + 1)$-space. We remark that the fundamental group does not place any restriction on the existence of e…

A fast binary embedding method preserves Euclidean distances in high-dimensional data.

Topolow embeds dissimilarity data into Euclidean space robustly against non-metricity and sparsity.

Proves local isometric embedding of low-differentiability metrics in 3D space.

Acoustic Neighbor Embeddings map speech and text to fixed dimensions for phonetic confusability.

New obstruction found for embedding Riemannian manifolds into Euclidean spaces.

Universal triangulation for flat tori with 2434 triangles.

Hyperbolic embeddings have recently gained attention in machine learning due to their ability to represent hierarchical data more accurately and succinctly than their Euclidean analogues. However, multi-relational knowledge graphs often exhibit multiple simultaneous hierarchies, which current hyperbolic models do not c…

We discuss constant mean curvature bubbletons in Euclidean 3-space via dressing with simple factors, and prove that single bubbletons are not embedded.

We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the Hölder space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $α> \frac{1}{2}$ the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, wh…

Curvature regularization prevents distortion in graph embeddings.

The paper analyzes side effects of learning from low-dimensional data embedded in a Euclidean space.

We prove that an m-dimensional unit ball D^m in the Euclidean space {\mathbb R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B_r^d \subset {\mathbb R}^d of radius r < 1/2 unless one of two conditions is met -- (1)The embedding manifold has dimension d >= 2m. (2) The embedding is not smoot…

The study compares Euclidean and cosine distances in medical drug prescription prediction.

The notion of ideal embeddings was introduced in [B.-Y. Chen, {Strings of Riemannian invariants, inequalities, ideal immersions and their applications.} The Third Pacific Rim Geometry Conference (Seoul, 1996), 7-60, Int. Press, Cambridge, MA, 1998]. Roughly speaking, an ideal embedding (or a best of living) is an isome…

Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. …

Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.

The space of graphs is often characterised by a non-trivial geometry, which complicates learning and inference in practical applications. A common approach is to use embedding techniques to represent graphs as points in a conventional Euclidean space, but non-Euclidean spaces have often been shown to be better suited f…

Recent work has demonstrated that embeddings of tree-like graphs in hyperbolic space surpass their Euclidean counterparts in performance by a large margin. Inspired by these results and scale-free structure in the word co-occurrence graph, we present an algorithm for learning word embeddings in hyperbolic space from fr…