arXiv research

We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…

In this paper are given examples of tori T^2 embedded in S^3 with all their asymptotic lines dense.

In this paper are given examples of tori T2 embedded in R3 with all their principal lines dense. These examples are obtained by stereographic projection of deformations of the Clifford torus in S3.

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.

ViCE uses superpixels to enhance self-supervised learning for better dense visual embeddings.

The square-peg problem is solved using configuration spaces and multijet transversality.

We study the problem of large-scale network embedding, which aims to learn latent representations for network mining applications. Previous research shows that 1) popular network embedding benchmarks, such as DeepWalk, are in essence implicitly factorizing a matrix with a closed form, and 2)the explicit factorization o…

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an …

New method uses product embeddings to predict bundle success.

In this paper we construct a properly embedded holomorphic disc in the unit ball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on $\mathbb{B}^2$; on the other hand, its boundary curve is eve…

Inducing sparseness while training neural networks has been shown to yield models with a lower memory footprint but similar effectiveness to dense models. However, sparseness is typically induced starting from a dense model, and thus this advantage does not hold during training. We propose techniques to enforce sparsen…

In this paper we construct a complete injective holomorphic immersion $\mathbb{C}\to\mathbb{C}^2$ whose image is dense in $\mathbb{C}^2$. The analogous result is obtained for any closed complex submanifold $X\subset \mathbb{C}^n$ for $n>1$ in place of $\mathbb{C}\subset\mathbb{C}^2$. We also show that, if $X$ intersect…

We prove that every proper $n$-dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space $\mathbb{R}^{3n+6,1}$. By an "approximate isometric embedding" we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.

SOLAR improves search efficiency and accuracy with sparse, orthogonal embeddings.

In this paper, we prove (1): for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a $C^\infty$-generic Riemannian metric, the union of closed geodesics is dense. The key observation is $C^\infty$-closing lemma for 3D R…

A dense amalgam connects boundaries of groups split by finite subgroups.

The Nash-Kuiper Theorem states that the collection of $C^1$-isometric embeddings from a Riemannian manifold $M^n$ into $\mathbb{E}^N$ is $C^0$-dense within the collection of all smooth 1-Lipschitz embeddings provided that $n < N$. This result is now known to be a consequence of Gromov's more general $h$-principle. Ther…

The paper studies PCA of probability measures with varying sample sizes and finds optimal convergence rates.

This work lists and describes the main recent strategies for building fixed-length, dense and distributed representations for words, based on the distributional hypothesis. These representations are now commonly called word embeddings and, in addition to encoding surprisingly good syntactic and semantic information, ha…

A new method uses sinusoidal functions to represent timestamps as dense vectors for improving irregularly sampled time series learning.

Study on stable Hamiltonian topology finds non-density of certain structures.

We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing…

The study constructs a dense orbit in the universal commensurability augmented Teichmüller space.

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclid…

One of the key challenges of visual perception is to extract abstract models of 3D objects and object categories from visual measurements, which are affected by complex nuisance factors such as viewpoint, occlusion, motion, and deformations. Starting from the recent idea of viewpoint factorization, we propose a new app…

Deep metric learning algorithms have been utilized to learn discriminative and generalizable models which are effective for classifying unseen classes. In this paper, a novel noise tolerant deep metric learning algorithm is proposed. The proposed method, termed as Density Aware Metric Learning, enforces the model to le…

New examples of embeddings defy Anosov representation limits.

Let $M$ be a projective toric manifold. We prove two results concerning respectively Kaehler-Einstein submanifolds of M and symplectic embeddings of the standard euclidean ball in M. Both results use the well-known fact that M contains an open dense subset biholomorphic to C^n.

We create interpretable word embeddings through sparse coding.

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces thus proving a conjecture of Yau (1982) for generic …

This work proposes a method to learn sparse representations that are more efficient for large-scale data retrieval.

Let $G$ be a simply connected, solvable Lie group and $Γ$ a lattice in $G$. The deformation space $\mathcal{D}(Γ,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(Γ,G)$ of all lattice embeddings of $Γ$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev…

New subgroup found in Lie groups with unusual properties.

Constructs equivariant embeddings of Hermitian symmetric spaces into tangent spaces.

We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex …

We present a method for training multi-label, massively multi-class image classification models, that is faster and more accurate than supervision via a sigmoid cross-entropy loss (logistic regression). Our method consists in embedding high-dimensional sparse labels onto a lower-dimensional dense sphere of unit-normed …

Modern deep learning-based recommendation systems exploit hundreds to thousands of different categorical features, each with millions of different categories ranging from clicks to posts. To respect the natural diversity within the categorical data, embeddings map each category to a unique dense representation within a…

Real-time traffic volume inference is key to an intelligent city. It is a challenging task because accurate traffic volumes on the roads can only be measured at certain locations where sensors are installed. Moreover, the traffic evolves over time due to the influences of weather, events, holidays, etc. Existing soluti…

Very recently Ben Andrews and Haizhong Li showed that every embedded cmc torus in the three dimensional sphere is axially symmetric. There is a two-parametric family of axially symmetric cmc surfaces; more precisely, for every real number H and every C > 2 (H+\sqrt{1+H^2}) there is an axially symmetry surface Σ_{H,C} w…

In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance betwee…

This note optimizes distributions using kernel mean embeddings with a new parameterization.

DCE learns customer embeddings from digital activity and financial context.

We prove the existence of a minimal (all leaves dense) foliation of codimension one, on every closed manifold of dimension at least 4 whose Euler characteristic is null, in every homotopy class of hyperplanes distributions, in every homotopy class of Haefliger structures, in every differentiability class, under the obv…

The study finds infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.

Constructs examples of domains divided by groups in dimensions 3 and above.

Proves optimal regularity for sphere minimizers in 3-sphere.

Embedding large and high dimensional data into low dimensional vector spaces is a necessary task to computationally cope with contemporary data sets. Superseding latent semantic analysis recent approaches like word2vec or node2vec are well established tools in this realm. In the present paper we add to this line of res…

We propose and evaluate new techniques for compressing and speeding up dense matrix multiplications as found in the fully connected and recurrent layers of neural networks for embedded large vocabulary continuous speech recognition (LVCSR). For compression, we introduce and study a trace norm regularization technique f…