Proves local isometric embedding of low-differentiability metrics in 3D space.
problem Isometric embedding of metrics of low differentiability in Euclidean 3-space.
method Simplified notation, geodesic and level parameters, solutions of initial value problems for first order non-linear PDEs, classical linear algebraic systems.
result Local isometric embedding exists for metrics of C1 differentiability.
Condition for embedding metric spaces into curved manifolds.
problem Embedding conditions for metric spaces in curved manifolds.
method If-and-only-if condition on five-point metric spaces.
result Five-point metric spaces admit embeddings into nonnegatively curved Riemannian manifolds.
Deep metric learning employs deep neural networks to embed instances into a metric space such that distances between instances of the same class are small and distances between instances from different classes are large. In most existing deep metric learning techniques, the embedding of an instance is given by a featur…
The paper extends isometric embedding results to null cones and spheres.
problem Isometric embedding of metrics on spheres in null cones.
method Extending Li-Wang's result to compact manifolds, specializing to 2D, developing existence and uniqueness theorems, and proving foliations.
result Existence and uniqueness of isometric embeddings in null cones.
We prove that every proper n-dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space R3n+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.
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 C1 Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also sh…
A new method estimates multi-dimensional value distributions using Hilbert space embeddings.
problem Estimating value distributions in complex, multi-dimensional reinforcement learning settings.
method Hilbert space mappings and kernel mean embeddings to estimate the kernel mean embedding of multi-dimensional value distributions.
result Uniform convergence guarantees and robust off-policy evaluation demonstrated in simulations.
Small neural networks embed arbitrary metric spaces into Gaussian mixtures.
problem Embedding arbitrary metric spaces into a fixed space with low distortion.
method Probabilistic transformers of small depth and width.
result Embeddings with low metric distortion for various metric spaces.
In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-k-curved metrics}. Quasi-k-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity.
Proves existence of at least two minimal spheres in any 3D space.
problem Existence of minimal spheres in arbitrary 3D spaces.
method Iterative relative min-max constructions.
result Proves existence of at least two embedded minimal spheres.
This paper tightens the generalization error bound for graph embedding in non-Euclidean spaces.
problem High generalization error in non-Euclidean graph embedding, preventing practical applications.
method Novel upper bound of graph embedding's generalization error using local Rademacher complexity.
result The new bound is tighter and faster, allowing better performance in non-Euclidean spaces.
An embedding of a metric graph (G,d) on a closed hyperbolic surface is \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic su…
Consider the sum of the first N eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for N sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree N to be thos…
The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest pa…
Study conic singular manifolds, proving Lipschitz normal embedding.
problem Understanding metric properties of conic singular manifolds.
method Analyzing interplay between conic and asymptotically conic behavior.
result Proves Lipschitz normal embedding for conic singular sub-manifolds.
Study of metrics on spheres and their complex structure properties.
problem Identifying metrics on spheres and their complex structure properties.
method Identify metrics via Nash isometric embeddings, use isotopic extension theorem, and analyze extrinsic quantities.
result No sphere of dimensions 6 or higher can be diffeomorphic to a complex manifold.
Paper presents a new method for learning hyperbolic representations using tree structures.
problem Learning faithful low-dimensional hyperbolic embeddings of data.
method Metric-first approach to learn tree structure, then embed into hyperbolic manifold.
result Novel fast algorithm TreeRep learns tree approximating original metric.
New KQEs improve probability metrics without mean function constraints.
problem Improving probability metrics without relying on mean function representations.
method Kernel quantile embeddings (KQEs) to construct new distances.
result KQEs offer a competitive alternative to MMD with near-linear cost.
We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric …
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…
New metric found for 4-manifolds with specific properties.
problem Finding metrics on 4-manifolds with embedded spheres.
method Constructing a Riemannian metric with anti-self-dual harmonic forms.
result Existence of a metric representing a cohomology class of a sphere.
We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in 3+1 dimensions as a hypersurface in R4,1. For the Schwarzschild metric the…
Rigidity theorem for discrete metric spaces embedded in Riemannian surfaces.
problem Understanding the rigidity of discrete metric spaces embedded in Riemannian surfaces.
method Proving that certain discrete metric spaces are rigidly embedded in the Euclidean plane or other Riemannian surfaces.
result Riemannian embeddings of certain discrete metric spaces are rigid, meaning they cannot be deformed without changing distances.
We obtain a compact Sobolev embedding for H-invariant functions in compact metric-measure spaces, where H is a subgroup of the measure preserving bijections. In Riemannian manifolds, H is a subgroup of the volume preserving diffeomorphisms: a compact embedding for the critical exponents follows. The results can b…
Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow thes…
A new method embeds distributions in a common space for optimal transport comparison.
problem Comparing distributions in different metric spaces.
method Sub-embedding robust Wasserstein (SERW) distance.
result SERW mimics GW distance properties and provides a cost relation.
The distance metric plays an important role in nearest neighbor (NN) classification. Usually the Euclidean distance metric is assumed or a Mahalanobis distance metric is optimized to improve the NN performance. In this paper, we study the problem of embedding arbitrary metric spaces into a Euclidean space with the goal…
The distance from the origin in the word metric for generalizations F(p) of Thompson's group F is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of F(p). This interpretation of the metric is used to prove that every F(p) admits a quasi-isometric embedding into ever…
The paper proves local isometric embeddings for singular metrics near a point.
problem Existence of local isometric embeddings for singular Riemannian metrics.
method Ramified local isometric embeddings using Leray's ramified Cauchy-Kovalevskaya Theorem.
result Existence of local analytic isometric embeddings into Euclidean space.
Unified framework for hyperbolic embeddings from mixed data types.
problem Computing hyperbolic embeddings from noisy metric and non-metric data.
method Semidefinite programming and spectral factorization methods.
result Efficient computation of hyperbolic embeddings from arbitrary data.
BERT embeddings improve sequence quality metrics.
problem Measuring the quality of generated sequences against references.
method Employ contextual BERT embeddings for sequence-level reward.
result Contextual embeddings provide a more effective learning signal.
Continuous vector representations of words and objects appear to carry surprisingly rich semantic content. In this paper, we advance both the conceptual and theoretical understanding of word embeddings in three ways. First, we ground embeddings in semantic spaces studied in cognitive-psychometric literature and introdu…
Smoothly approximates embeddings in Lorentzian manifolds.
problem Approximating embeddings in Lorentzian manifolds.
method C^0 approximation of embeddings.
result Approximated embeddings can be made smooth.
We show that any metric on S2 with Gauss curvature K≥−κ admits a C1,1-isometric embedding into the hyperbolic space with sectional curvature −κ. We also give a sufficient condition for a metric on S2 to be isometrically embedded into anti-de Sitter spacetime with the prescribed cosmological time fun…
We construct smooth metrics on 2-manifold with nonpositive Gauss curvature which cannot be (C^3) locally isometrically embedded in R^3. Moreover, the Gauss curvature of the metric can be made negative except for one point.
Study geodesic properties of time series data using Wasserstein metric.
problem Modeling nonlinear time series with transport-based metrics.
method Generalized Wasserstein metric and signed cumulative distribution transforms.
result Geodesic properties provide added interpretability and robustness in time series classifiers.
CAMEL enhances manifold embedding and learning with curvature metrics.
problem High-dimensional data classification, dimension reduction, and visualization.
method CAMEL uses a Riemannian manifold with curvature metrics for enhanced expressibility and interpretability.
result CAMEL outperforms state-of-the-art methods on high-dimensional datasets.
Identifies metrics on manifolds and their embeddings into spheres, characterizing constant curvature metrics.
problem Characterizing metrics on manifolds and their embeddings into spheres.
method Identifies metrics on manifolds and their embeddings into spheres, characterizes metrics of constant scalar curvature, and uses Yamabe metrics and almost Hermitian structures.
result Characterizes metrics of constant scalar curvature by properties of extrinsic quantities of their embeddings.
Topolow embeds dissimilarity data into Euclidean space robustly against non-metricity and sparsity.
problem Embedding dissimilarity data into Euclidean space when dissimilarities are non-metric or sparse.
method Topolow uses a physics-inspired, gradient-free optimization framework to maximize likelihood under a Laplace error model.
result Topolow outperforms standard MDS methods in reconstructing sparse and non-Euclidean data.
The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.
problem Conditions for branched minimal immersions of spheres into ellipsoids to be embedded.
method Using eigenfunctions with respect to a critical metric, the study provides sufficient conditions for embeddings and constructions of non-planar minimal spheres.
result Conditions for embeddings and constructions of non-planar minimal spheres using eigenfunctions.
We show that for each n\ge 2 there is a quasi-isometric embedding of the hyperbolic space H^n in the product T^n=Tx...xT of n copies of a (simplicial) metric tree T. On the other hand, we prove that there is no quasi-isometric embedding H^2 --> TxR^m for any metric tree T and any m\ge 0.
Random walks on metric spaces embed quasi-isometrically into the space.
problem Embedding random subgroups of metric spaces quasi-isometrically.
method Analyzing random walks and contracting elements in metric spaces.
result Random subgroups of isometry groups are quasi-isometrically embedded.
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.
We show that any infinite order element g of a virtually cyclic hyperbolically embedded subgroup of a group G is Morse, that is to say any quasi-geodesic connecting points in the cyclic group C generated by g stays close to C. This answers a question of Dahmani-Guirardel-Osin. What is more, we show that hyper…
Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.
problem Understanding non-linear Hopf manifolds and their properties.
method Holomorphic embeddings and LCK metrics.
result Non-linear Hopf manifolds admit LCK metrics.
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.
Deep metric learning is often used to learn an embedding function that captures the semantic differences within a dataset. A key factor in many problem domains is how this embedding generalizes to new classes of data. In observing many triplet selection strategies for Metric Learning, we find that the best performance …
Generalizes embedding formalism for CFTs on curved backgrounds.
problem Capturing CFTs on curved backgrounds and non-trivial states.
method Using ambient metric and geometric invariants of the ambient space.
result Exact agreement with holographic computations and thermal OPEs.