Isometric Embedding Research Articles

Asymptotics of the n-widths of a Wiener spiral in complex Hilbert space

We describe isometric embeddings of the Wiener spiral in complex Hilbert space and obtain the asymptotics of the Kolmogorov n-widths of specific embeddings. We note the difference with the asymptotics of the n-widths of the Wiener spiral in real Hilbert space.

Isometric Embeddings in Imaging and Vision: Facts and Fiction

We explore the practicability of Nash’s Embedding Theorem in vision and imaging sciences. In particular, we investigate the relevance of a result of Burago and Zalgaller regarding the existence of PL isometric embeddings of polyhedral surfaces in ℝ3 and we show that their proof does not extended directly to higher dimensions.

Almost isometric embeddings into configuration-compacta

We show that a Polish metric space X which has ‘compact configuration spaces’ is free of almost isometric embeddings, i.e. given such an embedding into X one gets an isometric embedding. By assuming a continuous version of ℵ 0 -homogeneity we get the converse statement, and also prove almost isometry uniqueness.

Hyperbolic non-Euclidean elastic strips and almost minimal surfaces

We study equilibrium configurations of thin and elongated non-Euclidean elastic strips with hyperbolic two-dimensional reference metrics ā which are invariant along the strip. In the vanishing thickness limit energy minima are obtained by minimizing the integral of the mean curvature squared among all isometric embeddings of ā. For narrow strips these minima are very close to minimal surfaces regardless of the specific form of the metric. We study the properties of these "almost minimal" surfaces and find a rich range of three-dimensional stable configurations. We provide some explicit solutions as well as a framework for the incorporation of additional forces and constraints.

The Bañados, Teitelboim, and Zanelli spacetime as an algebraic embedding

A simple algebraic global isometric embedding is presented for the nonrotating Bañados, Teitelboim, and Zanelli black hole and its counterpart of Euclidean signature. The image of the embedding, in Minkowski space of two extra dimensions, is the intersection of two quadric hypersurfaces. Furthermore an embedding into AdS4 or H4 is also obtained, showing that the spacetime is of embedding class one with respect to maximally symmetric space of negative curvature. The rotating solution of Euclidean signature is also shown to admit a quadratic algebraic embedding, but seemingly requires more than two extra dimensions.

The Fibonacci Dimension of a Graph

The Fibonacci dimension ${\rm fdim}(G)$ of a graph $G$ is introduced as the smallest integer $f$ such that $G$ admits an isometric embedding into $\Gamma_f$, the $f$-dimensional Fibonacci cube. We give bounds on the Fibonacci dimension of a graph in terms of the isometric and lattice dimension, provide a combinatorial characterization of the Fibonacci dimension using properties of an associated graph, and establish the Fibonacci dimension for certain families of graphs. From the algorithmic point of view, we prove that it is NP-complete to decide whether ${\rm fdim}(G)$ equals the isometric dimension of $G$, and show that no algorithm to approximate ${\rm fdim}(G)$ has approximation ratio below $741/740$, unless P$=$NP. We also give a $(3/2)$-approximation algorithm for ${\rm fdim}(G)$ in the general case and a $(1+\varepsilon)$-approximation algorithm for simplex graphs.

Maximal Linear Embedding for Dimensionality Reduction

Over the past few decades, dimensionality reduction has been widely exploited in computer vision and pattern analysis. This paper proposes a simple but effective nonlinear dimensionality reduction algorithm, named Maximal Linear Embedding (MLE). MLE learns a parametric mapping to recover a single global low-dimensional coordinate space and yields an isometric embedding for the manifold. Inspired by geometric intuition, we introduce a reasonable definition of locally linear patch, Maximal Linear Patch (MLP), which seeks to maximize the local neighborhood in which linearity holds. The input data are first decomposed into a collection of local linear models, each depicting an MLP. These local models are then aligned into a global coordinate space, which is achieved by applying MDS to some randomly selected landmarks. The proposed alignment method, called Landmarks-based Global Alignment (LGA), can efficiently produce a closed-form solution with no risk of local optima. It just involves some small-scale eigenvalue problems, while most previous aligning techniques employ time-consuming iterative optimization. Compared with traditional methods such as ISOMAP and LLE, our MLE yields an explicit modeling of the intrinsic variation modes of the observation data. Extensive experiments on both synthetic and real data indicate the effectivity and efficiency of the proposed algorithm.

A Poset-based Approach to Embedding Median Graphs in Hypercubes and Lattices

A median graph G is a graph where, for any three vertices u, v and w, there is a unique node that lies on a shortest path from u to v, from u to w, and from v to w. While not obvious from the definition, median graphs are partial cubes; that is, they can be isometrically embedded in hypercubes and, consequently, in integer lattices. The isometric and lattice dimensions of G, denoted as dimI(G) and dimZ(G), are the smallest integers k and r so that G can be isometrically embedded in the k-dimensional hypercube and the r-dimensional lattice respectively. Motivated by recent results on the cover graphs of distributive lattices, we study these parameters through median semilattices, a class of ordered structures related to median graphs. We show that not only does this approach provide new combinatorial characterizations for dimI(G) and dimZ(G), they also have nice algorithmic consequences. Assume G has n vertices and m edges. We prove that dimI(G) can be computed in O(n + m) time, and an isometric embedding of G on a hypercube with dimension dimI(G) can be constructed in O(n × dimI(G)) time. The algorithms are extremely simple and the running times are optimal. We also show that dimZ(G) can be computed and an isometric embedding of G on a lattice with dimension dimZ(G) can be constructed in \(O( n \times dim_I(G) + dim_I(G)^{2.5})\) time by using an existing algorithm of Eppstein’s that performs the same tasks for partial cubes. We are able to speed up his algorithm by using our framework to provide a new “interpretation” to the algorithm. In particular, we note that its main part is essentially a generalization of Fulkerson’s method for finding a smallest-sized chain decomposition of a poset.

Metric characterization of apartments in dual polar spaces

Let Π be a polar space of rank n and let G k ( Π ) , k ∈ { 0 , … , n − 1 } be the polar Grassmannian formed by k-dimensional singular subspaces of Π. The corresponding Grassmann graph will be denoted by Γ k ( Π ) . We consider the polar Grassmannian G n − 1 ( Π ) formed by maximal singular subspaces of Π and show that the image of every isometric embedding of the n-dimensional hypercube graph H n in Γ n − 1 ( Π ) is an apartment of G n − 1 ( Π ) . This follows from a more general result concerning isometric embeddings of H m , m ⩽ n in Γ n − 1 ( Π ) . As an application, we classify all isometric embeddings of Γ n − 1 ( Π ) in Γ n ′ − 1 ( Π ′ ) , where Π ′ is a polar space of rank n ′ ⩾ n .

The intrinsic flat distance between Riemannian manifolds and other integral current spaces

Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance between oriented $m$ dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all common metric spaces. This is made rigorous by applying Ambrosio-Kirchheim's extension of Federer-Fleming's notion of integral currents to arbitrary metric spaces. We prove the intrinsic flat distance between two compact oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of Ambrosio-Kirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably $\mathcal{H}^m$ rectifiable spaces and present numerous examples.

Geometric Sampling of Infinite Dimensional Signals

We show that our geometric sampling method for images extends to a large class of infinitely dimensional signals relevant in Image Processing. We also recall classical triangulation results for infinitely dimensional manifolds, and apply them to endow P(X) – the space of probability measures on a set X – with a simple geometry. Furthermore, we show that there exists a natural isometric embedding of images in an infinitely dimensional function space and that this embedding admits an arbitrarily good bi-Lipshitz approximation by an embedding into a finitely dimensional space.

Construction of Local Conservation Laws by Generalized Isometric Embeddings of Vector Bundles

This article uses Cartan–Kahler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the article’s main result to construct conservation laws for covariant divergence free energy-momentum tensors. We also generalize the local isometric embedding of surfaces in the analytic case by applying the main result to vector bundles of rank two over any surface.

Polish ultrametric Urysohn spaces and their isometry groups

In this paper we give some new constructions of Polish ultrametric Urysohn spaces and investigate the universality properties of their isometry groups. It is shown that all isometry groups of Polish ultrametric Urysohn spaces, regardless of their distance sets, are embeddable into each other, and in particular universal for all isometry groups of Polish ultrametric spaces. We also consider a strengthening notion, called extensive isometric embedding, and show that any isometric embedding from a compact ultrametric space into a Polish ultrametric Urysohn space is extensive. It is shown that every isometry between two compact subsets of a Polish ultrametric Urysohn space can be extended to an isometry of the entire space. We introduce a notion of generalized trees to study Polish ultrametric spaces and prove a duality theorem between the categories of Polish ultrametric spaces and their generalized tree representations. Finally we draw some conclusions about the descriptive complexity of embeddability, biembeddability and isometry relations among Polish ultrametric spaces.

Exact point-distributions over the complex sphere

We study point-distributions over the surface of the unit sphere in unitary space that generate quadrature rules which are exact for spherical polynomials up to a certain bi-degree. In this first stage, we explore several different characterizations for this type of point sets using standard tools such as, positive definiteness, reproducing kernel techniques, linearization formulas, etc. We find bounds on the cardinality of a point-distribution, without discussing the deeper question regarding best bounds. We include examples, construction methods and explain, via isometric embeddings from real to complex spheres, the proper connections with the so-called spherical designs.

The Gauss map of minimal surfaces in the Anti-de Sitter space

It is proved that a pair of spinors satisfying a Dirac-type equation represent surfaces immersed in Anti-de Sitter space with prescribed mean curvature. Here, we consider Anti-de Sitter space as the Lie group SU 1 , 1 endowed with a one-parameter family of left-invariant metrics where only one of them is bi-invariant and corresponds to the isometric embedding of Anti-de Sitter space as a quadric in R 2 , 2 . We prove that the Gauss map of a minimal surface immersed in SU 1 , 1 is harmonic. Conversely, we exhibit a representation of minimal surfaces in Anti-de Sitter space in terms of a given harmonic map.

Isometric embeddings of Johnson graphs in Grassmann graphs

Let V be an n-dimensional vector space (4?n<?) and let ${\mathcal{G}}_{k}(V)$ be the Grassmannian formed by all k-dimensional subspaces of V. The corresponding Grassmann graph will be denoted by Γ k (V). We describe all isometric embeddings of Johnson graphs J(l,m), 1<m<l?1 in Γ k (V), 1<k<n?1 (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of J(n,k) in Γ k (V) is an apartment of ${\mathcal{G}}_{k}(V)$ if and only if n=2k. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in Γ k (V), 1<k<n?1.

The dynamical neighborhood selection based on the sampling density and manifold curvature for isometric data embedding

The construction of the neighborhood is a critical problem of manifold learning. Most of manifold learning algorithms use a stable neighborhood parameter (such as k-NN), but it may not work well for the entire manifold, since manifold curvature and sampling density may vary over the manifold. Although some dynamical neighborhood algorithms have been proposed, they are limited by either another global parameter or an assumption. This paper proposes a new approach to select the dynamical neighborhood for each point while constructing the tangent subspace based on the sampling density and the manifold curvature. And the parameters of the approach can be automatically determined by computing the correlation coefficient of the matrices of geodesic distances between pairs of points in input and output spaces. When we apply it to ISOMAP, the results of experiments on the synthetic data as well as the real world patterns demonstrate that the proposed approach can efficiently maintain an accurate low dimensional representation of the manifold data with less distortion, and give higher average classification rate compared to others.

Sobolev Mappings: Lipschitz Density is not an Isometric Invariant of the Target

If $M$ is a compact smooth manifold and $X$ is a compact metric space, the Sobolev space $W^{1,p}(M,X)$ is defined through an isometric embedding of $X$ into a Banach space. We prove that the answer to the question whether Lipschitz mappings ${\rm Lip}\,(M,X)$ are dense in $W^{1,p}(M,X)$ may depend on the isometric embedding of the target.

The Local Isometric Embedding Problem for 3-Dimensional Riemannian Manifolds with Cleanly Vanishing Curvature

We prove the following result: let (M, g) be a 3-dimensional C ∞ Riemannian manifold for which there exists a p ∈ M and a v ∈ T p M such that Then there exists a C ∞ isometric embedding from a neighborhood of p into ℝ6.

A Property of Isometric Mappings Between Dual Polar Spaces of Type $${DQ (2n, {\mathbb K})}$$

Let f be an isometric embedding of the dual polar space $${\Delta = DQ(2n, {\mathbb K})}$$ into $${\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}$$ . Let P denote the point-set of Δ and let $${e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})}$$ denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that $${f(H) = f(P) \cap H^\prime}$$ . We use this to show that there exists a subgeometry $${\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})}$$ of Σ′ such that: (i) $${e^\prime \circ f (x) \in \Sigma}$$ for every point x of $${\Delta; ({\rm ii})\,e := e^\prime \circ f}$$ defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.