Isometric Embedding Research Articles

Tensor distance based multilinear globality preserving embedding: A unified tensor based dimensionality reduction framework for image and video classification

Image and video classification tasks often suffer from the problem of high-dimensional feature space. How to discover the meaningful, low-dimensional representations of such high-order, high-dimensional observations remains a fundamental challenge. In this paper, we present a unified framework for tensor based dimensionality reduction including a new tensor distance (TD) metric and a novel multilinear globality preserving embedding (MGPE) strategy. Different with the traditional Euclidean distance, which is constrained by orthogonality assumption, TD measures the distance between data points by considering the relationships among different coordinates of high-order data. To preserve the natural tensor structure in low-dimensional space, MGPE directly works on the high-order form of input data and employs an iterative strategy to learn the transformation matrices. To provide faithful global representation for datasets, MGPE intends to preserve the distances between all pairs of data points. According to the proposed TD metric and MGPE strategy, we further derive two algorithms dubbed tensor distance based multilinear multidimensional scaling (TD-MMDS) and tensor distance based multilinear isometric embedding (TD-MIE). TD-MMDS finds the transformation matrices by keeping the TDs between all pairs of input data in the embedded space, while TD-MIE intends to preserve all pairwise distances calculated according to TDs along shortest paths in the neighborhood graph. By integrating tensor distance into tensor based embedding, TD-MMDS and TD-MIE perform tensor based dimensionality reduction through the whole learning procedure and achieve obvious performance improvement on various standard datasets.

Rigidity for local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ up to conformal factors

In this article, we study local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok, and Mok and Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map between balls. Applying a linearity criterion of Huang, we conclude the total geodesy of non-constant components.

Linearization of isometric embeddings between Banach spaces: An elementary approach

Mazur-Ulam’s classical theorem states that any isometric onto map between normed spaces is linear. This result has been generalized by T. Figiel [F] who showed that ifΦ is an isometric embedding from a Banach space X to a Banach space Y such that Φ(0) = 0 and vect [φ(X )] = Y , there exists a linear quotient map Q such that ‖Q‖ = 1 and Q ◦Φ= I dX . The third chapter of this short story is [GK] where it is shown that if a quotient map Q from a Banach space Y onto a separable Banach space X has a Lipschitz lifting, then it actually has a continuous linear lifting. Combining this statement with Figiel’s theorem provides another result from [GK] : if a separable Banach space X isometrically embeds into a Banach space Y , there exists an isometric linear embedding from X into Y . Nigel Kalton had an outstanding ability to set theorems and proofs in their proper frame. His articles are therefore fountains of ideas, irrigating each of the many fields to which he contributed : a non exhaustive survey on these contributions is [G]. The paper [GK] is no exception to this rule, and among other things it prepared the ground for far-reaching extensions, where e.g. the Lipschitz assumption is weakened to the Holder condition, leading to very different conclusions ( [K1], [K2], [K3] ). However, some readers could find daunting some arguments from [GK], whichwould probably turn down undergraduate students. Since the extension from [GK] of Mazur-Ulam’s theorem which is recalled above has a statement which is understandable to any student in mathematics, it seems appropriate to provide an elementary proof. This is the purpose of the present short note, where only basic functional analysis (elementary duality theory) and calculus (culminating at Fubini’s theorem for continuous functions on Rn) are used. Moreover this note is fully self-contained : even the generic smoothness of convex fuctions on Rn is shown, and a detailed proof of Figiel’s theorem is provided (following [BL], although Lemma 2 is not stated there). Themain result of the note is Theorem 5, whose proof follows the strategy of the proof of ( [GK], Corollary 3.2) but in such an elementary way that diagrams, free spaces and infinitedimensional integration or differentiation arguments are avoided. Thus our approach makes it clear that the core of the proof consists of finite-dimensional considerations. We can therefore teach at an undergraduate level Mazur-Ulam’s theorem and its natural extensions from [F] and [GK].

The Nash-Kuiper process for curves

A strictly short embedding is an embedding of a Riemannian manifold into an Euclidean space that strictly shortens distances. From such an embedding, the Nash-Kuiper process builds a sequence of maps converging toward an isometric embedding. In that paper, we describe this Nash-Kuiper process in the case of curves. We state an explicit formula for the limit normal map and perform its Fourier series expansion. We then address the question of Holder regularity of the limit map.

On a construction of Burago and Zalgaller

The purpose of this note is to scrutinize the proof of Burago and Zalgaller regarding the existence of $PL$ isometric embeddings of $PL$ compact surfaces into $\mathbb{R}^3$. We conclude that their proof does not admit a direct extension to higher dimensions. Moreover, we show that, in general, $PL$ manifolds of dimension $n \ge 3$ admit no nontrivial $PL$ embeddings in $\mathbb{R}^{n+1}$ that are close to conformality. We also extend the result of Burago and Zalgaller to a large class of noncompact $PL$ 2-manifolds. The relation between intrinsic and extrinsic curvatures is also examined, and we propose a $PL$ version of the Gauss compatibility equation for smooth surfaces.

Contractive Hilbert modules and their dilations

In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z,w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S −1(z, w)k(z, w) is a positive kernel function, where S(z,w) is the Szegö kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale [8] and Ambrozie, Englis and Müller [2]. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in ℂm. Some consequences of this more general result are then explored in the case of several natural function algebras.

Embeddings of Grassmann graphs

Let V and V′ be vector spaces of dimension n and n′, respectively. Let k∈{2,…,n-2} and k′∈{2,…,n′-2}. We describe all isometric and l-rigid isometric embeddings of the Grassmann graph Γk(V) in the Grassmann graph Γk′(V′).

An efficient tree-based computation of a metric comparable to a natural diffusion distance

Using diffusion to define distances between points on a manifold (or a sampled data set) has been successfully employed in various applications such as data organization and approximately isometric embedding of high dimensional data in low dimensional Euclidean space. Recently, P. Jones has proposed a diffusion distance which is both intuitively appealing and scales appropriately with increasing time. In the first part of our paper, we present an efficient tree-based approach to computing an approximation to Jonesʼs diffusion distance. We also show our approximation is comparable to Jonesʼs distance. Neither Jonesʼs distance, nor our approximation, satisfies the triangle inequality; in particular, in the case of heat flow on Rn, Jonesʼs separation distance gives a scaled square of the Euclidean distance. In the second part of our paper, we present a general construction to obtain an “almost” metric from a general distance. We also discuss a numerical procedure to implement our construction. Additionally, we show that in the case of heat flow on Rn, we recover (scaled) Euclidean distance from Jonesʼs distance.

Auerbach Bases and Minimal Volume Sufficient Enlargements

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

The linearized system for isometric embeddings and its characteristic variety

In this paper we prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 4. This is accomplished by introducing a smaller yet equivalent linearized system, in an appropriate way, which facilitates analysis of the characteristic variety.

Isometric embedding of semi-Riemannian metrics into Minkowski space

Given a two-dimensional semi-Riemannian metric with positive Gaussian curvature, we construct local isometric embeddings into Minkowski space ℝ2,1. Using reasonable geometric initial conditions, we control the number of solutions and deduce symmetry properties of the solutions.

A ŠVARC–MILNOR LEMMA FOR MONOIDS ACTING BY ISOMETRIC EMBEDDINGS

We continue our program of extending key techniques from geometric group theory to semigroup theory, by studying monoids acting by isometric embeddings on spaces equipped with asymmetric, partially defined distance functions. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Švarc–Milnor lemma to this setting.

Evaluating Quasilocal Energy and Solving Optimal Embedding Equation at Null Infinity

We study the limit of quasilocal energy defined in [7] and [8] for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi-Sachs energy-momentum at a retarded time. The quasilocal mass in [7] and [8] is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases. Assuming analyticity, the equation can be solved and the solution is locally minimizing in all orders. In particular, this produces an optimal reference hypersurface in the Minkowski space for the spatial or null exterior region of an asymptotically flat spacetime.

A geometry of the correlation space and a nonlocal degenerate parabolic equation from isotropic turbulence

AbstractConsidering the metric tensor ds2(t), associated with the two‐point velocity correlation tensor field (parametrized by the time variable t) in the space 𝒦3of correlation vectors, at the first part of the paper we construct the Lagrangian system (Mt,ds2(t)) in the extended space 𝒦3 × R+ for homogeneous isotropic turbulence. This allows to introduce into the consideration common concept and technics of Lagrangian mechanics for the application in turbulence. Dynamics in time of (Mt,ds2(t)) (a singled out fluid volume equipped with a family of pseudo‐Riemannian metrics) is described in the frame of the geometry generated by the 1‐parameter family of metrics ds2(t) whose components are the correlation functions that evolve according to the von Kármán‐Howarth equation. This is the first step one needs to get in a future analysis the physical realization of the evolution of this volume. It means that we have to construct isometric embedding of the manifold Mt equipped with metric ds2(t) into R3 with the Euclidean metric. In order to specify the correlation functions, at the second part of this paper we study in details an initial‐boundary value problem to the closure model [19,26] for the von Kármán‐Howarth equation in the case of large Reynolds numbers limit.

Bifurcation of Constant Mean Curvature Tori in Euclidean Spheres

We use equivariant bifurcation theory to show the existence of infinite sequences of isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.

An investigation of embeddings for spherically symmetric spacetimes into Einstein manifolds

Embeddings into higher dimensions are very important in the study of higher-dimensional theories of our Universe and in high-energy physics. Theorems which have been developed recently guarantee the existence of embeddings of pseudo-Riemannian manifolds into Einstein spaces and more general pseudo-Riemannian spaces. These results provide a technique that can be used to determine solutions for such embeddings. Here we consider local isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein manifolds. Difficulties in solving the five-dimensional equations for given four-dimensional spaces motivate us to investigate embedded spaces that admit bulks of a specific type. We show that the general Schwarzschild–de Sitter spacetime and Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space of a particular form, and we discuss their five-dimensional solutions.

Isometric embeddings in cosmology and astrophysics

Recent interest in higher-dimensional cosmological models has prompted some significant work on the mathematical technicalities of how one goes about embedding spacetimes into some higher-dimensional space. We survey results in the literature (existence theorems and simple explicit embeddings); briefly outline our work on global embeddings as well as explicit results for more complex geometries; and provide some examples. These results are contextualized physically, so as to provide a foundation for a detailed commentary on several key issues in the field such as: the meaning of ‘Ricci equivalent’ embeddings; the uniqueness of local (or global) embeddings; symmetry inheritance properties; and astrophysical constraints.

Extrinsic analysis on manifolds is computationally faster than intrinsic analysis with applications to quality control by machine vision

In our technological era, non‐Euclidean data abound, especially because of advances in digital imaging. Patrangenaru (‘Asymptotic statistics on manifolds’, PhD Dissertation, 1998) introduced extrinsic and intrinsic means on manifolds, as location parameters for non‐Euclidean data. A large sample nonparametric theory of inference on manifolds was developed by Bhattacharya and Patrangenaru (J. Stat. Plann. Inferr., 108, 23–35, 2002; Ann. Statist., 31, 1–29, 2003; Ann. Statist., 33, 1211–1245, 2005). A flurry of papers in computer vision, statistical learning, pattern recognition, medical imaging, and other computational intensive applied areas using these concepts followed. While pursuing such location parameters in various instances of data analysis on manifolds, scientists are using intrinsic means, almost without exception. In this paper, we point out that there is no unique intrinsic analysis because the latter depends on the choice of the Riemannian metric on the manifold, and in dimension two or higher, there are infinitely such nonisometric choices. Also, using John Nash's celebrated isometric embedding theorem and an equivariant version, we show that for each intrinsic analysis there is an extrinsic counterpart that is computationally faster and give some concrete examples in shape and image analysis. The computational speed is important, especially in automated industrial processes. In this paper, we mention two potential applications in the industry and give a detailed presentation of one such application, for quality control in a manufacturing process via 3D projective shape analysis from multiple digital camera images. Copyright © 2011 John Wiley & Sons, Ltd.

Embedding function spaces into [formula omitted

We show that if the space ℓ ∞ / c 0 contains an isometric copy of every function space over a first countable compactum or every function space over a Corson compactum of weight not exceeding the continuum then every subset of R 2 belongs to the σ-field generated by sets of the form A 1 × A 2 . We prove a similar result about isomorphic rather than isometric embeddings into ℓ ∞ / c 0 in terms of the σ-field of subsets of R k generated by sets of the form A 1 × ⋯ × A k for other positive integers k.

Isometric embeddings of infinite-dimensional Grassmannians

We investigate geometric properties of the (Sato-Segal-Wilson) Grassmannian and its submanifolds, with special attention to the orbits of the KP flows. We use a coherentstates model, by which Spera and Wurzbacher gave equations for the image of a product of Grassmannians using the Powers-Stormer purification procedure. We extend to this product Sato’s idea of turning equations that define the projective embedding of a homogeneous space into a hierarchy of partial differential equations. We recover the BKP equations from the classical Segre embedding by specializing the equations to finite-dimensional submanifolds.