Smooth Embeddings Research Articles

System identification of auditory processes using noise probes in isolation or embedded into soundscapes

This talk will cover large experimental datasets spanning different levels of auditory analysis from low-level to higher-level, with specific focus on two inter-related problems: (1) how to exploit perceptual weights and their derivatives to infer the structure of the underlying auditory process; (2) how to transfer these tools from identification of isolated processes to situations where the process is embedded within complex sounds (e.g., speech). I will highlight connections with recent developments in both auditory and vision literature, for example (in relation to number 1 and 2 above, respectively) detailed understanding of how important nonlinearities are reflected in identifiable modulations of perceptual weights, and construction of noise probes for smooth embedding within structured contexts together with associated tasks specifically designed to meanigfully probe contextual phenomena. These developments demonstrate very substantial progress over the past two decades, and offer new exciting dire...

Remarks on the space of volume preserving embeddings

Let (N,g) be a Riemannian manifold. Given a compact, connected and oriented submanifold M of N, we define the space of volume preserving embeddings Embμ(M,N) as the set of smooth embeddings f:M↪N such that f⁎μf=μ, where μf (resp. μ) is the Riemannian volume form on f(M) (resp. M) induced by the ambient metric g (the orientation on f(M) being induced by f).In this article, we use the Nash–Moser inverse function Theorem to show that the set of volume preserving embeddings in Embμ(M,N) whose mean curvature is nowhere vanishing forms a tame Fréchet manifold, and determine explicitly the Euler–Lagrange equations of a natural class of Lagrangians.As an application, we generalize the Euler equations of an incompressible fluid to the case of an “incompressible membrane” of arbitrary dimension moving in N.

SSE: Semantically Smooth Embedding for Knowledge Graphs

This paper considers the problem of embedding Knowledge Graphs (KGs) consisting of entities and relations into low-dimensional vector spaces. Most of the existing methods perform this task based solely on observed facts. The only requirement is that the learned embeddings should be compatible within each individual fact. In this paper, aiming at further discovering the intrinsic geometric structure of the embedding space, we propose <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Semantically Smooth Embedding</i> (SSE). The key idea of SSE is to take full advantage of additional semantic information and enforce the embedding space to be semantically smooth, i.e., entities belonging to the same semantic category will lie close to each other in the embedding space. Two manifold learning algorithms Laplacian Eigenmaps and Locally Linear Embedding are used to model the smoothness assumption. Both are formulated as geometrically based regularization terms to constrain the embedding task. Two lines of embedding strategies are tested, i.e., strategies based on latent distance models and strategies based on tensor factorization techniques. We empirically evaluate SSE on two benchmark tasks of link prediction and triple classification, and achieve significant and consistent improvements over state-of-the-art methods. The results demonstrate the superiority and generality of SSE.

On the local isometric embedding in ℝ3 of surfaces with zero sets of Gaussian curvature forming cusp domains

ABSTRACTWe study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a point, then local sufficiently smooth isometric embeddings exist.

EMBEDDINGS, NORMAL INVARIANTS AND FUNCTOR CALCULUS

This paper investigates the space of codimension zero embeddings of a Poincaré duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincaré immersions to a certain space of “unlinked” Poincaré embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie’s homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.

On the maximal mean curvature of a smooth surface

Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume.

Smooth embeddings of the Long Line and other non-paracompact manifolds into locally convex spaces

We show that every real finite dimensional Hausdorff (not necessarily paracompact, not necessarily second countable) Cr-manifold can be Cr-embedded into a weakly complete vector space, i.e. a locally convex space of the form RI for an uncountable index set I and determine the minimal cardinality of I for which such an embedding is possible.

On the notions of suborbifold and orbifold embedding

The purpose of this article is to investigate the relationship between suborbifolds and orbifold embeddings. In particular, we give natural definitions of the notion of suborbifold and orbifold embedding and provide many examples. Surprisingly, we show that there are (topologically embedded) smooth suborbifolds which do not arise as the image of a smooth orbifold embedding. We are also able to characterize those suborbifolds which can arise as the images of orbifold embeddings. As an application, we show that a length-minimizing curve (a geodesic segment) in a Riemannian orbifold can always be realized as the image of an orbifold embedding.

Bilinear Embedding Theorems for Differential Operators in ℝ2

Here and in what follows, we write "a b" instead of "a ≤ Cb for some uniform constant C" for brevity; we also write ab when a b and b a .T he symbol∂j, j =1 , 2, denotes the differentiation with respect to jth variable. To be more precise, we study estimates of the scalar product of two functions in some Hilbert space (in this paper, some Sobolev space of fractional order) in terms of the product of L1- norms of some differential polynomials applied to these functions. For the author, the interest in inequalities of such type originated from the work on nonisomorphism problems for Banach spaces of smooth functions and embedding theorems used there, see the short report (5) and preprint (6). We are going to use some formalism to make our statements shorter. Let k and l be natural numbers, let α and β be real nonnegative numbers, and let σ and τ be complex nonzero numbers. The symbol BE(k, l, α, β, σ, τ) means the statement that the inequality �

Isometric embedding of negatively curved complete surfaces in Lorentz–Minkowski space

Hilbert-Efimov theorem states that any complete surface with curvature bounded above by a negative constant can not be isometrically imbedded in $\mathbb{R}^3.$ We demonstrate that any simply-connected smooth complete surface with curvature bounded above by a negative constant admits a smooth isometric embedding into the Lorentz-Minkowski space $\mathbb{R}^{2,1}$.

EQUIVARIANT COHOMOLOGY OF RATIONALLY SMOOTH GROUP EMBEDDINGS

We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. These embeddings are the projectivizations of reductive monoids. Our main result describes their equivariant cohomology in terms of roots, idempotents, and underlying monoid data. Also, we characterize those embeddings whose equivariant cohomology ring is obtained via restriction to the associated toric variety. Such characterization is given in terms of the closed orbits.

Local positivity of line bundles on smooth toric varieties and Cayley polytopes

For any positive integer k the k-th osculating dimension at a given point x of a variety X embedded in projective space gives a measure of the local positivity of order k at that point. In this paper we show that a smooth toric embedding having the property that at every point the t-th osculating dimension is maximal if and only if t≤k, is associated to a Cayley polytope of order k. This result generalises an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly k at every point of X, generalising a result of Atsushi Ito.

Multiple disjunction for spaces of smooth embeddings

We obtain multirelative connectivity statements about spaces of smooth embeddings, deducing these from analogous results about spaces of Poincare embeddings that were established in our previous paper.

Riemannian simplices and triangulations

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the “thickness” or “fatness” of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.

Global isometric embeddings of multiple warped product metrics into quadrics

In this paper, we construct smooth isometric embeddings of multiple warped product manifolds in quadrics of semi-Euclidean spaces. Our main theorem generalizes previous results as given by Blanuša, Rozendorn, Henke and Azov.

Desingularization preserving stable simple normal crossings

The subject is partial resolution of singularities. Given an algebraic variety X (not necessarily equidimensional) in characteristic zero (or, more generally, a pair (X, D), where D is a divisor on X), we construct a functorial desingularization of all but stable simple normal crossings (stable-snc) singularities, by smooth blowings-up that preserve such singularities. A variety has stable simple normal crossings at a point if, locally, its irreducible components are smooth and transverse in some smooth embedding variety. We also show that our main assertion is false for more general simple normal crossings singularities.

Categorical frameworks for generalized functions

We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Frolicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frolicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frolicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Frolicher spaces) as a framework for Schwartz distributions, and prove that in the category of diffeological spaces, both the special and the full Colombeau algebras are smooth differential algebras, with a smooth embedding of Schwartz distributions and smooth pointwise evaluations of Colombeau generalized functions.

On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target

We study the question: When are Lipschitz mappings dense in the Sobolev space W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) ? Here M M denotes a compact Riemannian manifold with or without boundary, while H n \mathbb {H}^n denotes the n n th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) for all 1 ≤ p &gt; ∞ 1\le p&gt;\infty if dim ⁡ M ≤ n \dim M \le n , but that Lipschitz maps are not dense in W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) if dim ⁡ M ≥ n + 1 \dim M \ge n+1 and n ≤ p &gt; n + 1 n\le p&gt;n+1 . The proofs rely on the construction of smooth horizontal embeddings of the sphere S n \mathbb {S}^n into H n \mathbb {H}^n . We provide two such constructions, one arising from complex hyperbolic geometry and the other arising from symplectic geometry. The nondensity assertion can be interpreted as nontriviality of the n n th Lipschitz homotopy group of H n \mathbb {H}^n . We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.

When does the Hawking into Unruh mapping for global embeddings work?

We discuss for which smooth global embeddings of a metric into a Minkowskian spacetime the Hawking into Unruh mapping takes place. There is a series of examples of global embeddings into the Minkowskian spacetime (GEMS) with such mapping for physically interesting metrics. These examples use Fronsdal-type embeddings for which timelines are hyperbolas. In the present work we show that for some new embeddings (non Fronsdal-type) of the Schwarzschild and Reissner-Nordstrom metrics there is no mapping. We give also the examples of hyperbolic and non hyperbolic type embeddings for the de Sitter metric for which there is no mapping. For the Minkowski metric where there is no Hawking radiation we consider a non trivial embedding with hyperbolic timelines, hence in the ambient space the Unruh effect takes place, and it follows that there is no mapping too. The considered examples show that the meaning of the Hawking into Unruh mapping for global embeddings remains still insufficiently clear and requires further investigations.

Supervised locality discriminant manifold learning for head pose estimation

In this paper, we propose a novel supervised manifold learning approach, supervised locality discriminant manifold learning (SLDML), for head pose estimation. Traditional manifold learning methods focus on preserving only the intra-class geometric properties of the manifold embedded in the high-dimensional ambient space, so they cannot fully utilize the underlying discriminative knowledge of the data. The proposed SLDML aims to explore both geometric structure and discriminant information of the data, and yields a smooth and discriminative low-dimensional embedding by adding the local discriminant terms in the optimization objectives of manifold learning. Moreover, for efficiently handling out-of-sample extension and learning with the local consistency, we decompose the manifold learning as a two-step approach. We incorporate the manifold learning and the regression with a learned discriminant manifold-based projection function obtained by discriminatively Laplacian regularized least squares. The SLDML provides both the low-dimensional embedding and projection function with better intra-class compactness and inter-class separability, therefore preserves the local geometric structures more effectively. Meanwhile, the SLDML is supervised by both biased distance and continuous head pose angle information when constructing the graph, embedding the graph and learning the projection function. Our experiments demonstrate the superiority of the proposed SLDML over several current state-of-art approaches for head pose estimation on the publicly available FacePix dataset.