Showing papers on "Distribution (differential geometry) published in 2017"

Geometric nonlinear thermoelasticity and the time evolution of thermal stresses

Abstract: In this paper we formulate a geometric theory of nonlinear thermoelasticity that can be used to calculate the time evolution of temperature and thermal stress fields in a nonlinear elastic body. In particular, this formulation can be used to calculate residual thermal stresses. In this theory the material manifold (natural stress-free configuration of the body) is a Riemannian manifold with a temperature-dependent metric. Evolution of the geometry of the material manifold is governed by a generalized heat equation. As examples, we consider an infinitely long circular cylindrical bar with a cylindrically symmetric temperature distribution and a spherical ball with a spherically-symmetric temperature distribution. In both cases we assume that the body is made of an arbitrary incompressible isotropic solid. We numerically solve for the evolution of thermal stress fields induced by thermal inclusions in both a cylindrical bar and a spherical ball, and compare the linear and nonlinear solutions for a generaliz...

Bayesian nonparametric inference on the Stiefel manifold

Abstract: The Stiefel manifold $V_{p,d}$ is the space of all $d \times p$ orthonormal matrices, with the $d-1$ hypersphere and the space of all orthogonal matrices constituting special cases. In modeling data lying on the Stiefel manifold, parametric distributions such as the matrix Langevin distribution are often used; however, model misspecification is a concern and it is desirable to have nonparametric alternatives. Current nonparametric methods are Frechet mean based. We take a fully generative nonparametric approach, which relies on mixing parametric kernels such as the matrix Langevin. The proposed kernel mixtures can approximate a large class of distributions on the Stiefel manifold, and we develop theory showing posterior consistency. While there exists work developing general posterior consistency results, extending these results to this particular manifold requires substantial new theory. Posterior inference is illustrated on a real-world dataset of near-Earth objects.

Lee–Yang zeros for the DHL and 2D rational dynamics, I. Foliation of the physical cylinder

Abstract: In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal–Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the renormalization transformation). We prove that R is partially hyperbolic on an invariant cylinder C . The Lee–Yang zeros are organized in a transverse measure for the central-stable foliation of R | C . Their distribution is absolutely continuous. Its density is C ∞ (and non-vanishing) below the critical temperature. Above the critical temperature, it is C ∞ on a open dense subset, but it vanishes on the complementary set of positive measure.

Statistics on the (compact) Stiefel manifold: Theory and Applications

Abstract: A Stiefel manifold of the compact type is often encountered in many fields of Engineering including, signal and image processing, machine learning, numerical optimization and others. The Stiefel manifold is a Riemannian homogeneous space but not a symmetric space. In previous work, researchers have defined probability distributions on symmetric spaces and performed statistical analysis of data residing in these spaces. In this paper, we present original work involving definition of Gaussian distributions on a homogeneous space and show that the maximum-likelihood estimate of the location parameter of a Gaussian distribution on the homogeneous space yields the Frechet mean (FM) of the samples drawn from this distribution. Further, we present an algorithm to sample from the Gaussian distribution on the Stiefel manifold and recursively compute the FM of these samples. We also prove the weak consistency of this recursive FM estimator. Several synthetic and real data experiments are then presented, demonstrating the superior computational performance of this estimator over the gradient descent based non-recursive counter part as well as the stochastic gradient descent based method prevalent in literature.

Multiscale singular value manifold for rotating machinery fault diagnosis

Abstract: Time-frequency distribution of vibration signal can be considered as an image that contains more information than signal in time domain. Manifold learning is a novel theory for image recognition that can be also applied to rotating machinery fault pattern recognition based on time-frequency distributions. However, the vibration signal of rotating machinery in fault condition contains cyclical transient impulses with different phrases which are detrimental to image recognition for time-frequency distribution. To eliminate the effects of phase differences and extract the inherent features of time-frequency distributions, a multiscale singular value manifold method is proposed. The obtained low-dimensional multiscale singular value manifold features can reveal the differences of different fault patterns and they are applicable to classification and diagnosis. Experimental verification proves that the performance of the proposed method is superior in rotating machinery fault diagnosis.

Algebraic embeddings of smooth almost complex structures

Abstract: The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. It is shown that every almost complex structure can be realized by the transverse structure to an algebraic distribution on an affine algebraic variety, namely an algebraic subbundle of the tangent bundle. In fact, there even exist universal embedding spaces for this problem, and their dimensions grow quadratically with respect to the dimension of the almost complex manifold to embed. We give precise variation formulas for the induced almost complex structures and study the related versality conditions. At the end, we discuss the original question raised by F.Bogomolov: can one embed every compact holomorphic manifold as a C infinity smooth subvariety that is transverse to an algebraic foliation on a complex projective algebraic variety?

Deforming Lie algebras to Frobenius integrable non-autonomous Hamiltonian systems

Abstract: Motivated by the theory of Painleve equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stackel systems.

New relations between G2 geometries in dimensions 5 and 7

Abstract: There are two well-known parabolic split G2 geometries in dimension 5, (2, 3, 5) distributions and G2 contact structures. Here we link these two geometries with yet another G2 related contact structure, which lives on a 7-manifold. More precisely, we present a natural geometric construction that associates to a (2, 3, 5) distribution a 7-dimensional bundle endowed with a canonical Lie contact structure. We further study the relation between the canonical normal Cartan connections associated with the two structures and we show that the Cartan holonomy of the induced Lie contact structure reduces to G2. This motivates the study of the curved orbit decomposition associated with a G2 reduced Lie contact structure on a 7-manifold. It is shown that, provided an additional curvature condition is satisfied, in a neighborhood of each point in the open curved orbit the structure descends to a (2, 3, 5) distribution on a local leaf space. The closed orbit carries an induced G2 contact structure.

Stochastic Development Regression on Non-linear Manifolds

Abstract: We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is based on stochastic development of Euclidean diffusion processes to the manifold. Defining the data distribution as the transition distribution of the mapped stochastic process, parameters of the model, the non-linear analogue of design matrix and intercept, are found via maximum likelihood. The model is intrinsically related to the geometry encoded in the connection of the manifold. We propose an estimation procedure which applies the Laplace approximation of the likelihood function. A simulation study of the performance of the model is performed and the model is applied to a real dataset of Corpus Callosum shapes.

A manifold framework of multiple-kernel learning for hyperspectral image classification

Abstract: Manifold learning is a promising intelligent data analysis method, and the manifold learning preserves the local embedding features of the data in manifold mapping space. Manifold learning has its limitations on extracting the nonlinear features of the data in many applications. For example, hyperspectral image classification needs to seek the nonlinear local relationships between spectral curves. For that, researchers applied the kernel trick to manifold learning in the previous works. The kernel-based manifold learning was developed, but still endures the problem that the inappropriate kernel model reduces the system performance. In order to solve the problem of kernel model selection, we propose a manifold framework of multiple-kernel learning for the application of hyperspectral image classification. In this framework, the quasiconformal mapping-based multiple-kernel model is optimized based on the optimization objective equation, which maximizes the class discriminant ability of data. Accordingly, the discriminative structure of data distribution is achieved for classification with the quasiconformal mapping-based multiple-kernel model.

Eigenvalues and poles of nonlinear systems: A geometric approach

Abstract: The classical notions of eigenvalue and of pole of a linear system are revisited using a geometric approach and extended to nonlinear systems exploiting the notions of f — reducing manifold and of f-reducing distribution. The proposed definitions contribute to a deeper understanding of the model reduction problem at isolated singularities, recently studied in a companion paper by the authors, and may prove useful in the solution of stabilization problems for nonlinear systems. Simple worked-out examples illustrate the theory.

Stochastic Development Regression on Non-Linear Manifolds

Abstract: We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is based on stochastic development of Euclidean diffusion processes to the manifold. Defining the data distribution as the transition distribution of the mapped stochastic process, parameters of the model, the non-linear analogue of design matrix and intercept, are found via maximum likelihood. The model is intrinsically related to the geometry encoded in the connection of the manifold. We propose an estimation procedure which applies the Laplace approximation of the likelihood function. A simulation study of the performance of the model is performed and the model is applied to a real dataset of Corpus Callosum shapes.

Contact structures on Lie algebroids

Abstract: In this paper we generalize the main notions from the geometry of (almost) contact manifolds in the category of Lie algebroids. Also, using the framework of generalized geometry, we obtain an (almost) contact Riemannian Lie algebroid structure on a vertical Liouville distribution over the big-tangent manifold of a Riemannain manifold.

Representation of complex probabilities and complex Gibbs sampling

Abstract: Complex weights appear in Physics which are beyond a straightforward importance sampling treatment, as required in Monte Carlo calculations. This is the well-known sign problem. The complex Langevin approach amounts to effectively construct a posi\-tive distribution on the complexified manifold reproducing the expectation values of the observables through their analytical extension. Here we discuss the direct construction of such positive distributions paying attention to their localization on the complexified manifold. Explicit localized repre\-sentations are obtained for complex probabilities defined on Abelian and non Abelian groups. The viability and performance of a complex version of the heat bath method, based on such representations, is analyzed.

Coverings and integrable pseudosymmetries of differential equations

Abstract: We study the problem on the construction of coverings by a given system of differential equations and the description of systems covered by it. This problem is of interest in view of its relationship with the computation of nonlocal symmetries, recursion operators, B¨acklund transformations, and decompositions of systems. We show that the distribution specified by the fibers of the covering is determined by a pseudosymmetry of the system and is integrable in the infinite-dimensional sense. Conversely, every integrable pseudosymmetry of a system defines a covering by this system. The vertical component of the pseudosymmetry is a matrix analog of the evolution differentiation, and the corresponding generating matrix satisfies a matrix analog of the linearization of an equation.

Initial boundary-value problem for the spherically symmetric Einstein equations with fluids with tangential pressure

Abstract: We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial spacelike hypersurface with a timelike boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.

Isometries of almost-Riemannian structures on Lie groups

Abstract: A simple Almost-Riemannian Structure on a Lie group G is defined by a linear vector field (that is an infinitesimal automorphism) and dim(G) -- 1 left-invariant ones. It is first proven that two different ARSs are isometric if and only if there exists an isometry between them that fixes the identity. Such an isometry preserves the left-invariant distribution and the linear field. If the Lie group is nilpotent it is an automorphism. These results are used to state a complete classification of the ARSs on the 2D affine and the Heisenberg groups.

Non-integrated defect relation for meromorphic mappings from a K\"ahler manifold with hypersurfaces of a projective variety in subgeneral position

Abstract: In this paper, we establish a truncated non-integrated defect relation for meromorphic mappings from a complete Kahler manifold into a projective variety intersecting a family of hypersurfaces located in subgeneral position, where the truncation level of the defect is explicitly estimated. Our result generalizes and improves previous ones. In particular, when the family of hypersurfaces located in general position, our theorem will implies the previous result of Min Ru-Sogome. In the last part of this paper we will apply ours to study the distribution of the Gauss map of minimal surfaces.

A family of anisotropic distributions on the hyperbolic plane

Abstract: Most of the parametric families of distributions on manifold are constituted of radial distributions. The main reason is that quantifying the anisotropy of a distribution on a manifold is not as straightforward as in vector spaces and usually leads to numerical computations. Based on a simple definition of the covariance on manifolds, this paper presents a way of constructing anisotropic distributions on the hyperbolic space whose covariance matrices are explicitly known. The approach remains valid on every manifold homeomorphic to vector spaces.

On Pseudo-Einstein Real Hypersurfaces

Abstract: Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c eq0$, $n\geq 3$. We show that the Ricci tensor $S$ of $M$ satisfies $S(X,Y)=ag(X,Y)$ for any vector fields $X$ and $Y$ on the holomorphic distribution, $a$ being a constant, if and only if $M$ is a pseudo-Einstein real hypersurface.

Constant Curvature Models in Sub-Riemannian Geometry

Abstract: Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curvature invariants in the cases where the background structure is one of the parabolic geometries. As an illustration, constant curvature models are discussed for certain sub-Riemannian geometries.

Moments of Sectional Curvature

Abstract: The sectional curvature of a compact Riemannian manifold M can be seen as a random variable on the Grassmann bundle of 2-planes in TM endowed with the Fubini-Study volume density. In this article we calculate the moments of this random variable by integrating suitable local Riemannian invariants and discuss the distribution of the sectional curvature of Riemannian products. Moreover we calculate the moments and the distribution of the sectional curvature for all compact symmetric spaces of rank 1 explicitly and derive a formula for the moments of general symmetric spaces. Interpolating the explicit values for the moments obtained we prove a weak version of the Hitchin-Thorpe Inequality.

Laplacians on smooth distributions as $C^*$-algebra multipliers

Abstract: In this paper we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold, initiated in a previous paper. Under assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distribution defines an unbounded regular self-adjoint operator in some Hilbert module over the foliation $C^*$-algebra.

Cognitive feature and manifold ranking-based image retrieval method

Abstract: The invention relates to the field of image retrieval in computer vision, and discloses a cognitive feature and manifold ranking-based image retrieval method. The method comprises the following steps of: respectively extracting visual feature vectors of a query sample and database images; verifying manifold structure preserving characteristics of visual features by utilizing a manifold learning algorithm, and assuming sample manifold distribution as basis of a manifold ranking algorithm; and calculating similarity scores between the database images and the query sample by utilizing the manifold ranking algorithm, and outputting the database images with the highest scores as retrieval results. According to the image retrieval method, the most similar image retrieval results can be found according to manifold structures between sample features.

Frobenius integrability and Finsler metrizability for 2-dimensional sprays

Abstract: For a 2-dimensional non-flat spray we associate a Berwald frame and a 3-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is provided by a closed, homogeneous 1-form from the annihilator of the Berwald distribution. We discuss both the degenerate and non-degenerate cases using the fact that the regularity of the Finsler function is encoded into a regularity condition of a 2-form, canonically associated to the given spray. The integrability of the Berwald distribution and the regularity of the 2-form have simple and useful expressions in terms of the Berwald frame.

From deformations of Lie algebras to Frobenius integrable non-autonomous Hamiltonian systems

Abstract: Motivated by the theory of Painleve equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stackel systems.

A numerical study of Gibbs $u$-measures for partially hyperbolic diffeomorphisms on $\mathbb T^3$

Abstract: We consider a hyperbolic automorphism $A\colon\mathbb T^3\to\mathbb T^3$ of the 3-torus whose 2-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold $A$ into two one-parameter families of Anosov diffeomorphisms --- a conservative family and a dissipative one. For diffeomorphisms in these families we numerically calculate the strong unstable manifold of the fixed point. Our calculations strongly suggest that the strong unstable manifold is dense in $\mathbb T^3$. Further, we calculate push-forwards of the Lebesgue measure on a local strong unstable manifold. These numeric data indicate that the sequence of push-forwards converges to the SRB measure.

Geometric structure of the joint N‐voM distribution manifold and its applications to sensor networks

Abstract: In this paper, the geometric structure for normal distribution manifold, von Mises distribution manifold and their joint distribution manifold are firstly given by the metric, curvature, and divergence, respectively. Furthermore, the active detection with sensor networks is presented by a classical measurement model based on metric manifold, and the information resolution is presented for the range and angle measurements sensor networks. The preliminary analysis results introduced in this paper indicate that our approach is able to offer consistent and more comprehensive means to understand and solve sensor network problems containing sensors management and target detection, which are not easy to be handled by conventional analysis methods.

Curavture-Aware Non-Negative Matrix Factorization for Clustering

Abstract: Nonnegative matrix factorization (NMF) is one of the most successful techniques in pattern recognition and computer vision The main advantage of NMF is the parts-based representation of the input, which is consistent with how the human brain recognizes objects On the other hand, the high dimensional data usually resides on a low dimensional manifold The performance of the standard NMF can be significantly improved by incorporating with the manifold regularization However, most existing manifold methods fail to take the extrinsic geometry into account, ie, how the manifold embedded in the original high dimensional data space, which can discriminative the proximal points from different clusters In this paper, we propose a novel algorithm, named curvature-aware nonnegative matrix factorization (CANMF), to explicitly consider the extrinsic geometrical structures of the data distribution First, we build an affinity graph to encode the intrinsic geometrical structure of the data Then, an anisotropic diffusion process is utilized to exploit the extrinsic curvature, which rescales the weights of the affinity graph by the diffusivity operator Thus, the weights from different classes can be compressed, and enhance the discriminative ability of the proposed CANMF The experimental results on several real-world datasets show the advantages of our algorithm

Connections in sub-Riemannian geometry of parallelizable distributions

Abstract: The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this struc...