Showing papers on "Distribution (differential geometry) published in 2011"

Subspaces Indexing Model on Grassmann Manifold for Image Search

Abstract: Conventional linear subspace learning methods like principal component analysis (PCA), linear discriminant analysis (LDA) derive subspaces from the whole data set. These approaches have limitations in the sense that they are linear while the data distribution we are trying to model is typically nonlinear. Moreover, these algorithms fail to incorporate local variations of the intrinsic sample distribution manifold. Therefore, these algorithms are ineffective when applied on large scale datasets. Kernel versions of these approaches can alleviate the problem to certain degree but face a serious computational challenge when data set is large, where the computing involves Eigen/QP problems of size N × N. When N is large, kernel versions are not computationally practical. To tackle the aforementioned problems and improve recognition/searching performance, especially on large scale image datasets, we propose a novel local subspace indexing model for image search termed Subspace Indexing Model on Grassmann Manifold (SIM-GM). SIM-GM partitions the global space into local patches with a hierarchical structure; the global model is, therefore, approximated by piece-wise linear local subspace models. By further applying the Grassmann manifold distance, SIM-GM is able to organize localized models into a hierarchy of indexed structure, and allow fast query selection of the optimal ones for classification. Our proposed SIM-GM enjoys a number of merits: 1) it is able to deal with a large number of training samples efficiently; 2) it is a query-driven approach, i.e., it is able to return an effective local space model, so the recognition performance could be significantly improved; 3) it is a common framework, which can incorporate many learning algorithms. Theoretical analysis and extensive experimental results confirm the validity of this model.

Algorithm comparison for Karcher mean computation of rotation matrices and diffusion tensors

Abstract: This paper concerns the computation, by means of gradient and Newton methods, of the Karcher mean of a finite collection of points, both on the manifold of 3×3 rotation matrices endowed with its usual bi-invariant metric and on the manifold of 3×3 symmetric positive definite matrices endowed with its usual affine invariant metric. An explicit expression for the Hessian of the Riemannian squared distance function of these manifolds is given. From this, a condition on the step size of a constant step gradient method that depends on the data distribution is derived. These explicit expressions make a more efficient implementation of the Newton method possible and it is shown that the Newton method outperforms the gradient method in some cases.

Differential Forms on Wasserstein Space and Infinite-Dimensional Hamiltonian Systems

Abstract: Let M denote the space of probability measures on R D endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced in (5). In this paper we develop a calculus for the corresponding class of differential forms on M. In particular we prove an analogue of Green's theorem for 1-forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For D = 2d we then define a symplectic distribution on M in terms of this calculus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as introduced in (3). Throughout the paper we emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of R D.

Kinematics for rolling a Lorentzian sphere

Abstract: We derive the equations of motion for the n-dimensional Lorentzian sphere (one-sheet hyperboloid) rolling, without slipping and twisting, over the affine tangent space at a point. Both manifolds are endowed with semi-Riemannian metrics, induced by the Lorentzian metric on the embedding manifold which is the generalized Minkowski space. The kinematic equations turn out to be a nonlinear control system evolving on a connected subgroup of the Poincare group. The controls correspond to the choice of the curves along which the Lorentzian sphere rolls. Controllability of this rolling system will be proved by showing that the corresponding distribution is bracket-generating.

Integrable random matrix ensembles

Abstract: We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner–Dyson random matrices and Poisson statistics. The construction is based on integrable N-body classical systems with a random distribution of momenta and coordinates of the particles. The Lax matrices of these systems yield random matrix ensembles whose joint distribution of eigenvalues can be calculated analytically thanks to the integrability of the underlying system. Formulae for spacing distributions and level compressibility are obtained for various instances of such ensembles.

Contact structures and supersymmetric mechanics

Abstract: Weestablisharelationbetweencontactstructuresonsupermanifoldsandsupersymmetricmechanics in the superspace formulation. This allows one to use the language of contactgeometrywhendealingwithsupersymmetricmechanics. 1 Introduction In this work we construct a contact form associated with N= 2 supersymmetric mechanics. Thisallows the interpretation of the SUSY transformations in superspace as strict contactomorphisms.A similar relation between superconformal field theory and contact complex geometry was ex-plored by Schwarz [7].With the relation between SUSY and contact structures being our primary goal here, let uspresent a lightning review of classical contact structures highlighting the elements we need later.Recall that a precontact structure on a manifold is a one-form that is nowhere vanishing 1 . As-sociated with every precontact structure on a manifold is a hyperplane distribution, that is asubbundle of the tangent bundle of corank 1. The hyperplane distribution is defined to be thespan of the kernel of the precontact structure. That is if we denote the precontact structure asα∈ Ω

Discretization of Parametrizable Signal Manifolds

Abstract: Transformation-invariant analysis of signals often requires the computation of the distance from a test pattern to a transformation manifold. In particular, the estimation of the distances between a transformed query signal and several transformation manifolds representing different classes provides essential information for the classification of the signal. In many applications, the computation of the exact distance to the manifold is costly, whereas an efficient practical solution is the approximation of the manifold distance with the aid of a manifold grid. In this paper, we consider a setting with transformation manifolds of known parameterization. We first present an algorithm for the selection of samples from a single manifold that permits to minimize the average error in the manifold distance estimation. Then we propose a method for the joint discretization of multiple manifolds that represent different signal classes, where we optimize the transformation-invariant classification accuracy yielded by the discrete manifold representation. Experimental results show that sampling each manifold individually by minimizing the manifold distance estimation error outperforms baseline sampling solutions with respect to registration and classification accuracy. Performing an additional joint optimization on all samples improves the classification performance further. Moreover, given a fixed total number of samples to be selected from all manifolds, an asymmetric distribution of samples to different manifolds depending on their geometric structures may also increase the classification accuracy in comparison with the equal distribution of samples.

On the k-nullity foliations in Finsler geometry and completeness

Abstract: Here, a Finsler manifold (M,F) is considered with corresponding cur-vature tensor, regarded as 2-forms on the bundle of non-zero tangentvectors. Certain subspaces of the tangent spaces of M determined bythe curvature are introduced and called k-nullity foliations of the curva-ture operator. It is shown that if the dimension of foliation is constantthen the distribution is involutive and each maximal integral manifold istotally geodesic. Characterization of the k-nullity foliation is given, aswell as some results concerning constancy of the flag curvature, and com-pleteness of their integral manifolds, providing completeness of (M,F).The introduced k-nullity space is a natural extension of nullity space inRiemannian geometry, introduced by S. S. Chern and N. H. Kuiper andenlarged to Finsler setting by H. Akbar-Zadeh and contains it as a specialcase.Keywords: Foliation, k-nullity, Finsler manifolds, Curvature operator.MSC: 2000 Mathematics subject Classification: 58B20, 53C60, 53C12.

On the k-nullity foliations in finsler geometry

Abstract: Here, a Finsler manifold (M,F) is considered with cor- responding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k- nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant, then the distribution is involutive and each maximal integral manifold is totally geodesic. Character- ization of the k-nullity foliation is given, as well as some results concerning constancy of the flag curvature, and completeness of their integral manifolds, providing completeness of (M,F). The introduced k-nullity space is a natural extension of nullity space in Riemannian geometry, introduced by Chern and Kuiper and en- larged to Finsler setting by Akbar-Zadeh and contains it as a special case.

Pneumatic distribution machine

Abstract: The pneumatic distributor has a horizontal feed pipe (4) for supplying the grit. A manifold (5) is provided at an angle of 90[deg] with respect to horizontal feed pipe. A corrugated pipe riser (6) is provided adjoint to the manifold. A header (7) with radial hose lines is arranged at upper end of manifold for supplying the grit. A guide element is provided in inner wall of manifold for guiding the flow of grit toward the center of manifold. The guide element is arranged to close the space between the element and inner wall of manifold.

Metric foliations on hyperbolic spaces

Abstract: . On the hyperbolic space D n , codimension-one totally geo-desic foliations of class C k are classified. Except for the unique parabolichomogeneous foliation, the set of all such foliations is in one-one corre-spondence (up to isometry) with the set of all functions z : [0,π] → S n−1 of class C k−1 with z(0) = e 1 = z(π) satisfying|z ′ (r)| ≤ 1for all r, modulo an isometric action by O(n − 1) × R× Z 2 .Since 1-dimensional metric foliations on D n are always either homoge-neous or flat (that is, their orthogonal distributions are integrable), thisclassifies all 1-dimensional metric foliations as well.Equations of leaves for a non-trivial family of metric foliations on D 2 (called “fifth-line”) are found. 1. IntroductionLet Fbe a foliation on a Riemannian manifold (M,g). The tangent vectorfield (vertical) and the complementary vector field (horizontal) of Fare denotedby Vand H, respectively. The foliation Fis said to be metric if ∇ v : H×H→Vis skew-symmetric, or equivalently, the leaves of Fare equi-distant locally.Such a foliation is flat if the orthogonal distribution is integrable (and henceforms a totally geodesic foliation); is homogeneous if it consists of the orbits ofa free action of a subgroup of the isometry group. Gromoll-Grove ([4]) showedthat 1-dimensional metric foliations on constant curvature spaces are either flator homogeneous. As a consequence, the only 1-dimensional metric foliations ofEuclidean spheres are the Hopf fibrations S

Foliations and Complemented Framed Structures on an Almost Contact Metric Manifold

Abstract: On an odd dimensional manifold, we define a structure which generalizes several known structures on almost contact manifolds, namely Sasakian, trans-Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic structures. This structure, hereinafter called a generalized quasi-Sasakian, shortly G.Q.S. structure, is defined on an almost contact metric manifold and satisfies an additional condition. Then we consider a distribution \({\mathcal{D}_{1}}\) wich allows a suitable decomposition of the tangent bundle of a G.Q.S. manifold. Necessary and sufficient conditions for the normality of the complemented framed structure on the distribution \({\mathcal{D}_{1}}\) defined on a G.Q.S manifold are studied. The existence of the foliation on G.Q.S. manifolds and of bundle-like metrics are also proven. It is shown that under certain circumstances a new foliation arises and its properties are investigated. Some examples illustrating these results are given in the final part of this paper.

Fluctuation geometry: A counterpart approach of inference geometry

Abstract: Starting from an axiomatic perspective, \emph{fluctuation geometry} is developed as a counterpart approach of inference geometry. This approach is inspired on the existence of a notable analogy between the general theorems of \emph{inference theory} and the the \emph{general fluctuation theorems} associated with a parametric family of distribution functions $dp(I|\theta)=\rho(I|\theta)dI$, which describes the behavior of a set of \emph{continuous stochastic variables} driven by a set of control parameters $\theta$. In this approach, statistical properties are rephrased as purely geometric notions derived from the \emph{Riemannian structure} on the manifold $\mathcal{M}_{\theta}$ of stochastic variables $I$. Consequently, this theory arises as an alternative framework for applying the powerful methods of differential geometry for the statistical analysis. Fluctuation geometry has direct implications on statistics and physics. This geometric approach inspires a Riemannian reformulation of Einstein fluctuation theory as well as a geometric redefinition of the information entropy for a continuous distribution.

Some framed f -structures on transversally Finsler foliations

Abstract: Some problems concerning to Liouville distribution and framed \(f\)-structures are studied on the normal bundle of the lifted Finsler foliation to its normal bundle. It is shown that the Liouville distribution of transversally Finsler foliations is an integrable one and some natural framed \(f(3,\varepsilon)\)- structures of corank 2 exist on the normal bundle of the lifted Finsler foliation.

On the Geometry of Almost -Manifolds

Abstract: An 𝑓 -structure on a manifold 𝑀 is an endomorphism field 𝜑 satisfying 𝜑 3 + 𝜑 = 0 . We call an f-structure regular if the distribution 𝑇 = k e r 𝜑 is involutive and regular, in the sense of Palais. We show that when a regular f-structure on a compact manifold M is an almost 𝒮 -structure, it determines a torus fibration of M over a symplectic manifold. When rank 𝑇 = 1 , this result reduces to the Boothby-Wang theorem. Unlike similar results for manifolds with 𝒮 -structure or 𝒦 -structure, we do not assume that the f-structure is normal. We also show that given an almost 𝒮 -structure, we obtain an associated Jacobi structure, as well as a notion of symplectization.

A manifold and distribution manifold assembly

Abstract: A manifold (50) and distribution manifold assembly (44) is described for use in dividing the flow of an air-entrained material from a primary distribution line into a plurality of secondary distribution lines. The manifold (50) is formed of a single molded piece thereby reducing the overall tooling cost. One application of such a distribution manifold assembly is in an air seeder.

Involutive distributions and dynamical systems of second-order type

Abstract: We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define associated connections and we give a coordinate-independent criterion for determining whether the vector field is of quadratic type. Further, we investigate the underlying global bundle structure of the manifold under consideration, induced by the vector field and the involutive distribution.

On the geometry of almost $\mathcal{S}$-manifolds

Abstract: An $f$-structure on a manifold $M$ is an endomorphism field $\phi$ satisfying $\phi^3+\phi=0$. We call an $f$-structure {\em regular} if the distribution $T=\ker\phi$ is involutive and regular, in the sense of Palais. We show that when a regular $f$-structure on a compact manifold $M$ is an almost $\S$-structure, as defined by Duggal, Ianus, and Pastore, it determines a torus fibration of $M$ over a symplectic manifold. When $\rank T = 1$, this result reduces to the Boothby-Wang theorem. Unlike similar results due to Blair-Ludden-Yano and Soare, we do not assume that the $f$-structure is normal. We also show that given an almost $\mathcal{S}$-structure, we obtain an associated Jacobi structure, as well as a notion of symplectization.

Properties and applications of Fisher distribution on the rotation group

Abstract: We study properties of Fisher distribution (von Mises-Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011), and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data.

Melt distribution manifold

Abstract: A melt distribution manifold for use with first and second mold portions moveable with respect to each other, the melt distribution manifold including a melt inlet means for receiving melt from an injection molding machine, a first manifold portion connected to the first mold portion, wherein the first manifold portion is stationary with respect to the first mold portion, a second manifold portion connecting the first manifold portion to the hot runner nozzle, wherein the second manifold portion is moveable with respect to the first manifold portion such that when the first mold portion moves with respect to the second mold portion, the second manifold portion remains connected to the first manifold portion and to the hot runner nozzle, a hinged joint connecting the first manifold portion to the second manifold portion. Each of the first manifold portion, the second manifold portion, and the hinged joint having respective melt distribution bores in fluid communication with each other.

Data-driven importance distributions for articulated tracking

Abstract: We present two data-driven importance distributions for particle filterbased articulated tracking; one based on background subtraction, another on depth information. In order to keep the algorithms efficient, we represent human poses in terms of spatial joint positions. To ensure constant bone lengths, the joint positions are confined to a non-linear representation manifold embedded in a high-dimensional Euclidean space. We define the importance distributions in the embedding space and project them onto the representation manifold. The resulting importance distributions are used in a particle filter, where they improve both accuracy and efficiency of the tracker. In fact, they triple the effective number of samples compared to the most commonly used importance distribution at little extra computational cost.

On the distribution of the coefficients of normal forms for Frobenius expansions

Abstract: Frobenius expansions are representations of integers to an algebraic base which are sometimes useful for efficient (hyper)elliptic curve cryptography. The normal form of a Frobenius expansion is the polynomial with integer coefficients obtained by reducing a Frobenius expansion modulo the characteristic polynomial of Frobenius. We consider the distribution of the coefficients of reductions of Frobenius expansions and non-adjacent forms of Frobenius expansions (NAFs) to normal form. We give asymptotic bounds on the coefficients which improve on naive bounds, for both genus one and genus two. We also discuss the non-uniformity of the distribution of the coefficients (assuming a uniform distribution for Frobenius expansions).

Involutive distributions and dynamical systems of second-order type

Abstract: We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define associated connections and we give a coordinate-independent criterion for determining whether the vector field is of quadratic type. Further, we investigate the underlying global bundle structure of the manifold under consideration, induced by the vector field and the involutive distribution.

On almost-Riemannian surfaces

Abstract: An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting its novelties with respect to Riemannian geometry. We present some results that investigate topological, metric and geometric aspects of almostRiemannian surfaces from a local and global point of view.

Tamed Symplectic forms and Generalized Geometry

Abstract: We show that symplectic forms taming complex structures on compact manifolds are related to special types of almost generalized K\"ahler structures. By considering the commutator $Q$ of the two associated almost complex structures $J_{\pm}$, we prove that if either the manifold is 4-dimensional or the distribution ${Im} \, Q$ is involutive, then the manifold can be expressed locally as a disjoint union of twisted Poisson leaves.

Deforming metrics of foliations

Abstract: Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies conformally along one of the distributions. Then we introduce the Extrinsic Geometric Flow depending on the mean curvature vector field of the distribution, and show existence/uniquenes and convergence of a solution as $t\to\infty$, when the complementary distribution is integrable with compact leaves. We apply the method to the problem of prescribing mean curvature vector field of a foliation, and give examples for harmonic and umbilical foliations and for the double-twisted product metrics, including the codimension-one case.

Discretization of Parametrizable Signal Manifolds

Abstract: Transformation-invariant analysis of signals often requires the computation of the distance from a test pattern to a transformation manifold. In particular, the estimation of the distances between a transformed query signal and several transformation manifolds representing different classes provides essential information for the classification of the signal. In many applications the computation of the exact distance to the manifold is costly, whereas an efficient practical solution is the approximation of the manifold distance with the aid of a manifold grid. In this paper, we consider a setting with transformation manifolds of known parameterization. We first present an algorithm for the selection of samples from a single manifold that permits to minimize the average error in the manifold distance estimation. Then we propose a method for the joint discretization of multiple manifolds that represent different signal classes, where we optimize the transformation-invariant classification accuracy yielded by the discrete manifold representation. Experimental results show that sampling each manifold individually by minimizing the manifold distance estimation error outperforms baseline sampling solutions with respect to registration and classification accuracy. Performing an additional joint optimization on all samples improves the classification performance further. Moreover, given a fixed total number of samples to be selected from all manifolds, an asymmetric distribution of samples to different manifolds depending on their geometric structures may also increase the classification accuracy in comparison with the equal distribution of samples.

Flow field plate assembly with two or more parallel flow channels

Abstract: PROBLEM TO BE SOLVED: To provide a flow field plate promoting enhancement of efficiency, or facilitation of distribution of fluid. SOLUTION: The flow field plate assembly is provided with a first manifold 11, a second manifold 12, a first flow channel and a second flow channel. The first manifold 11 is equipped with a fluid inlet for accepting inflow fluid, and provides a flow channel for transferring the inflow fluid in a first direction. The first manifold 11 is provided with at least two distribution outlets and releases at least part of the inflow fluid from each distribution outlet as a discharge fluid. The second manifold 12 is provided with a fluid outlet for discharging discharge fluid containing part of the inflow fluid. The second manifold provides a flow channel for transferring the discharge fluid along a second direction. The second manifold accepts the discharge fluid from at least two discharge fluid inlets. COPYRIGHT: (C)2011,JPO&INPIT

On a Class of Nilpotent Distributions

Abstract: This paper presents a sufficient condition for two vector fields $X$ and $Y$ to have the squares noncommutative, i.e. $[X^2, Y^2] ot=0$. We prove that if the vector fields $X$, $Y$ span a nilpotent distribution with nilpotence class 2, then the squares of the vector fields do not commute.