Showing papers on "Distribution (differential geometry) published in 2016"

Testing the manifold hypothesis

Abstract: The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for fitting a manifold to an unknown probability distribution supported in a separable Hilbert space, only using i.i.d samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution $P$ supported on the unit ball of a separable Hilbert space $H$. Let $G(d, V, \tau)$ be the set of submanifolds of the unit ball of $H$ whose volume is at most $V$ and reach (which is the supremum of all $r$ such that any point at a distance less than $r$ has a unique nearest point on the manifold) is at least $\tau$. Let $L(M, P)$ denote mean-squared distance of a random point from the probability distribution $P$ to $M$. We obtain an algorithm that tests the manifold hypothesis in the following sense. The algorithm takes i.i.d random samples from $P$ as input, and determines which of the following two is true (at least one must be): (a) There exists $M \in G(d, CV, \frac{\tau}{C})$ such that $L(M, P) \leq C \epsilon.$ (b) There exists no $M \in G(d, V/C, C\tau)$ such that $L(M, P) \leq \frac{\epsilon}{C}.$ The answer is correct with probability at least $1-\delta$.

Spectral convergence of the connection Laplacian from random samples

Abstract: Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Vector Diffusion Maps and showed that the connection Laplacian of the tangent bundle of the manifold can be approximated from random samples. In this paper, we present a unified framework for approximating other connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many independent random samples. We generalize the spectral convergence results to the case where the data points are sampled from a non-uniform distribution, and for manifolds with and without boundary.

Pure spinors, intrinsic torsion and curvature in odd dimensions

Abstract: We study the geometric properties of a ( 2 m + 1 ) -dimensional complex manifold M admitting a holomorphic reduction of the frame bundle to the structure group P ⊂ Spin ( 2 m + 1 , C ) , the stabiliser of the line spanned by a pure spinor at a point. Geometrically, M is endowed with a holomorphic metric g, a holomorphic volume form, a spin structure compatible with g, and a holomorphic pure spinor field ξ up to scale. The defining property of ξ is that it determines an almost null structure, i.e. an m-plane distribution N ξ along which g is totally degenerate. We develop a spinor calculus, by means of which we encode the geometric properties of N ξ and of its rank- ( m + 1 ) orthogonal complement N ξ ⊥ corresponding to the algebraic properties of the intrinsic torsion of the P-structure. This is the failure of the Levi-Civita connection ∇ of g to be compatible with the P-structure. In a similar way, we examine the algebraic properties of the curvature of ∇. Applications to spinorial differential equations are given. Notably, we relate the integrability properties of N ξ and N ξ ⊥ to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when ( M , g ) has prescribed curvature. We discuss applications of this work to the study of real pseudo-Riemannian manifolds.

Universal Nonlinear Regression on High Dimensional Data Using Adaptive Hierarchical Trees

Abstract: We study online sequential regression with nonlinearity and time varying statistical distribution when the regressors lie in a high dimensional space. We escape the curse of dimensionality by tracking the subspace of the underlying manifold using a hierarchical tree structure. We use the projections of the original high dimensional regressor space onto the underlying manifold as the modified regressor vectors for modeling of the nonlinear system. By using the proposed algorithm, we reduce the computational complexity to the order of the depth of the tree and the memory requirement to only linear in the intrinsic dimension of the manifold. The proposed techniques are specifically applicable to high dimensional streaming data analysis in a time varying environment. We demonstrate the significant performance gains in terms of mean square error over the other state of the art techniques through simulated as well as real data.

The family problem: hints from heterotic line bundle models

Abstract: Within the class of heterotic line bundle models, we argue that $$ \mathcal{N}=1 $$ vacua which lead to a small number of low-energy chiral families are preferred. By imposing an upper limit on the volume of the internal manifold, as required in order to obtain finite values of the four-dimensional gauge couplings, and validity of the supergravity approximation we show that, for a given manifold, only a finite number of line bundle sums are consistent with supersymmetry. By explicitly scanning over this finite set of line bundle models on certain manifolds we show that, for a sufficiently small volume of the internal manifold, the family number distribution peaks at small values, consistent with three chiral families. The relation between the maximal number of families and the gauge coupling is discussed, which hints towards a possible explanation of the family problem.

On the structure of codimension 1 foliations with pseudoeffective conormal bundle.

Abstract: Let X be a projective manifold equipped with a codimension 1 (maybe singular) distribution whose conormal sheaf is assumed to be pseudoeffective. By a theorem of Jean–Pierre Demailly, this distribution is actually integrable and thus defines a codimension 1 holomorphic foliation \( \mathcal{F} \). We aim at describing the structure of such a foliation, especially in the non-abundant case: It turns out that \( \mathcal{F} \) is the pull-back of one of the “canonical foliations” on a Hilbert modular variety. This result remains valid for “logarithmic foliated pairs.”

Articulated tracking with manifold regularized particle filter

Abstract: In this paper, we investigate articulated human motion tracking from video sequences using Bayesian approach. We derive a generic particle-based filtering procedure with a low-dimensional manifold. The manifold can be treated as a regularizer that enforces a distribution over poses during tracking process to be concentrated around the low-dimensional embedding. We refer to our method as manifold regularized particle filter. We present a particular implementation of our method based on back-constrained gaussian process latent variable model and gaussian diffusion. The proposed approach is evaluated using the real-life benchmark dataset HumanEva. We show empirically that the presented sampling scheme outperforms sampling-importance resampling and annealed particle filter procedures.

Moduli spaces for topologically quasi-homogeneous functions.

Abstract: We consider the topological class of a germ of 2-variables quasi-homogeneous complex analytic function. Each element f in this class induces a germ of foliation (f = constants) and a germ of curve (f = 0). We first describe the moduli space of the foliations in this class and give analytic normal forms. The classification of curves induces a distribution on this moduli space. By studying the infinitesimal generators of this distribution, we can compute the generic dimension of the moduli space for the curves, and we obtain the corresponding generic normal forms.

The Einstein-Hilbert type action on foliated pseudo-Riemannian manifolds

Abstract: We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained EulerLagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.

Viability theorem for deterministic mean field type control systems

Abstract: A mean field type control system is a dynamical system in the Wasserstein space describing an evolution of a large population of agents with mean-field interaction under a control of a unique decision maker. We develop the viability theorem for the mean field type control system. To this end we introduce a set of tangent elements to the given set of probabilities. Each tangent element is a distribution on the tangent bundle of the phase space. The viability theorem for mean field type control systems is formulated in the classical way: the given set of probabilities on phase space is viable if and only if the set of tangent distributions intersects with the set of distributions feasible by virtue of dynamics.

A groupoid approach to pseudodifferential calculi

Abstract: We give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural R × -action. Specif- ically, a properly supported semiregular distribution on M × M is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibres of the tangent groupoid which is homogeneous for the R × -action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Further, with the appropriate generalization of the tangent bundle, the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.

On generic submanifolds of manifolds endowed with metric mixed 3-structures

Abstract: We introduce the concept of generic submanifold in a manifold equipped with a metric mixed 3-structure and investigate the canonical distributions induced on such submanifold. In particular, we obtain necessary and sufficient conditions for the integrability of these distributions and discuss the geometry of leaves. Moreover, some examples are given.

Complex Variable Meshless Manifold Method for Elastic Dynamic Problems

Abstract: Combining the finite covering technical and complex variable moving least square, the complex variable meshless manifold method can handle the discontinuous problem effectively. In this paper, the complex variable meshless method is applied to solve the problem of elastic dynamics, the complex variable meshless manifold method for dynamics is established, and the corresponding formula is derived. The numerical example shows that the numerical solutions are in good agreement with the analytical solution. The CVMMM for elastic dynamics and the discrete forms are correct and feasible. Compared with the traditional meshless manifold method, the CVMMM has higher accuracy in the same distribution of nodes.

Fuel Cell Stack Manifold Optimization through Modelling and Simulation

Abstract: A Polymer Electrolyte Membrane (PEM) fuel cell uses the chemical energy of hydrogen to cleanly and efficiently produce electricity. If hydrogen is the fuel, electricity, water, and heat are the only products. Also, fuel cells are quiet during operation as they have fewer moving parts. For the better performance of the fuel cell stack, uniform distribution of reactants, minimum pressure drop and flow losses are desired. Uneven distribution of reactants from the manifold to cells will lead to the poor performance of the fuel cell. Thus it is very much essential to give importance to the inlet geometry of fuel cell stack manifold. In this paper, 5-pass serpentine channel fuel cell stack consisting of 20 cells and 50 cells has been considered for modelling of three geometries for the inlet manifold namely rectangular, short diffuser and long diffuser. The proposed model is governed by mass and momentum conservation equations. Suitable convergence criteria for mass and momentum calculation were set. Velocity boundary condition for the inlet of the manifold has been calculated by using suitable formula. The model is numerically implemented using commercial CFD tool ANSYS-FLUENT 12.1. Flow and pressure distribution for the manifold and channel regions with three different inlet geometries have been simulated and analysed thereafter to reveal the influence of manifold inlet geometry on the flow distribution in the manifold and the channels.

Geodesics in Geometry with Constraints and Applications

Abstract: In this course we carefully define the notion of a non-holonomic manifold which is a manifold with a certain non-integrable distribution. We describe such concepts as horizontal distribution, the Ehresmann connection, bracket generating condition for a distribution, sub-Riemannian structure and sub-Riemannian metric, Hamiltonian system, normal and abnormal geodesics, principal bundle and others.

Screen conformal lightlike submanifolds of semi-Riemannian manifolds

Abstract: Since the induced objects on a lightlike submanifold depend on its screen distribution which, in general, is not unique and hence we can not use the classical submanifold theory on a lightlike submanifold in the usual way. Therefore, in present paper, we study screen conformal lightlike submanifolds of a semi-Riemannian manifold, which are essential for the existence of unique screen distribution. We obtain a characterization theorem for the existence of screen conformal lightlike submanifolds of a semi-Riemannian manifold. We prove that if the differential operator Ds is a metric Otsuki connection on transversal lightlike bundle for a screen conformal lightlike submanifold then semi-Riemannian manifold is a semi-Euclidean space. We also obtain some characterization theorems for a screen conformal totally umbilical lightlike submanifold of a semi-Riemannian space form. Further, we obtain a necessary and sufficient condition for a screen conformal lightlike submanifold of constant curvature to be a semi-Euclidean space. Finally, we prove that for an irrotational screen conformal lightlike submanifold of a semi-Riemannian space form, the induced Ricci tensor is symmetric and the null sectional curvature vanishes.

Classification of holomorphic foliations on Hopf manifolds

Abstract: We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. We prove that any nonsingular codimension one distribution on an intermediary or generic Hopf manifold is integrable and admits a holomorphic first integral. Also, we prove some results about singular holomorphic distributions on Hopf manifolds.

Freedom of h(2)-variationality and metrizability of sprays

Abstract: In this paper we are investigating variational homogeneous second order differential equations by considering the questions of how many different variational principles exist for a given spray. We focus our attention on h(2)-variationality; that is, the regular Lagrange function is homogeneous of degree two in the directional argument. Searching for geometric objects characterizing the degree of freedom of h(2)-variationality of a spray, we show that the holonomy distribution generated by the tangent direction to the parallel translations can be used to calculate it. As a working example, the class of isotropic sprays is considered.

Deep water is control allocation unit and manifold integrated system under water

Abstract: The utility model provides an integrated system that deep water is control allocation unit and manifold under water for the development of deep sea oil gas field, its characterized in that: it constitutes to prevent fishing net protective frame, manifold, substructure basis including upper portion. The manifold control allocation unit under water that has integrateed wherein. Integrated system by control allocation unit under water, prevent the fishing net and pull protective frame, manifold structural framework, substructure basis, manifold pipeline, lug, anode block, structure pipe fitting, valve, person in charge's line, joint system, transition forging etc. And constitute. Integrated system not only possess the effect of ordinary manifold, insert when can realize many well heads, the main is the control allocation unit under water that has integrateed in this manifold system, through this distribution unit, can with the signal of telecommunication, hydraulic power and the chemical agent distribution that are used for control with the monitoring to the subsea tree who closes on, has saved and has set up control allocation modular construction under water alone. The utility model discloses integrateed manifold and under water the control allocation unit and can be on active service in fishery activity district.

Cross-Manifold Guidance in Deformable Registration of Brain MR Images

Abstract: Manifold is often used to characterize the high-dimensional distribution of individual brain MR images. The deformation field, used to register the subject with the template, is perceived as the geodesic pathway between images on the manifold. Generally, it is non-trivial to estimate the deformation pathway directly due to the intrinsic complexity of the manifold. In this work, we break the restriction of the single and complex manifold, by short-circuiting the subject-template pathway with routes from multiple simpler manifolds. Specifically, we reduce the anatomical complexity of the subject/template images, and project them to the virtual and simplified manifolds. The projected simple images then guide the subject image to complete its journey toward the template image space step by step. In the final, the subject-template pathway is computed by traversing multiple manifolds of lower complexity, rather than depending on the original single complex manifold only. We validate the cross-manifold guidance and apply it to brain MR image registration. We conclude that our method leads to superior alignment accuracy compared to state-of-the-art deformable registration techniques.

$J$-tangent affine hyperspheres with an involutive contact distribution

Abstract: In this paper we study J-tangent affine hyperspheres. The main purpose of this paper is to give a local characterization of J-tangent affine hyperspheres of arbitrary dimension with an involutive contact distribution. Some new examples of such hyperspheres are also given.

Inteliggent manifold assemblies for a light source, light sources including intelligent manifold assemblies, and methods of operating the same

Abstract: A manifold assembly for distribution of a cooling fluid configured for use with a light source is provided. The manifold assembly includes a fluid manifold for providing a cooling fluid to a lamp head assembly of the light source, at least one sensor for sensing at least one characteristic of the cooling fluid in the fluid manifold, and a microprocessor for receiving information related to the at least one characteristic from the at least one sensor.

On the extremal function of the modulus of a foliation

Abstract: We investigate the properties of the modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of the extremal function and state some of its properties. We obtain an integral formula which, in a sense, combines the integral over the manifold with the integral over the leaves. We state a relation between the extremal function and the geometry of the distribution orthogonal to a foliation.

Singularities for analytic continuations of holonomy germs of Riccati foliations

Abstract: In this paper we study the problem of analytic extension of germs of holonomy of algebraic foliations. More precisely we prove that for a Riccati foliation associated to a branched projective structure over a finite type surface which is non-elementary and parabolic, all the germs of holonomy between a fiber and a holomorphic section of the bundle are led to singularities by almost every developed geodesic ray. We study in detail the distribution of these singularities and prove in particular that they are included and dense in the limit set and uncountable, giving another negative answer to a conjecture of Loray.

Manifold design for uniform fluid distribution in parallel microchannels

Abstract: In this paper, we use an electrical circuit analogy to design a microfluidic manifold achieving a uniform distribution of flow across multiple channels with minimal pressure gradient throughout the flow. We investigate the impact of manifold design to the mass flow distribution derived from the electrical circuit analogy and then analyzed the flow distribution in microchannels using computational fluid dynamics including thermal modelling. We compared the flow uniformity in the parallel channels with other manifold designs and found that step manifold design has improved the flow and temperature uniformity of the parallel channels.