Manifold

Nonnegative matrix factorization (NMF), a popular part-based representation technique, does not capture the intrinsic local geometric structure of the data space. Graph regularized NMF (GNMF) was recently proposed to avoid this limitation by regularizing NMF with a nearest neighbor graph constructed from the input data set. However, GNMF has two main bottlenecks. First, using the original feature space directly to construct the graph is not necessarily optimal because of the noisy and irrelevant features and nonlinear distributions of data samples. Second, one possible way to handle the nonlinear distribution of data samples is by kernel embedding. However, it is often difficult to choose the most suitable kernel. To solve these bottlenecks, we propose two novel graph-regularized NMF methods, AGNMF FS and AGNMF MK , by introducing feature selection and multiple-kernel learning to the graph regularized NMF, respectively. Instead of using a fixed graph as in GNMF, the two proposed methods learn the nearest neighbor graph that is adaptive to the selected features and learned multiple kernels, respectively. For each method, we propose a unified objective function to conduct feature selection/multi-kernel learning, NMF and adaptive graph regularization simultaneously. We further develop two iterative algorithms to solve the two optimization problems. Experimental results on two challenging pattern classification tasks demonstrate that the proposed methods significantly outperform state-of-the-art data representation methods.

Emerging from music and the visual arts, questions about hearing and seeing deeply affected Hermann Helmholtz's and Bernhard Riemann's contributions to what became called the ''problem of space [Raumproblem],'' which in turn influenced Albert Einstein's approach to general relativity. Helmholtz's physiological investigations measured the time dependence of nerve conduction and mapped the three-dimensional manifold of color sensation. His concurrent studies on hearing illuminated musical evidence through experiments with mechanical sirens that connect audible with visible phenomena, especially how the concept of frequency unifies motion, velocity, and pitch. Riemann's critique of Helmholtz's work on hearing led Helmholtz to respond and study Riemann's then-unpublished lecture on the foundations of geometry. During 1862-1870, Helmholtz applied his findings on the manifolds of hearing and seeing to the Raumproblem by supporting the quadratic distance relation Riemann had assumed as his fundamental hypothesis about geometrical space. Helmholtz also drew a ''close analogy … in all essential relations between the musical scale and space.'' These intersecting studies of hearing and seeing thus led to reconsideration and generalization of the very concept of ''space,'' which Einstein shaped into the general manifold of relativistic space-time.

Rendón, The fuzzy classifier system: motivations and first results, Proc. First Intl. Conf. on Parallel Problem Solving from Nature-PPSN I, Springer, Berlin, 1991, pp. 330-334 (scatter Mamdani fuzzy rules for control/modeling problems) M. Valenzuela-Rendón, Reinforcement learning in the fuzzy classifier system, Expert Systems with Applications 14 (1998) 237-247 (scatter Mamdani fuzzy rules for control/modeling problems) J.R. Velasco, Genetic-based on-line learning for fuzzy process control, IJIS 13 (10-11) (1998) 891-903 (scatter Mamdani fuzzy rules for control problems)

We present an accessible account of the local geometry of the parameter space of Calabi-Yau manifolds. It is shown that the parameter space decomposes, at least locally, into a product with the space of parameters of the complex structure as one factor and a complex extension of the parameter space of the Kfihler class as the other. It is also shown that each of these spaces is itself a Kfihler manifold and is moreover a gdihler manifold of restricted type. There is a remarkable symmetry in the intrinsic structures of the two parameter spaces and the relevance of this to the conjectured existence of mirror manifolds is discussed. The two parameter spaces behave differently with respect to modular transformations and it is argued that the role of quantum corrections is to restore the symmetry between the two types of parameters so as to enforce modular invariance.

L'expression explicite de la formalité de M. Kontsevich sur R d est la base de la preuve du théorème de formalité pour une variété quelconque [K1 § 7], qui impliqueà son tour l'existence d'étoile-produits sur une variété de Poisson quelconque. Nous proposons ici un choix cohérent d'orientations et de signes qui permet de reprendre la démonstration du théorème pour R d en tenant compte des signes qui apparaissent devant les différents termes de l'équation de formalité.

Recently it has been shown that there are paths on the moduli space between two Calabi-Yau manifolds with different topology that have finite length. A priori, there exists the possibility that the singular manifold that is common to two different moduli spaces has two different Ricci-flat Kiihler metrics, each being the limit of the Ricci-flat Kähler metric of the respective topologically distinct Calabi-Yau manifold. In this paper we show that this is not the case and the topology changing path is Continuous even in the space of Ricci-flat Kähler metrics. The explicit form of the Ricci-flat Kähler metrics is calculated in the vicinity of the nodes for the conifold, the resolution and the deformation. A preliminary discussion of global issues is presented and it is shown that, owing to a topological obstruction, the manifold obtained as the result of independently resolving and deforming the nodes of a conifold in general cannot be Ki~hler.

We discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits (cycles) of maps and their implementation in matcont, a matlab toolbox for continuation and bifurcation analysis of dynamical systems. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points (i.e. LP, PD and NS), and their continuation in two control parameters, as well as detection and location of all codimension 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed, both numerically with finite directional differences and using symbolic derivatives of the original map. Using a parameter-dependent center manifold reduction, explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period. These asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations. We provide two examples illustrating the developed techniques: a generalized Hénon map and a juvenile/adult competition model from mathematical biology.

Does inner sense, like outer sense, provide inner sensations or, in other words, a sensory manifold of its own? Advocates of the disparity thesis on inner and outer sense claim that it does not. This interpretation, which is dominant in the preexisting literature, leads to several inconsistencies when applied to Kant’s doctrine of inner experience. Yet, while so, the parity thesis, which is the contrasting view, is also unable to provide a convincing interpretation of inner sensations. In this paper, I argue that this deadlock can be traced back to an inadequate understanding of inner sense shared by both sides. Drawing upon an analysis of the notion of obscure representations, I offer an alternative interpretation of inner sense with a special regard to self-affection, apprehension, and attention. From this basis, I will infer that outer sense delivers sensory content that is initially and intrinsically unaccompanied by phenomenal consciousness; inner sense contributes by endowing such content with phenomenal consciousness. Therefore, phenomenal qualities can be regarded as the sensory manifold of inner sense. This alternative interpretation solves the long-standing dispute concerning inner sensations and would further illuminate Kant’s notion of inner experience.

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in $\R^3$ with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry,

In general C * -algebras, elements with minimal norm in some equivalence class are introduced and characterized. We study the set of minimal hermitian matrices, in the case where the C * -algebra consists of 3×3 complex matrices, and the quotient is taken by the subalgebra of diagonal matrices. We thoroughly study the set of minimal matrices particularly because of its relation to the geometric problem of finding minimal curves in flag manifolds. For the flag manifold of 'four mutually orthogonal complex lines' in C 4 , it is shown that there are infinitely many minimal curves joining arbitrarily close points. In the case of the flag manifold of 'three mutually orthogonal complex lines' in C 3 , we show that the phenomenon of multiple minimal curves joining arbitrarily close points does not occur.

We define super Riemann surfaces as smooth 2{2-dimensional supermanifolds equipped with a reduction of their structure group to the group of invertible upper triangular 2 x 2 complex matrices. The integrability conditions for such a reduction turn out to be (most of) the torsion constraints of 2d supergravity. We show that they are both necessary and sufficient for a frame to admit local superconformal coordinates. The other torsion constraints are merely conditions to fix some of the gauge freedom in this description, or to specify a particular connection on such a manifold, analogous to the Levi-Civita connection in Riemannian geometry. Unlike ordinary Riemann surfaces, a super Riemann surface cannot be regarded as having only one complex dimension. Nevertheless, in certain important aspects super Riemann surfaces behave as nicely as if they had only one dimension. In particular they posses an analog ~ of the Cauchy-Riemann operator on ordinary Riemann surfaces, a differential operator taking values in the bundle of half-volume forms. This operator furnishes a short resolution of the structure sheaf, making possible a Quillen theory of determinant line bundles. Finally we show that the moduli space of super Riemann surfaces is embedded in the larger space of complex curves of dimension 111.

In this article, we investigate the response of a thin superconducting shell to an arbitrary external magnetic field. We identify the intensity of the applied field that forces the emergence of vortices in minimizers, the so-called first critical field H c1 in Ginzburg-Landau theory, for closed simply connected manifolds and arbitrary fields. In the case of a simply connected surface of revolution and vertical and constant field, we further determine the exact number of vortices in the sample as the intensity of the applied field is raised just above H c1 . Finally, we derive via Γ -convergence similar statements for three-dimensional domains of small thickness, where in this setting point vortices are replaced by vortex lines.

A morph between two Riemannian n-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian n-manifolds and, with respect to these classes, prove the existence of minimal distortion morphs and diffeomorphisms. In particular, we consider the class of time-dependent vector fields (on an open subset Ω of R n+1 in which the manifolds are embedded) that generate morphs between two manifolds M and N via an evolution equation, define the bending and the morphing distortion energies for these morphs, and prove the existence of minimizers of the corresponding functionals in the set of time-dependent vector fields that generate morphs between M and N and are L 2 functions from [0, 1] to the Sobolev space W k,2 0 (Ω, R n+1 ).

A symplectic groupoid is a manifold T with a partially defined multiplication (satisfying certain axioms) and a compatible symplectic structure. The identity elements in T turn out to form a Poisson manifold To? and the correspondence between symplectic groupoids and Poisson manifolds is a natural extension of the one between Lie groups and Lie algebras.

Many real-world polygonal surfaces contain topological singularies that represent a challenge for processes such as simplification, compression, smoothing, etc. We present an algorithm for removing such singularities, thus converting non-manifold sets of polygons to manifold polygonal surfaces (orientable if necessary).

In this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images. To this end, we use a space of 3 by 3 high-contrast patches M. We develop a theoretical model for the high-density 2-dimensional submanifold of M showing that it has the topology of the Klein bottle. Using our topological software package PLEX we experimentally verify our theoretical conclusions. We use polynomial representation to give coordinatization to various subspaces of M. We find the bestfitting embedding of the Klein bottle into the ambient space of M. Our results are currently being used in developing a compression algorithm based on a Klein bottle dictionary.

Statistical analysis on landmark-based shape spaces has diverse applications in morphometrics, medical diagnostics, machine vision, robotics and other areas. These shape spaces are non-Euclidean quotient manifolds, often the quotient of the unit sphere under a group of transformations. To conduct nonparametric inferences, one may define notions of center and spread of a probability distribution on an arbitrary manifold and work with their estimates. There has been a significant amount of work done in this direction. However, it is useful to consider full likelihood-based methods, which allow nonparametric estimation of the probability density. This article proposes a class of mixture models constructed using suitable kernels on a general compact non-Euclidean manifold and then on the planar shape space in particular. Following a Bayesian approach with a nonparametric prior on the mixing distribution, conditions are obtained under which the Kullback-Leibler property holds, implying large support and weak posterior consistency. Gibbs sampling methods are developed for posterior computation, and the methods are applied to problems in density estimation on shape space and classification with shape-based predictors.

We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine a straightforward formula for the computation of the degree of the associated tangent vector field that does not require any explicit knowledge of the manifold. We use this formula to study the set of harmonic solutions to periodic perturbations of our equations. Two different classes of applications are provided.

In recent years differential systems whose solutions evolve on manifolds of matrices have acquired a certain relevance in numerical analysis. A classical example of such a differential system is the well-known Toda flow. This paper is a partial survey of numerical methods recently proposed for approximating the solutions of ordinary differential systems evolving on matrix manifolds. In particular, some results recently obtained by the author jointly with his co-workers will be presented. We will discuss numerical techniques for isospectral and isodynamical flows where the eigenvalues of the solutions are preserved during the evolution and numerical methods for ODEs on the orthogonal group or evolving on a more general quadratic group, like the symplectic or Lorentz group. We mention some results for systems evolving on the Stiefel manifold and also review results for the numerical solution of ODEs evolving on the general linear group of matrices.

Pipelines with multiple outlets are used extensively for irrigation under various types of surface, sprinkler and trickle irrigation systems. In this research, a lateral pipeline with two diameters (tapered), laid on horizontal, uphill and downhill slopes that usually faced in design were investigated using back stepwise method. Effect of different parameters such as variation of friction factor along the pipeline, variation of discharge with head and velocity head on the total head loss and pressure along the pipeline was studied. Location of average operating pressure head along the pipeline for different slopes and tapering length ratios, which is important for computing the inlet and end pressures head, was estimated. Furthermore, empirical equations for estimating the ratio of head loss at location of average pressure head to the total head loss were developed. Then the pressure head at the upstream end or downstream end can be computed using friction correction factor method. The results showed that neglecting the effect of variation of friction factor underestimate the value of head loss along the lateral and neglecting the effect of variation in discharge overestimate the value of head loss, therefore, neglecting effects of the two parameters at the same time leads to a value of head loss very near to the value of exact solution. The velocity head have small effect on the head loss and can be neglected at sprinkler laterals but it has effect on the total head. Finally, example is given to show how to use the proposed empirical equations at inlet and end pressure computation.

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function. A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

Recently, some research efforts have shown that face images possibly reside on a nonlinear sub-manifold. Though Laplacianfaces method considered the manifold structures of the face images, it has limits to solve face recognition problem. This paper proposes a new feature extraction method, Two Dimensional Laplacian EigenMap (2DLEM), which especially considers the manifold structures of the face images, and extracts the proper features from face image matrix directly by using a linear transformation. As opposed to Laplacianfaces, 2DLEM extracts features directly from 2D images without a vectorization preprocessing. To test 2DLEM and evaluate its performance, a series of experiments are performed on the ORL database and the Yale database. Moreover, several experiments are performed to compare the performance of three 2D methods. The experiments show that 2DLEM achieves the best performance.

In this article, we investigate the response of a thin superconducting shell to an arbitrary external magnetic field. We identify the intensity of the applied field that forces the emergence of vortices in minimizers, the so-called first critical field H c1 in Ginzburg-Landau theory, for closed simply connected manifolds and arbitrary fields. In the case of a simply connected surface of revolution and vertical and constant field, we further determine the exact number of vortices in the sample as the intensity of the applied field is raised just above H c1. Finally, we derive via Γ-convergence similar statements for three-dimensional domains of small thickness, where in this setting point vortices are replaced by vortex lines.

We investigate how a Lorentzian manifold can be reconstructed from a more fundamental structure, in the context ofa proposal for a quantum theory ofgravity. We first show that some collections of countable sets of points of the manifold, together with their causal relations, contain all the information ofthe geometrical structure ofthe manifold. We then define a notion ofcloseness between Lorentzian metrics, and formulate a conjecture regarding how a Lorentzian manifold can be approximated by a locally finite subset of points.

We generalize the classical Bochner-Yano theorems of Riemannian geometry to pseudo-Riemannian manifolds in order to obtain information on higher dimensional space-times with symmetries. The results are used to shed some light on questions of consistency and the zero-mode ansatz in Kaluza-Klein theories. Some applications to Einstein spaces are given.

Let J be a semisimple Lie group with all simple factors of real rank at least two. Let Γ < J be a lattice. We prove a very general local rigidity result about actions of J or Γ. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of Γ by toral automorphisms is locally rigid. More generally, given a manifold M on which Γ acts isometrically and a torus T n on which it acts by automorphisms, we show that the diagonal action on T n ×M is locally rigid.

A unitarizing measure is a probability measure such that the associated L 2 space contains a closed subspace of holomorphic functionals on which the Virasoro algebra acts unitarily. It has been shown that the unitarizing property is equivalent to an a priori given formula of integration by parts, which has been computed explicitly. We show in this Note that unitarizing measures must be supported by the quotient of the homeomorphism group of the circle by the subgroup of Möbius transformations. To cite this article: H. Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621-626.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS THEOREM 1.1. -(i) The cocycle ω is invariant under the adjoint action of γ ; (ii) The Hilbert transform J considered in [2, (1.2.5)] is invariant under the adjoint action of γ ; (iii) denoting M 1 := Diff S 1 / , then by passing to the quotient, J and ω define on M 1 the structure of a homogeneous Kähler manifold. (iv) The function K defined in [2, 4.1] satisfies K(gg 0 ) = K(g), ∀g 0 ∈ ,

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function. A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

For an almost contact metric manifold N , we find conditions for which either the total space of an S 1-bundle over N or the Riemannian cone over N admits a strong Kähler with torsion (SKT) structure. In this way we construct new 6-dimensional SKT manifolds. Moreover, we study the geometric structure induced on a hypersurface of an SKT manifold, and use such structures to construct new SKT manifolds via appropriate evolution equations. Hyper-Kähler with torsion (HKT) structures on the total space of an S 1-bundle over manifolds with three almost contact structures are also studied.

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration M → S 1 is the modular elliptic curve E = X 0 (49), which admits multiplication by the ring of integers of Q[ √ −7]. We first base change the holomorphic differential on E to a cusp form on GL(2) over K = Q[ √ −3], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of O * D of level 7. To analyze the topological properties of M, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a non-compact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument. CONTENTS 1. Introduction 2. Fibered faces of the Thurston norm ball 3. The geometric theorem 4. The base manifold M 5. The automorphic structure of the cohomology of M 6. The Thurston norm and fibered faces of M 7. Proof of the main result 8. The Whitehead link complement 9. Work of Long and Reid References

We discuss the problem of global invertibility of nonlinear maps defined on the finite dimensional Euclidean space via differential tests. We provide a generalization of the Fujisawa-Kuh global inversion theorem and introduce a generalized ratio condition which detects when the pre-image of a certain class of linear manifolds is non-empty and connected. In particular, we provide conditions that also detect global injectivity.

Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary. We also compare it with the refined analytic torsion on manifolds with boundary. As a byproduct of the gluing formula for refined analytic torsion and the comparison theorem for the Cappell-Miller analytic torsion and the refined analytic torsion, we establish the gluing formula for the Cappell-Miller analytic torsion in the case that the Hermitian metric is flat.

Resume. Nous donnons une methode explicite pour construire des varietes symplectiques munies d'une pseudo-metrique compatible. Biles apparaissent comme espace total d'une c1asse speciale de fibrations etudiee en [1]. Parmi elles , quelques exemples de varietes pseudo-kahleriennes compactes acourbure non nulle sont exhibes. © Academie des ScienceslElsevier, Paris Examples ofpseudo-metries compatible with a symplecticfon»; particular case of pseudo-Kahlerian manifolds Abstract. Wegive an explicit method to construct symplectic manifolds endowed with a compatible pseudo-metric. They appear as the total space of a special kind offibrations that were studied in [I]. Among them, some examples of compact pseudo-Kiihlerian manifolds with non-vanishing curvaturen are showed. © Academic des Sciences/Elsevier, Paris Toutes les structures considerees sont de cIasse Coo. Soit M une variete differentiable munie d'une forme symplectique w et d'une pseudo-metrique 9 et soit .J Ie champ d'endomorphismes defini par la relation:

A class of Virasoro algebras with continuously varying central charge are obtained by the addition of terms linear in Kač-Moody generators. The change of the spectrum of Virasoro primary fields and that of characters are investigated. A lattice model is used to show how finite size effects are altered. The underlying two-dimensional field theory can be constructed on a curved manifold.

For the flag manifold X = G/B of a complex semi-simple Lie group G, we make connections between the Kostant harmonic forms on G/B and the geometry of the Bruhat Poisson structure. We show that on each Schubert cell, the corresponding Kostant harmonic form can be described using only data coming from the Bruhat Poisson structure. We do this by using an explicit set of coordinates on the Schubert cell.

New computational procedures are proposed for experimentally evaluating air-fuel ratio and mass fractions of exhaust emissions as well as EGR rate, oxygen mass fraction and thermal capacity of the inducted charge in IC engines running with diesel oil, gasoline or any alternative liquid or gaseous fuel, such as LPG or CNG. Starting from the chemical reaction of fuel with air, from gaseous and smoke level measurements in the raw gases, the procedures calculate the volume fractions of oxygen in the combustion air and of compounds in the exhaust gases, including those that are not usually measured, such as water, nitrogen and hydrogen. The methods also take the effects of various fuel and combustion air compositions into account, as well as to the presence of water vapor, CO 2 , Ar and He in the combustion air. The algorithms are applied to four different automotive engines under wide ranges of steady-state operating conditions: three turbocharged diesel engines featuring high-pressure cooled EGR systems, and an SI naturally aspirated bi-fuel engine running on either gasoline or CNG. The computed air-fuel ratios are compared to those obtained from directly measured air and fuel mass-flow rates as well as from more conventional UEGO sensor data. The mass emissions are worked out in terms of both brake specific mass emissions and emission indexes of each pollutant species, and the results are compared to those obtained by applying SAE and ISO recommended practices. The computed oxygen mass fraction of the inducted charge was then compared to that derived from direct measurement of O 2 concentration in inlet manifold. Finally, the sensitivity of results to the main engine working parameters, the influence of environmental conditions (in particular the effect of air humidity on NO x formation) and the experimental uncertainties are determined.

We propose a novel regularizer when training an auto-encoder for unsupervised feature extraction. We explicitly encourage the latent representation to contract the input space by regularizing the norm of the Jacobian (analytically) and the Hessian (stochastically) of the encoder&#x27;s output with respect to its input, at the training points. While the penalty on the Jacobian&#x27;s norm ensures robustness to tiny corruption of samples in the input space, constraining the norm of the Hessian extends this robustness when moving further away ...