regression on manifolds estimation of the exterior derivative collinearity and near collinearity of predictors cause difficulties when doing regression in these cases variable selection becomes untenable because of mathematical issues concerning the existence and numerical stability of the regression coefficients and interpretation of the coefficients is ambiguous because gradients are not defined using a differential geometric interpretation in which the regression coefficients are interpreted as estimates of the exterior derivative of a function we develop a new method to do regression in the presence of collinearities our regularization scheme can improve estimation error and it can be easily modified to include lasso type regularization these estimators also have simple extensions to the large p small n context

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

manifold topological multi resolution analysis method in this paper two significant weaknesses of locally linear embedding lle applied to computer vision are addressed intrinsic dimension and eigenvector meanings topological embedding and multi resolution nonlinearity capture are introduced based on mathematical analysis of topological manifolds and lle the manifold topological analysis mta method is described and is based on topological embedding mta is a more robust method to determine the intrinsic dimension of a manifold with typical topology which is important for tracking and perception understanding the manifold multi resolution analysis mma method is based on multi resolution nonlinearity capture mma defines lle eigenvectors as features for pattern recognition and dimension reduction both mta and mma are proved mathematically and several examples are provided applications in 3d object recognition and 3d object viewpoint space partitioning are also described

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

how to count zeroes of polynomials on quadrature domains using the bezout matrix abstract classically the bezout matrix or simply bezoutian of two polynomials is used to locate the roots of the polynomial and in particular test for stability in this paper we develop the theory of bezoutians on real riemann surfaces of dividing type the main result connects the signature of the bezoutian of two real meromorphic functions to the topological data of their quotient which can be seen as the generalization of the classical cauchy index as an application we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the bezoutian we provide examples of our method in the case of simply connected quadrature domains

spectrum separation and inertia for operator polynomials abstract problems on spectrum separation and inertia of operator polynomials are solved using the concept of bezoutian of operator polynomials the main results extend those previously known for matrix polynomials applications of the main results include differential equations with constant coefficients and some problems concerning orthogonal operator polynomials

manifold learning the price of normalization we analyze the performance of a class of manifold learning algorithms that find their output by minimizing a quadratic form under some normalization constraints this class consists of locally linear embedding lle laplacian eigenmap local tangent space alignment ltsa hessian eigenmaps hlle and diffusion maps we present and prove conditions on the manifold that are necessary for the success of the algorithms both the finite sample case and the limit case are analyzed we show that there are simple manifolds in which the necessary conditions are violated and hence the algorithms cannot recover the underlying manifolds finally we present numerical results that demonstrate our claims

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

quantitative analysis of nonlinear embedding a lot of nonlinear embedding techniques have been developed to recover the intrinsic low dimensional manifolds embedded in the high dimensional space however the quantitative evaluation criteria are less studied in literature the embedding quality is usually evaluated by visualization which is subjective and qualitative the few existing evaluation methods to estimate the embedding quality neighboring preservation rate for example are not widely applicable in this paper we propose several novel criteria for quantitative evaluation by considering the global smoothness and co directional consistence of the nonlinear embedding algorithms the proposed criteria are geometrically intuitive simple and easy to implement with a low computational cost experiments show that our criteria capture some new geometrical properties of the nonlinear embedding algorithms and can be used as a guidance to deal with the embedding of the out of samples

parallel vector field embedding we propose a novel local isometry based dimensionality reduction method from the perspective of vector fields which is called parallel vector field embedding pfe we first give a discussion on local isometry and global isometry to show the intrinsic connection between parallel vector fields and isometry the problem of finding an isometry turns out to be equivalent to finding orthonormal parallel vector fields on the data manifold therefore we first find orthonormal parallel vector fields by solving a variational problem on the manifold then each embedding function can be obtained by requiring its gradient field to be as close to the corresponding parallel vector field as possible theoretical results show that our method can precisely recover the manifold if it is isometric to a connected open subset of euclidean space both synthetic and real data examples demonstrate the effectiveness of our method even if there is heavy noise and high curvature

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

boundary layers for the subcritical modes of the 3d primitive equations in a cube abstract in this article we study the boundary layers for the subcritical modes of the viscous linearized primitive equations lpes in a cube at small viscosity the boundary layers include the parabolic boundary layers ordinary boundary layers and their interaction corner layers the boundary layer correctors are determined by a phenomenological study reminiscent of the prandtl corrector approach and then a rigorous convergence result is proved which a posteriori justifies the phenomenological study

bezout and hankel matrices associated with row reduced matrix polynomials barnett type formulas abstract based on the approach introduced by b d o anderson and e i jury in 1976 the definition of finite hankel and bezout matrices corresponding to matrix polynomials is extended to the case where the denominator of the corresponding rational matrix function is not necessarily monic but is row reduced the matrices introduced keep most of the well known properties that hold in the monic case in particular we derive extensions of formulas giving a connection with polynomials in the companion matrix usually called barnett formulas of the inversion theorem and of formulas concerning alternating products of hankel and bezout matrices

schroedinger eigenmaps for the analysis of biomedical data we introduce schroedinger eigenmaps se a new semi supervised manifold learning and recovery technique this method is based on an implementation of graph schroedinger operators with appropriately constructed barrier potentials as carriers of labeled information we use our approach for the analysis of standard biomedical datasets and new multispectral retinal images

common eigenvalues divisors and multiples of matrix polynomials a review abstract the main objective of this paper is to review some results obtained in the last decade using methods of spectral theory and concerning common multiples common divisors and common eigenvalues for matrix polynomials this includes discussion of resultant and bezoutian matrices for matrix polynomials and descriptions of their kernels in terms of spectral data some extensions to operator polynomials whose coefficients are bounded linear operators on a hilbert space are also included

generalized bezoutian and matrix equations abstract a natural generalization of the classical bezout matrix of two polynomials is introduced for a family of several matrix polynomials the main aim of the paper is to show that this generalized bezoutian serves as an adequate connecting link between the class of equations in matrix polynomials m y z l r and the class of linear matrix equations ax xb c each equation in one of these classes is coupled with a certain equation in the other class so that for each couple the generalized bezoutian corresponding to a solution y z of the equation in matrix polynomials is a solution of the matrix equation and conversely any solution x of the matrix equation is a generalized bezoutian corresponding to a certain solution of the equation in matrix polynomials in particular either both equations are solvable or both have no solutions explicit formulas connecting the solutions of the two equations are given also various representation formulas for the generalized bezoutian are derived and its relation to the resultant matrix and the greatest common divisor of several matrix polynomials is discussed

bezoutian and schur cohn problem for operator polynomials abstract a generalized bezout operator bezoutian for a pair of operator polynomials is introduced and its kernel is described in terms of common spectral data of the underlying polynomials the location of the spectrum of an operator polynomial with compact spectrum with respect to the unit circle infinite dimensional version of the schur cohn problem is expressed via the inertia of a suitable bezoutian an application to the geometric dichotomy problem for difference equations with operator coefficients is given as well

matrix representations of bezoutians abstract an approach is presented to get matrix representations for classical and more general r bezoutians with entries from a noncommutative ring formulas due to ammar and gader barnett gohberg and semencul gohberg and heinig kravitsky and russakovskii lerer and tismentsky ptak and wimmer as well as some generalizations and modifications are obtained in a unified way

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

the linearized 2d inviscid shallow water equations in a rectangle boundary conditions and well posedness we consider the linearized 2d inviscid shallow water equations in a rectangle a set of boundary conditions is proposed which make these equations well posed several different cases occur depending on the relative values of the reference velocities u 0 v 0 and reference height phi 0 sub or super critical flow at each part of the boundary

principal manifolds and nonlinear dimensionality reduction via tangent space alignment we present a new algorithm for manifold learning and nonlinear dimensionality reduction based on a set of unorganized data points sampled with noise from a parameterized manifold the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold we also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases we illustrate our algorithm using curves and surfaces both in two dimensional three dimensional 2d 3d euclidean spaces and in higher dimensional euclidean spaces we also address several theoretical and algorithmic issues for further research and improvements

on the convergence of maximum variance unfolding maximum variance unfolding is one of the main methods for nonlinear dimensionality reduction we study its large sample limit providing specific rates of convergence under standard assumptions we find that it is consistent when the underlying submanifold is isometric to a convex subset and we provide some simple examples where it fails to be consistent

bezout equation vandermonde operators and common multiples for operator polynomials soit h un espace de hilbert complexe on considere des polynomes d operateurs de type p j 0 m i p j ou p j sont des operateurs agissant dans h et est un parametre complexe on considere des p pour lequel le spectre p est un ensemble compact

the theory of v bezoutians and its applications abstract some propositions of the theory of v bezoutians are presented as an application the following problem is considered to construct an appropriate pontryagin space in which a given matrix boundary value problem containing the eigenvalue parameter in the boundary conditions is self adjoint the solution of this problem is given in three special cases with the help of v bezoutians of matrix polynomials entering the boundary conditions

boundary layers for the 3d primitive equations in a cube the supercritical modes abstract in this article we study the boundary layers for the viscous linearized primitive equations lpes when the viscosity is small the lpes are considered here in a cube besides the usual boundary layers that we analyze here too corner layers due to the interaction between the different boundary layers are also studied