on the comparisons of decorrelation approaches for non gaussian neutral vector variables as a typical non gaussian vector variable a neutral vector variable contains nonnegative elements only and its l norm equals one in addition its neutral properties make it significantly different from the commonly studied vector variables e g the gaussian vector variables due to the aforementioned properties the conventionally applied linear transformation approaches e g principal component analysis pca and independent component analysis ica are not suitable for neutral vector variables as pca cannot transform a neutral vector variable which is highly negatively correlated into a set of mutually independent scalar variables and ica cannot preserve the bounded property after transformation in recent work we proposed an efficient nonlinear transformation approach i e the parallel nonlinear transformation pnt for decorrelating neutral vector variables in this article we extensively compare pnt with pca and ica through both theoretical analysis and experimental evaluations the results of our investigations demonstrate the superiority of pnt for decorrelating the neutral vector variables

finite size scaling of the three state potts model on a simple cubic lattice finite size scaling is studied for the three state potts model on a simple cubic lattice we show that the specific heat and the magnetic susceptibility scale accurately as the volume the correlation length exhibits behaviors expected for a genuine first order transition the one extracted from the unsubtracted correlation function shows a characteristic finite size behavior whereas the physical correlation length that characterizes the first excited state stays at a finite value and is discontinuous at the transition point

a rigorous theory of finite size scaling at first order phase transitions a large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered the dependence of the finite volume magnetizationm per h l in cubes of sizel d under periodic boundary conditions on the external fieldh is analyzed for the case where two phases coexist at the infinite volume transition pointh t we prove that independent of the details of the model the finite volume magnetization per lattice site behaves likem per h t m tanh ml d h ht withm per h denoting the infinite volume magnetization and m 1 2 m per h t 0 m per h t 0 introducing the finite size transition pointh m l as the point where the finite volume susceptibility attains the maximum we show that in the case of asymmetric field driven transitions its shift ish t h m l o l 2d in contrast to claims in the literature starting from the obvious observation that the number of stable phases has a local maximum at the transition point we propose a new way of determining the pointh t from finite size data with a shift that is exponentially small inl finally the finite size effects are discussed also in the case where more than two phases coexist

finite size scaling for potts models recently borgs and koteck developed a rigorous theory of finite size effects near first order phase transitions here we apply this theory to the ferromagneticq state potts model which forq large andd 2 undergoes a first order phase transition as the inverse temperature is varied we prove a formula for the internal energy in a periodic cube of side lengthl which describes the rounding of the infinite volume jump e in terms of a hyperbolic tangent and show that the position of the maximum of the specific heat is shifted by m l inq e l d o l 2d with respect to the infinite volume transition point t we also propose an alternative definition of the finite volume transition temperature t l which might be useful for numerical calculations because it differs only by exponentially small corrections from t

high dimensional semiparametric scale invariant principal component analysis we propose a new high dimensional semiparametric principal component analysis pca method named copula component analysis coca the semiparametric model assumes that after unspecified marginally monotone transformations the distributions are multivariate gaussian coca improves upon pca and sparse pca in three aspects i it is robust to modeling assumptions ii it is robust to outliers and data contamination iii it is scale invariant and yields more interpretable results we prove that the coca estimators obtain fast estimation rates and are feature selection consistent when the dimension is nearly exponentially large relative to the sample size careful experiments confirm that coca outperforms sparse pca on both synthetic and real world data sets

three circle theorem and dimension estimate for holomorphic functions on kahler manifolds the classical hadamard three circle theorem is generalized to complete kahler manifolds more precisely we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem two sharp monotonicity formulae are derived as corollaries among applications we obtain sharp dimension estimates with rigidity of holomorphic functions with polynomial growth when the holomorphic sectional curvature is nonnegative when the bisectional curvature is nonnegative the sharp dimension estimate was due to ni

polynomial growth harmonic functions on groups of polynomial volume growth we consider harmonic functions of polynomial growth of some order d on cayley graphs of discrete groups of polynomial volume growth of order d w r t the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions more precisely the dimension of this space of harmonic functions is at most of order d d 1 as in the already known riemannian case this estimate is polynomial in the growth degree more generally our techniques also apply to graphs roughly isometric to cayley graphs of groups of polynomial volume growth

measuring and testing dependence by correlation of distances distance correlation is a new measure of dependence between random vectors distance covariance and distance correlation are analogous to product moment covariance and correlation but unlike the classical definition of correlation distance correlation is zero only if the random vectors are independent the empirical distance dependence measures are based on certain euclidean distances between sample elements rather than sample moments yet have a compact representation analogous to the classical covariance and correlation asymptotic properties and applications in testing independence are discussed implementation of the test and monte carlo results are also presented

the zeta functional determinants on manifolds with boundary abstract in this paper we derive some geometric formulas for the quotient of the zeta functional determinants for certain elliptic boundary value problems on riemannian 3 and 4 manifolds with smooth boundary

sharp dimension estimates of holomorphic functions and rigidity let m n be a complete noncompact kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature denote by o d m n the space of holomorphic functions of polynomial growth of degree at most d on m n in this paper we prove that dimcod m n dim c o d c n for all d 0 with equality for some positive integer d if and only if m n is holomorphically isometric to c n we also obtain sharp improved dimension estimates when its volume growth is not maximal or its ricci curvature is positive somewhere

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

constant q curvature metrics in arbitrary dimension working in a given conformal class we prove existence of constant q curvature metrics on compact manifolds of arbitrary dimension under generic assumptions the problem is equivalent to solving a nth order non linear elliptic differential or integral equation with variational structure where n is the dimension of the manifold since the corresponding euler functional is in general unbounded from above and below we use critical point theory jointly with a compactness result for the above equation

bayesian estimation of the von mises fisher mixture model with variational inference this paper addresses the bayesian estimation of the von mises fisher vmf mixture model with variational inference vi the learning task in vi consists of optimization of the variational posterior distribution however the exact solution by vi does not lead to an analytically tractable solution due to the evaluation of intractable moments involving functional forms of the bessel function in their arguments to derive a closed form solution we further lower bound the evidence lower bound where the bound is tight at one point in the parameter distribution while having the value of the bound guaranteed to increase during maximization we derive an analytically tractable approximation to the posterior distribution which has the same functional form as the assigned prior distribution the proposed algorithm requires no iterative numerical calculation in the re estimation procedure and it can potentially determine the model complexity and avoid the over fitting problem associated with conventional approaches based on the expectation maximization moreover we derive an analytically tractable approximation to the predictive density of the bayesian mixture model of vmf distributions the performance of the proposed approach is verified by experiments with both synthetic and real data

slavnov and gaudin korepin formulas for models without u 1 symmetry the xxx chain on the segment we consider the isotropic spin heisenberg chain with the most general integrable boundaries the scalar product between the on shell bethe vector and its off shell dual obtained by means of the modified algebraic bethe ansatz is given by a modified slavnov formula the corresponding gaudin korepin formula i e the square of the norm is also obtained

polynomial growth harmonic functions on connected sums of complete riemannian manifolds let m be a connected sum of complete riemannian manifolds satisfying the volume doubling condition and the poincare inequality we prove that the space of polynomial growth harmonic functions on m is finite dimensional whenever m has finitely many ends and satisfies the finite covering condition on each end this result directly generalizes that of tam and it also partially generalizes that of colding and minicozzi ii

harmonic functions and volume growth of fingerless ends let m be a noncompact complete riemannian manifold with finitely many ends in this paper we study the existence of green s function for the p laplace equation on m in terms of a certain volume growth we also show that the dimension of the linear space of polynomial growth harmonic functions is finite if a volume comparison condition and a mean value inequality for nonnegative subharmonic functions hold in sufficiently large parts of each end

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

compactness of solutions to some geometric fourth order equations we prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds the proof is based on a refined bubbling analysis for which the main estimates are given in integral form our result is used in a subsequent paper to find critical points via minimax arguments of some geometric functional which give rise to conformal metrics of constant q curvature as a byproduct of our method we also obtain compactness of such metrics

iterative gaussianization from ica to random rotations most signal processing problems involve the challenging task of multidimensional probability density function pdf estimation in this paper we propose a solution to this problem by using a family of rotation based iterative gaussianization rbig transforms the general framework consists of the sequential application of a univariate marginal gaussianization transform followed by an orthonormal transform the proposed procedure looks for differentiable transforms to a known pdf so that the unknown pdf can be estimated at any point of the original domain in particular we aim at a zero mean unit covariance gaussian for convenience rbig is formally similar to classical iterative projection pursuit algorithms however we show that unlike in pp methods the particular class of rotations used has no special qualitative relevance in this context since looking for interestingness is not a critical issue for pdf estimation the key difference is that our approach focuses on the univariate part marginal gaussianization of the problem rather than on the multivariate part rotation this difference implies that one may select the most convenient rotation suited to each practical application the differentiability invertibility and convergence of rbig are theoretically and experimentally analyzed relation to other methods such as radial gaussianization one class support vector domain description and deep neural networks is also pointed out the practical performance of rbig is successfully illustrated in a number of multidimensional problems such as image synthesis classification denoising and multi information estimation

three circles theorems for harmonic functions we proved two three circles theorems for harmonic functions on manifolds in integral sense as one application on manifold with nonnegative ricci curvature whose tangent cone at infinity is the unique metric cone with unique conic measure we showed the existence of nonconstant harmonic functions with polynomial growth this existence result recovered and generalized the former result of ding and led to a complete answer of ni s conjecture furthermore in similar context combining the techniques of estimating the frequency of harmonic functions with polynomial growth which were developed by colding and minicozzi we confirmed their conjecture about the uniform bound of frequency

isoperimetry for semilinear torsion problems in riemannian two manifolds abstract given m 2 g a smooth complete riemannian 2 manifold this note discusses three types of isoperimetric properties arising from u z o g o g z u d a g the non negative solution to the semilinear 0 1 torsion problem g u u in o subject to u o 0 for a relatively compact domain o m 2 with smooth boundary o

compactification of a class of conformally flat 4 manifold in this paper we generalize huber s result on complete surfaces of finite total curvature for complete locally conformally flat 4 manifolds of positive scalar curvature with q curvature integrable where q is a variant of the chern gauss bonnet integrand we first derive the cohn vossen inequality we then establish finiteness of the topology this allows us to provide conformal compactification of such manifolds

decorrelation of neutral vector variables theory and applications in this paper we propose novel strategies for neutral vector variable decorrelation two fundamental invertible transformations namely serial nonlinear transformation and parallel nonlinear transformation are proposed to carry out the decorrelation for a neutral vector variable which is not multivariate gaussian distributed the conventional principal component analysis cannot yield mutually independent scalar variables with the two proposed transformations a highly negatively correlated neutral vector can be transformed to a set of mutually independent scalar variables with the same degrees of freedom we also evaluate the decorrelation performances for the vectors generated from a single dirichlet distribution and a mixture of dirichlet distributions the mutual independence is verified with the distance correlation measurement the advantages of the proposed decorrelation strategies are intensively studied and demonstrated with synthesized data and practical application evaluations

on the chern gauss bonnet integral for conformal metrics on r4 where m is the euler number ofm later huber hu extended this inequality to metrics with much weaker regularity more importantly he proved that such a surface m is conformally equivalent to a closed surface with finitely many punctures the deficit in formula 1 1 has an interpretation as an isoperimetric constant on a complete and open surface with gaussian curvature absolutely integrable one may represent each end conformally as r2 k for some compact set k we consider the isoperimetric ratio

harmonic radius and concentration of energy hyperbolic radius and liouville s equations d u e u and d u u n 2 n 2 the conformal radius of a simply connected planar domain omega is defined as r x 1 f x where f denotes a riemann map from omega to the unit disk mapping the given point x to the origin beyond geometric function theory the conformal radius has various applications in partial differential equations therefore it is desirable to extend it to multiply connected and higher dimensional domains we discuss two natural extensions 1 the harmonic radius is defined in terms of the green s function its maximum points the harmonic centers play an important role in variational problems involving concentration of energy they are of the following form sup left int limits omega f u u in h 0 1 omega int omega nabla u 2 leqslant varepsilon 2 right as varepsilon to 0 the solutions concentrate at a harmonic center our examples include bernoulli s free boundary problem the plasma problem and the elastic membrane with a movable obstacle an efficient method

noise level estimation for model selection in kernel pca denoising one of the main challenges in unsupervised learning is to find suitable values for the model parameters in kernel principal component analysis kpca for example these are the number of components the kernel and its parameters this paper presents a model selection criterion based on distance distributions mdds this criterion can be used to find the number of components and the sigma 2 parameter of radial basis function kernels by means of spectral comparison between information and noise the noise content is estimated from the statistical moments of the distribution of distances in the original dataset this allows for a type of randomization of the dataset without actually having to permute the data points or generate artificial datasets after comparing the eigenvalues computed from the estimated noise with the ones from the input dataset information is retained and maximized by a set of model parameters in addition to the model selection criterion this paper proposes a modification to the fixed size method and uses the incomplete cholesky factorization both of which are used to solve kpca in large scale applications these two approaches together with the model selection mdd were tested in toy examples and real life applications and it is shown that they outperform other known algorithms

a first order phase transition in the three dimensional four state antiferromagnetic potts model studies the four state antiferromagnetic potts model with next nearest neighbour ferromagnetic interactions on the simple cubic lattice by the monte carlo method the ratio j2 j1 of the strength of the next nearest neighbour interactions to that of the nearest neighbour interactions is to be 10 the author analyses the finite size effects of the energy the specific heat the fourth order cumulant the magnetization the susceptibility and the effective transition temperatures with the help of the finite size scaling theory the author concludes that the transition of the model is assigned to be of first order

conformal modulus and its reduction via mean curvatures abstract this note shows that the conformal modulus of a condenser and the reduced conformal modulus of a compact subset in r n 2 are naturally but optimally tied with the corresponding graphic mass and the semi diameter through mean curvatures

bilayer ldpc convolutional codes for decode and forward relaying in this paper we present bilayer ldpc convolutional codes for half duplex relay channels two types of codes bilayer expurgated ldpc convolutional codes and bilayer lengthened ldpc convolutional codes are proposed for decode and forward df relaying in the case of the binary erasure relay channel we prove analytically that both code constructions achieve the capacities of the source relay link and the source destination link simultaneously provided that the channel conditions are known when designing the codes meanwhile both codes enable the highest transmission rate possible with df relaying for a wide range of channel parameters in addition the regular degree distributions can easily be computed from the channel parameters which significantly simplifies the code optimization the code construction and performance analysis are extended to the general binary memoryless symmetric channel where a capacity achieving performance is conjectured numerical results are provided for both types of codes with finite node degrees over binary erasure channels and binary input additive white gaussian noise channels which verify the aforementioned theoretical analysis

quantitative rigidity results for conformal immersions in this paper we prove several quantitative rigidity results for conformal immersions of surfaces in with bounded total curvature we show that branched conformal immersions which are close in energy to either a round sphere a conformal clifford torus an inverted catenoid an inverted enneper s minimal surface or an inverted chen s minimal graph must be close to these surfaces in the w norm moreover we apply these results to prove a corresponding rigidity result for complete connected and non compact surfaces

quantized kernel recursive least squares algorithm in a recent paper we developed a novel quantized kernel least mean square algorithm in which the input space is quantized partitioned into smaller regions and the network size is upper bounded by the quantization codebook size number of the regions in this paper we propose the quantized kernel least squares regression and derive the optimal solution by incorporating a simple online vector quantization method we derive a recursive algorithm to update the solution namely the quantized kernel recursive least squares algorithm the good performance of the new algorithm is demonstrated by monte carlo simulations

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

geometrical logarithmic capacitance abstract this paper is devoted to a novel and nontrivial exploration of eight aspects of the geometrical logarithmic capacitance a very key notion in mathematical physics quasiconformal geometry and variational calculus through 1 identifying with the reduced conformal module 2 evaluating the minimal log potential energy 3 relating to both the volume radius and the surface radius 4 linking with the n harmonic radius and the log capacity of the kevin image of a compact surface 5 finding the minkowski inequality and the general variational formula for the log capacity 6 pinching the log isocapacitary inequality from left and solving the left prescribed problem for the normalized log capacitary curvature measure 7 pinching the log isocapacitary inequality from right and handling the right prescribed problem for the normalized log capacitary curvature measure 8 handling an overdetermination of the n equilibrium potential of a given convex body via the log capacitary concavity

an efficient method to derive explicit klt kernel for first order autoregressive discrete process signal dependent karhunen loeve transform klt also called factor analysis or principal component analysis pca has been of great interest in applied mathematics and various engineering disciplines due to optimal performance however implementation of klt has always been the main concern therefore fixed transforms like discrete fourier dft and discrete cosine dct with efficient algorithms have been successfully used as good approximations to klt for popular applications spanning from source coding to digital communications in this paper we propose a simple method to derive explicit klt kernel or to perform pca in closed form for first order autoregressive ar 1 discrete process it is a widely used approximation to many real world signals the merit of the proposed technique is shown the novel method introduced in this paper is expected to make real time and data intensive applications of klt and pca more feasible

a least squares framework for component analysis over the last century component analysis ca methods such as principal component analysis pca linear discriminant analysis lda canonical correlation analysis cca locality preserving projections lpp and spectral clustering sc have been extensively used as a feature extraction step for modeling classification visualization and clustering ca techniques are appealing because many can be formulated as eigen problems offering great potential for learning linear and nonlinear representations of data in closed form however the eigen formulation often conceals important analytic and computational drawbacks of ca techniques such as solving generalized eigen problems with rank deficient matrices e g small sample size problem lacking intuitive interpretation of normalization factors and understanding commonalities and differences between ca methods this paper proposes a unified least squares framework to formulate many ca methods we show how pca lda cca lpp sc and its kernel and regularized extensions correspond to a particular instance of least squares weighted kernel reduced rank regression ls wkrrr the ls wkrrr formulation of ca methods has several benefits 1 provides a clean connection between many ca techniques and an intuitive framework to understand normalization factors 2 yields efficient numerical schemes to solve ca techniques 3 overcomes the small sample size problem 4 provides a framework to easily extend ca methods we derive weighted generalizations of pca lda sc and cca and several new ca techniques

optimal constants in the exceptional case of sobolev inequalities on riemannian manifolds let m g be a riemannian compact n manifold we know that for any e 0 there exists c e 0 such that for any u h n 1 m m e u dv g c e exp n e m u n dv g 1 vol m m udv g n being the smallest constant possible such that the inequality remains true for any u n 1 m we call n the first best constant we prove in this paper that it is possible to choose e 0 and keep c e a finite constant in other words we prove the existence of a second best constant in the exceptional case of sobolev inequalities on compact riemannian manifolds

toward best isoperimetric constants for h1 bmo normal conformal metrics on r n3 the aim of this article is a to establish the existence of the best isoperimetric constants for the h1 bmo normal conformal metrics e2u dx 2 on n3 i e the conformal metrics with the q curvature orientated conditions b to prove that is the optimal upper bound of the best isoperimetric constants for the complete h1 bmo normal conformal metrics with nonnegative scalar curvature c to find the optimal upper bound of the best isoperimetric constants via the quotients of two power integrals of green s functions for the n laplacian operators div u n 2u

binary independent component analysis with or mixtures independent component analysis ica is a computational method for separating a multivariate signal into subcomponents assuming the mutual statistical independence of the non gaussian source signals the classical independent components analysis ica framework usually assumes linear combinations of independent sources over the field of real valued numbers r in this paper we investigate binary ica for or mixtures bica which can find applications in many domains including medical diagnosis multi cluster assignment internet tomography and network resource management we prove that bica is uniquely identifiable under the disjunctive generation model and propose a deterministic iterative algorithm to determine the distribution of the latent random variables and the mixing matrix the inverse problem to infer the values of latent variables is also considered for noisy measurements we conduct an extensive simulation study to verify the effectiveness of the propose algorithm and present examples of real world applications where bica can be applied

the brezis nirenberg problem on s3 abstract in this paper we study existence and nonexistence of solutions to the brezis nirenberg problem for different values of in geodesic spheres on s 3 the picture differs considerably from the one in the euclidean space it is shown that large spheres containing the hemisphere have two different type of radial solutions for negative values of numerical results indicate that for very small the solutions have a maximum near the boundary whereas for larger values of the maximum is at the origin the techniques used are pohozaev type identities concentration compactness lemma and numerical methods

bayesian estimation of beta mixture models with variational inference bayesian estimation of the parameters in beta mixture models bmm is analytically intractable the numerical solutions to simulate the posterior distribution are available but incur high computational cost in this paper we introduce an approximation to the prior posterior distribution of the parameters in the beta distribution and propose an analytically tractable closed form bayesian approach to the parameter estimation the approach is based on the variational inference vi framework following the principles of the vi framework and utilizing the relative convexity bound the extended factorized approximation method is applied to approximate the distribution of the parameters in bmm in a fully bayesian model where all of the parameters of the bmm are considered as variables and assigned proper distributions our approach can asymptotically find the optimal estimate of the parameters posterior distribution also the model complexity can be determined based on the data the closed form solution is proposed so that no iterative numerical calculation is required meanwhile our approach avoids the drawback of overfitting in the conventional expectation maximization algorithm the good performance of this approach is verified by experiments with both synthetic and real data

variational bayesian matrix factorization for bounded support data a novel bayesian matrix factorization method for bounded support data is presented each entry in the observation matrix is assumed to be beta distributed as the beta distribution has two parameters two parameter matrices can be obtained which matrices contain only nonnegative values in order to provide low rank matrix factorization the nonnegative matrix factorization nmf technique is applied furthermore each entry in the factorized matrices i e the basis and excitation matrices is assigned with gamma prior therefore we name this method as beta gamma nmf bg nmf due to the integral expression of the gamma function estimation of the posterior distribution in the bg nmf model can not be presented by an analytically tractable solution with the variational inference framework and the relative convexity property of the log inverse beta function we propose a new lower bound to approximate the objective function with this new lower bound we derive an analytically tractable solution to approximately calculate the posterior distributions each of the approximated posterior distributions is also gamma distributed which retains the conjugacy of the bayesian estimation in addition a sparse bg nmf can be obtained by including a sparseness constraint to the gamma prior evaluations with synthetic data and real life data demonstrate the good performance of the proposed method

gromov hausdorff limits of kahler manifolds and the finite generation conjecture we study the uniformization conjecture of yau by using the gromovhausdorff convergence as a consequence we confirm yau s finite generation conjecture more precisely on a complete noncompact kahler manifold with nonnegative bisectional curvature the ring of polynomial growth holomorphic functions is finitely generated during the course of the proof we prove if m is a complete noncompact kahler manifold with nonnegative bisectional curvature and maximal volume growth then m is biholomorphic to an affine algebraic variety we also confirm a conjecture of ni on the existence of polynomial growth holomorphic functions on kahler manifolds with nonnegative bisectional curvature

a monotonicity formula on complete kahler manifolds with nonnegative bisectional curvature in y yau proposed to study the uniformization of complete kahler manifolds with nonnegative curvature in particular one wishes to determine whether or not a complete kahler manifold m with positive bisectional curvature is biholomorphic to cm see also gw si for this sake it was further asked in y whether or not the ring of the holomorphic functions with polynomial growth which we denote by op m is finitely generated and whether or not the dimension of the spaces of holomorphic functions of polynomial growth is bounded from above by the dimension of the corresponding spaces of polynomials on cm this paper addresses the latter questions we denote by 0 m the space of holomorphic functions of polynomial growth with degree d see section 3 for the precise definition then op m jd 0 od m in this paper we show that

a hybrid feature extraction selection approach for high dimensional non gaussian data clustering this paper presents an unsupervised approach for feature selection and extraction in mixtures of generalized dirichlet gd distributions our method defines a new mixture model that is able to extract independent and non gaussian features without loss of accuracy the proposed model is learned using the expectation maximization algorithm by minimizing the message length of the data set experimental results show the merits of the proposed methodology in the categorization of object images

kernel reconstruction ica for sparse representation independent component analysis with soft reconstruction cost rica has been recently proposed to linearly learn sparse representation with an overcomplete basis and this technique exhibits promising performance even on unwhitened data however linear rica may not be effective for the majority of real world data because nonlinearly separable data structure pervasively exists in original data space meanwhile rica is essentially an unsupervised method and does not employ class information motivated by the success of the kernel trick that maps a nonlinearly separable data structure into a linearly separable case in a high dimensional feature space we propose a kernel rica krica model to nonlinearly capture sparse representation in feature space furthermore we extend the unsupervised krica to a supervised one by introducing a class driven discrimination constraint such that the data samples from the same class are well represented on the basis of the corresponding subset of basis vectors this discrimination constraint minimizes inhomogeneous representation energy and maximizes homogeneous representation energy simultaneously which is essentially equivalent to maximizing between class scatter and minimizing within class scatter at the same time in an implicit manner experimental results demonstrate that the proposed algorithm is more effective than other state of the art methods on several datasets

n capacity n harmonic radius and n harmonic transplantation abstract for the n laplacian n on a regular domain the n green s function exists this allows us to define the n robin s function and the n harmonic radius we show their basic properties and extend the method of the harmonic transplantation to that of the n harmonic transplantation the method is used to estimate the first eigenvalue of the n laplacian n we also give another proof of the isoperimetric inequality for the n capacity and give the isoperimetric inequality for the n harmonic radius

integrability of scalar curvature and normal metric on conformally flat manifolds abstract on a manifold r n e 2 u d x 2 we say u is normal if the q curvature equation that u satisfies n 2 u q g e n u can be written as the integral form u x 1 c n r n log y x y q g y e n u y d y c in this paper we show that the integrability assumption on the negative part of the scalar curvature implies the metric is normal as an application we prove a bi lipschitz equivalence theorem for conformally flat metrics

support vector machine classifier with pinball loss traditionally the hinge loss is used to construct support vector machine svm classifiers the hinge loss is related to the shortest distance between sets and the corresponding classifier is hence sensitive to noise and unstable for re sampling in contrast the pinball loss is related to the quantile distance and the result is less sensitive the pinball loss has been deeply studied and widely applied in regression but it has not been used for classification in this paper we propose a svm classifier with the pinball loss called pin svm and investigate its properties including noise insensitivity robustness and misclassification error besides insensitive zone is applied to the pin svm for a sparse model compared to the svm with the hinge loss the proposed pin svm has the same computational complexity and enjoys noise insensitivity and re sampling stability

large scale behavior of kernels of schrodinger operators in this paper we study the asymptotic behavior of solutions to schrodinger equations on manifolds with nonnegative ricci curvature and euclidean volume growth as a first application we get a sharp asymptotic estimate on the green s function for the laplacian which sharpens results of li and yau next we establish a definite gap in the possible rates of decay of exterior harmonic functions on these manifolds when the manifold also has quadratic curvature decay we get supremum estimates on the c 2 norm of the green s function this has function theoretic implications for these manifolds we then give a generalization to manifolds with nonnegative ricci curvature outside a compact set finally we study schrodinger operators where the potential decays quadratically at infinity in this setting we obtain a geometric version at infinity of a classical theorem of bers

rate compatible ldpc convolutional codes achieving the capacity of the bec in this paper we propose a new family of rate compatible regular low density parity check ldpc convolutional codes the construction is based on graph extension i e the codes of lower rates are generated by successively extending the graph of the base code with the highest rate theoretically the proposed rate compatible family can cover all the rational rates from 0 to 1 in addition the regularity of degree distributions simplifies the code optimization we prove analytically that all the ldpc convolutional codes of different rates in the family are capable of achieving the capacity of the binary erasure channel bec the analysis is extended to the general binary memoryless symmetric channel for which a capacity approaching performance can be achieved analytical thresholds and simulation results for finite check and variable node degrees are provided for both becs and binary input additive white gaussian noise channels the results confirm that the decoding thresholds of the rate compatible codes approach the corresponding shannon limits over both channels

green s function harmonic transplantation and best sobolev constant in spaces of constant curvature ass rrac t we extend the method of harmonic transplantation ft om euclidean domains to spaces of constant positive or negative curvature to this end the structure of the green s function of the corresponding laplacebeltrami operator is investigated by means of isoperimetric inequalities we derive complementary estimates for its distribution function we apply the method of harmonic transplantation to the question of whether the best sobolev constant for the critical exponent is attained i e whether there is an extremal function for the best sobolev constant in spaces of constant curvature a fairly complete answer is given based on a concentration compactness argument and a pohozaev identity the result depends on the curvature

face recognition using an enhanced independent component analysis approach this paper is concerned with an enhanced independent component analysis ica and its application to face recognition typically face representations obtained by ica involve unsupervised learning and high order statistics in this paper we develop an enhancement of the generic ica by augmenting this method by the fisher linear discriminant analysis lda hence its abbreviation fica the fica is systematically developed and presented along with its underlying architecture a comparative analysis explores four distance metrics as well as classification with support vector machines svms we demonstrate that the fica approach leads to the formation of well separated classes in low dimension subspace and is endowed with a great deal of insensitivity to large variation in illumination and facial expression the comprehensive experiments are completed for the facial recognition technology feret face database a comparative analysis demonstrates that fica comes with improved classification rates when compared with some other conventional approaches such as eigenface fisherface and the ica itself

laplacian operators and q curvature on conformally einstein manifolds a new definition of canonical conformal differential operators p k k 1 2 with leading term a kth power of the laplacian is given for conformally einstein manifolds of any signature these act between density bundles and more generally between weighted tractor bundles of any rank by construction these factor into a power of a fundamental laplacian associated to einstein metrics there are natural conformal laplacian operators on density bundles due to graham jenne mason sparling gjms it is shown that on conformally einstein manifolds these agree with the p k operators and hence on einstein manifolds the gjms operators factor into a product of second order laplacian type operators in even dimension n the gjms operators are defined only for 1 k n 2 and so on conformally einstein manifolds the p k give an extension of this family of operators to operators of all even orders for n even and k n 2 the operators p k are each given by a natural formula in terms of an einstein metric but they are not natural conformally invariant operators in the usual sense they are shown to be nevertheless canonical objects on conformally einstein structures there are generalisations of these results to operators between weighted tractor bundles it is shown that on einstein manifolds the branson q curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature as part of development conformally invariant tractor equations equivalent to the conformal killing equation are presented

density dependent quantized least squares support vector machine for large data sets based on the knowledge that input data distribution is important for learning a data density dependent quantization scheme dqs is proposed for sparse input data representation the usefulness of the representation scheme is demonstrated by using it as a data preprocessing unit attached to the well known least squares support vector machine ls svm for application on big data sets essentially the proposed dqs adopts a single shrinkage threshold to obtain a simple quantization scheme which adapts its outputs to input data density with this quantization scheme a large data set is quantized to a small subset where considerable sample size reduction is generally obtained in particular the sample size reduction can save significant computational cost when using the quantized subset for feature approximation via the nystrom method based on the quantized subset the approximated features are incorporated into ls svm to develop a data density dependent quantized ls svm dqls svm where an analytic solution is obtained in the primal solution space the developed dqls svm is evaluated on synthetic and benchmark data with particular emphasis on large data sets extensive experimental results show that the learning machine incorporating dqs attains not only high computational efficiency but also good generalization performance

the emden fowler equation on a spherical cap of sn abstract let s n r n 1 n 3 be the unit sphere and let s s n be a geodesic ball with geodesic radius 0 we study the bifurcation diagram u r 2 of the radial solutions of the emden fowler equation on s s n u u p 0 in s u 0 on s u 0 in s where p 1 among other things we prove the following for each p p s n 2 n 2 there exists 0 such that the problem has a radial solution for and has no radial solution for 0 moreover this solution is unique in the space of radial functions if is close to if p s p p jl then there exists such that the problem has infinitely many radial solutions for where p jl 1 4 n 4 2 n 1 if n 11 if 2 n 10 asymptotic behaviors of the bifurcation diagram as p and p 1 are also studied

algebraic topological methods for the supercritical q curvature problem abstract we study the problem of existence of conformal metrics with prescribed q curvature on closed four dimensional riemannian manifolds this problem has a variational structure and in the case of interest here it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated euler lagrange functional the so called true critical points at infinity of the associated variational problem occur using the characterization of the critical points at infinity of the associated variational problem which is established in 42 combined with some arguments from morse theory some algebraic topological methods and some tools from dynamical system originating from conley s isolated invariant sets and isolated blocks theory we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold stable and unstable manifolds of some of the critical points at infinity of the associated euler lagrange functional

a multigrid approach to the scalar quantization problem a multigrid framework for the one dimensional scalar fixed rate quantization problem is presented the framework is based on the lloyd max iterative process which is a central building block in many quantization algorithms this process iteratively improves a given initial solution and generally converges to a local minimum of the quantization distortion contrary to the classical lloyd max process the convergence of the multigrid algorithm is practically independent of the number of representation levels sought using this approach a local minimum is reached at the cost of just a few lloyd max iterations the complexity of the proposed method is o n operations for the continuous case and 3n o n for the discrete case where n is the number of representation levels sought and n is the size of the discrete probability density function in addition to its independent attributes this work is a precursor to the more important vector quantization problem for which a multiscale framework is also being developed