an improved algorithm for the multidimensional moment constrained maximum entropy problem the maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with the least bias beyond specified constraints in the recent paper we introduced new computational framework for the moment constrained maximum entropy problem in a multidimensional domain and developed a simple numerical algorithm capable of computing maximum entropy problem in a two dimensional domain with moment constraints of order up to 4 here we design an improved numerical algorithm for computing the maximum entropy problem in a two and higher dimensional domain with higher order moment constraints the algorithm features multidimensional orthogonal polynomial basis in the dual space of lagrange multipliers to achieve numerical stability and rapid convergence of newton iterations the new algorithm is found to be capable of solving the maximum entropy problem in the two dimensional domain with moment constraints of order up to 8 in the three dimensional domain with moment constraints of order up to 6 and in the four dimensional domain with moment constraints of order up to 4 corresponding to the total number of moment constraints of 44 83 and 69 respectively the two and higher dimensional maximum entropy test problems in the current work are based upon long term statistics of numerical simulation of the real world geophysical model for wind stress driven oceanic currents such as the gulf stream and the kuroshio

a semi classical trace formula at a non degenerate critical level abstract we study the semi classical trace formula at a critical energy level for a h pseudodifferential operator whose principal symbol has a unique non degenerate critical point for that energy this leads to the study of hamiltonian systems near equilibrium and near the non zero periods of the linearized flow the contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals the new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface

information theoretic subspace clustering this paper addresses the problem of grouping the data points sampled from a union of multiple subspaces in the presence of outliers information theoretic objective functions are proposed to combine structured low rank representations lrrs to capture the global structure of data and information theoretic measures to handle outliers in theoretical part we point out that group sparsity induced measures ell 2 1 norm ell alpha norm and correntropy can be justified from the viewpoint of half quadratic hq optimization which facilitates both convergence study and algorithmic development in particular a general formulation is accordingly proposed to unify hq based group sparsity methods into a common framework in algorithmic part we develop information theoretic subspace clustering methods via correntropy with the help of parzen window estimation correntropy is used to handle either outliers under any distributions or sample specific errors in data pairwise link constraints are further treated as a prior structure of lrrs based on the hq framework iterative algorithms are developed to solve the nonconvex information theoretic loss functions experimental results on three benchmark databases show that our methods can further improve the robustness of lrr subspace clustering and outperform other state of the art subspace clustering methods

generalized correlation function definition properties and application to blind equalization with an abundance of tools based on kernel methods and information theoretic learning a void still exists in incorporating both the time structure and the statistical distribution of the time series in the same functional measure in this paper a new generalized correlation measure is developed that includes the information of both the distribution and that of the time structure of a stochastic process it is shown how this measure can be interpreted from a kernel method as well as from an information theoretic learning points of view demonstrating some relevant properties to underscore the effectiveness of the new measure a simple blind equalization problem is considered using a coded signal

robust subspace clustering with complex noise subspace clustering has important and wide applications in computer vision and pattern recognition it is a challenging task to learn low dimensional subspace structures due to complex noise existing in high dimensional data complex noise has much more complex statistical structures and is neither gaussian nor laplacian noise recent subspace clustering methods usually assume a sparse representation of the errors incurred by noise and correct these errors iteratively however large corruptions incurred by complex noise cannot be well addressed by these methods a novel optimization model for robust subspace clustering is proposed in this paper its objective function mainly includes two parts the first part aims to achieve a sparse representation of each high dimensional data point with other data points the second part aims to maximize the correntropy between a given data point and its low dimensional representation with other points correntropy is a robust measure so that the influence of large corruptions on subspace clustering can be greatly suppressed an extension of pairwise link constraints is also proposed as prior information to deal with complex noise half quadratic minimization is provided as an efficient solution to the proposed robust subspace clustering formulations experimental results on three commonly used data sets show that our method outperforms state of the art subspace clustering methods

hausdorff moment problem and maximum entropy a unified approach hausdorff moment problem in the approach of maximum entropy is reconsidered some aspects as existence condition entropy convergence entropy decreasing stability are studied through a unified approach results known in literature and some new ones are illustrated

an error entropy minimization algorithm for supervised training of nonlinear adaptive systems the paper investigates error entropy minimization in adaptive systems training we prove the equivalence between minimization of error s renyi 1970 entropy of order spl alpha and minimization of a csiszar 1981 distance measure between the densities of desired and system outputs a nonparametric estimator for renyi s entropy is presented and it is shown that the global minimum of this estimator is the same as the actual entropy the performance of the error entropy minimization criterion is compared with mean square error minimization in the short term prediction of a chaotic time series and in nonlinear system identification

robust compressive sensing of multichannel eeg signals in the presence of impulsive noise abstract latest studies have shown that combining cosparsity and low rank property usually results in efficient compressive sensing cs recovery approaches for multichannel electroencephalogram eeg signals however existing methods rarely consider noise or consider only the influence of gaussian noise generated during transmission when the measurement is corrupted by impulsive noise the performance of these cs approaches will degrade in this study a new robust cs approach is proposed to accurately recover multichannel eeg signals from noisy measurements in the presence of impulsive noise we first employ welsch estimator to depress the influence of impulsive noise in cs reconstruction we further develop an efficient iterative scheme based on half quadratic theory and alternating direction method of multipliers to solve the resulting nonconvex optimization problem experimental results show that our approach can obtain better reconstruction results than those of existing state of the art cs methods when measurements are corrupted by impulsive noise

generalized information potential criterion for adaptive system training we have previously proposed the quadratic renyi s error entropy as an alternative cost function for supervised adaptive system training an entropy criterion instructs the minimization of the average information content of the error signal rather than merely trying to minimize its energy in this paper we propose a generalization of the error entropy criterion that enables the use of any order of renyi s entropy and any suitable kernel function in density estimation it is shown that the proposed entropy estimator preserves the global minimum of actual entropy the equivalence between global optimization by convolution smoothing and the convolution by the kernel in parzen windowing is also discussed simulation results are presented for time series prediction and classification where experimental demonstration of all the theoretical concepts is presented

the semi classical trace formula and propagation of wave packets abstract we study spectral and propagation properties of operators of the form s h n j 0 h j p j where j p j is a differential operator of order j on a manifold m asymptotically as h 0 the estimates are in terms of the flow t of the classical hamiltonian h x p n j 0 p j x p on t m where p j is the principal symbol of p j we present two sets of results i the semiclassical trace formula on the asymptotic behavior of eigenvalues and eigenfunctions of s h in terms of periodic trajectors of h ii associated to certain isotropic submanifolds t m we define families of functions h and prove that t exp iths is a family of the same kind associated to t

square root algorithms for maximum correntropy estimation of linear discrete time systems in presence of non gaussian noise abstract recent developments in the realm of state estimation of stochastic dynamic systems in the presence of non gaussian noise have induced a new methodology called the maximum correntropy filtering the filters designed under the maximum correntropy criterion mcc utilize a similarity measure or correntropy between two random variables as a cost function they are shown to improve the estimators robustness against outliers or impulsive noises in this paper we explore the numerical stability of linear filtering technique proposed recently under the mcc approach the resulted estimator is called the maximum correntropy criterion kalman filter mcc kf the purpose of this study is two fold first the previously derived mcc kf equations are revised and the related kalman like equality conditions are proved based on this theoretical finding we improve the mcc kf technique in the sense that the new method possesses a better estimation quality with the reduced computational cost compared with the previously proposed mcc kf variant second we devise some square root implementations for the newly designed improved estimator the square root algorithms are well known to be inherently more stable than the conventional kalman like implementations which process the full error covariance matrix in each iteration step of the filter additionally following the latest achievements in the kf community all square root algorithms are formulated here in the so called array form it implies the use of orthogonal transformations for recursive update of the required filtering quantities and thereby no loss of accuracy is incurred apart from the numerical stability benefits the array form also makes the modern kalman like filters better suited to parallel implementation and to very large scale integration vlsi implementation all the mcc kf variants developed in this paper are demonstrated to outperform the previously proposed mcc kf version in two numerical examples

maximum correntropy criterion for robust face recognition in this paper we present a sparse correntropy framework for computing robust sparse representations of face images for recognition compared with the state of the art l1norm based sparse representation classifier src which assumes that noise also has a sparse representation our sparse algorithm is developed based on the maximum correntropy criterion which is much more insensitive to outliers in order to develop a more tractable and practical approach we in particular impose nonnegativity constraint on the variables in the maximum correntropy criterion and develop a half quadratic optimization technique to approximately maximize the objective function in an alternating way so that the complex optimization problem is reduced to learning a sparse representation through a weighted linear least squares problem with nonnegativity constraint at each iteration our extensive experiments demonstrate that the proposed method is more robust and efficient in dealing with the occlusion and corruption problems in face recognition as compared to the related state of the art methods in particular it shows that the proposed method can improve both recognition accuracy and receiver operator characteristic roc curves while the computational cost is much lower than the src algorithms

the c loss function for pattern classification this paper presents a new loss function for neural network classification inspired by the recently proposed similarity measure called correntropy we show that this function essentially behaves like the conventional square loss for samples that are well within the decision boundary and have small errors and l0 or counting norm for samples that are outliers or are difficult to classify depending on the value of the kernel size parameter the proposed loss function moves smoothly from convex to non convex and becomes a close approximation to the misclassification loss ideal 0 1 loss we show that the discriminant function obtained by optimizing the proposed loss function in the neighborhood of the ideal 0 1 loss function to train a neural network is immune to overfitting more robust to outliers and has consistent and better generalization performance as compared to other commonly used loss functions even after prolonged training the results also show that it is a close competitor to the svm since the proposed method is compatible with simple gradient based online learning it is a practical way of improving the performance of neural network classifiers highlightswe propose a new classification method the method is based on the recently proposed similarity measure called correntropy the method is robust against overfitting and less sensitive to system parameters we obtain better performance as compared to several conventional classifiers

robust multidimensional scaling using a maximum correntropy criterion multidimensional scaling mds refers to a class of dimensionality reduction techniques which represent entities as points in a low dimensional space so that the interpoint distances approximate the initial pairwise dissimilarities between entities as closely as possible the traditional methods for solving mds are susceptible to outliers here a unified framework is proposed where the mds is treated as maximization of a correntropy criterion which is solved by half quadratic optimization in either multiplicative or additive forms by doing so mds can cope with an initial dissimilarity matrix contaminated with outliers because the correntropy criterion is closely related to m estimators three novel algorithms are derived their performances are assessed experimentally against three state of the art mds techniques namely the scaling by majorizing a complicated function the robust euclidean embedding and the robust mds under the same conditions the experimental results indicate that the proposed algorithms perform substantially better than the aforementioned competing techniques

maximum entropy and lacunary stieltjes moment problem the lacunary stieltjes moment problem in the framework of the maximum entropy approach is considered a proof for the existence conditions of the solution is provided

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

m estimators for robust multidimensional scaling employing l 2 1 norm regularization abstract multidimensional scaling mds has been exploited to visualise the hidden structures among a set of entities in a reduced dimensional metric space here we are interested in cases whenever the initial dissimilarity matrix is contaminated by outliers it is well known that the state of the art algorithms for solving the mds problem generate erroneous embeddings due to the distortion introduced by such outliers to remedy this vulnerability a unified framework for the solution of mds problem is proposed which resorts to half quadratic optimization and employs potential functions of m estimators in combination with l2 1 norm regularization two novel algorithms are derived their performance is assessed for various m estimators against state of the art mds algorithms on four benchmark data sets the numerical tests demonstrate that the proposed algorithms perform better than the competing alternatives

detecting differences between delay vector distributions we propose a test for the null hypothesis that two sets of vectors are drawn from the same multidimensional probability distribution the application to delay vector distributions provides a test for the null hypothesis that two time series have been generated by the same mechanism

generalized mean for robust principal component analysis in this paper we propose a robust principal component analysis pca to overcome the problem that pca is prone to outliers included in the training set different from the other alternatives which commonly replace l2 norm by other distance measures the proposed method alleviates the negative effect of outliers using the characteristic of the generalized mean keeping the use of the euclidean distance the optimization problem based on the generalized mean is solved by a novel method we also present a generalized sample mean which is a generalization of the sample mean to estimate a robust mean in the presence of outliers the proposed method shows better or equivalent performance than the conventional pcas in various problems such as face reconstruction clustering and object categorization highlightswe propose a robust principal component analysis the generalized mean is used in the proposed method instead of the arithmetic mean a novel method is also presented to solve our optimization problem

robust nonnegative patch alignment for dimensionality reduction dimensionality reduction is an important method to analyze high dimensional data and has many applications in pattern recognition and computer vision in this paper we propose a robust nonnegative patch alignment for dimensionality reduction which includes a reconstruction error term and a whole alignment term we use correntropy induced metric to measure the reconstruction error in which the weight is learned adaptively for each entry for the whole alignment we propose locality preserving robust nonnegative patch alignment lp rna and sparsity preserviing robust nonnegative patch alignment sp rna which are unsupervised and supervised respectively in the lp rna we propose a locally sparse graph to encode the local geometric structure of the manifold embedded in high dimensional space in particular we select large p nearest neighbors for each sample then obtain the sparse representation with respect to these neighbors the sparse representation is used to build a graph which simultaneously enjoys locality sparseness and robustness in the sp rna we simultaneously use local geometric structure and discriminative information in which the sparse reconstruction coefficient is used to characterize the local geometric structure and weighted distance is used to measure the separability of different classes for the induced nonconvex objective function we formulate it into a weighted nonnegative matrix factorization based on half quadratic optimization we propose a multiplicative update rule to solve this function and show that the objective function converges to a local optimum several experimental results on synthetic and real data sets demonstrate that the learned representation is more discriminative and robust than most existing dimensionality reduction methods

the existence conditions for maximum entropy distributions having prescribed the first three moments the criteria for existence of maximum entropy distributions having prescribed the first three moments in a semi infinite range are given some relationships between moments and lagrange multipliers are also presented

correntropy properties and applications in non gaussian signal processing the optimality of second order statistics depends heavily on the assumption of gaussianity in this paper we elucidate further the probabilistic and geometric meaning of the recently defined correntropy function as a localized similarity measure a close relationship between correntropy and m estimation is established connections and differences between correntropy and kernel methods are presented as such correntropy has vastly different properties compared with second order statistics that can be very useful in non gaussian signal processing especially in the impulsive noise environment examples are presented to illustrate the technique

a second order method for fitting the canonical polyadic decomposition with non least squares cost the canonical polyadic decomposition cpd can be used to extract meaningful components from a tensor most existing optimization methods for fitting the cpd use as cost function the least squares distance between the tensor and its cpd while the minimum of this cost function coincides with the maximum likelihood estimator for data with additive i i d gaussian distributed noise for other noise distributions better suited cost functions exist for such cost functions first order gradient based optimization methods have been proposed however approximate second order methods which additionally use information from the hessian of the cost function to achieve faster convergence are still largely unexplored in this paper we generalize the gauss newton nonlinear least squares algorithm to twice differentiable entry wise cost functions the low rank structure of the problem is exploited to keep the computational cost low as a special case beta divergence cost functions are examined we show that quadratic convergence can be obtained close to the solution with a reasonable extra cost in memory and computation time making the proposed method particularly useful when high accuracy of the decomposition is desired

maximum entropy in the hamburger moments problem the maximum entropy approach to the solution of the hamburger inverse problem of moments in which one seeks to recover a positive density function p x where x from the values of a finite n 1 of its moments is considered the obtained results show that unexpected upper bounds for the moments do not exist in the general hamburger finite moment problem unlike in the symmetric case previously considered some physical examples illustrating the use of partial information to determine the approximate function are presented

contributions of non extremum critical points to the semi classical trace formula we study the semi classical trace formula at a critical energy level for a h pseudo differential operator on rn whose principal symbol has a totally degenerate critical point for that energy we compute the contribution to the trace formula of isolated non extremum critical points under a condition of real principal type the new contribution to the trace formula is valid for all time in a compact subset of r but the result is modest since we have restrictions on the dimension

robust guided image filtering using nonconvex potentials filtering images using a guidance signal a process called guided or joint image filtering has been used in various tasks in computer vision and computational photography particularly for noise reduction and joint upsampling this uses an additional guidance signal as a structure prior and transfers the structure of the guidance signal to an input image restoring noisy or altered image structure the main drawbacks of such a data dependent framework are that it does not consider structural differences between guidance and input images and that it is not robust to outliers we propose a novel sd for static dynamic filter to address these problems in a unified framework and jointly leverage structural information from guidance and input images guided image filtering is formulated as a nonconvex optimization problem which is solved by the majorize minimization algorithm the proposed algorithm converges quickly while guaranteeing a local minimum the sd filter effectively controls the underlying image structure at different scales and can handle a variety of types of data from different sensors it is robust to outliers and other artifacts such as gradient reversal and global intensity shift and has good edge preserving smoothing properties we demonstrate the flexibility and effectiveness of the proposed sd filter in a variety of applications including depth upsampling scale space filtering texture removal flash non flash denoising and rgb nir denoising

information theoretic shape matching in this paper we describe two related algorithms that provide both rigid and non rigid point set registration with different computational complexity and accuracy the first algorithm utilizes a nonlinear similarity measure known as correntropy the measure combines second and high order moments in its decision statistic showing improvements especially in the presence of impulsive noise the algorithm assumes that the correspondence between the point sets is known which is determined with the surprise metric the second algorithm mitigates the need to establish a correspondence by representing the point sets as probability density functions pdf the registration problem is then treated as a distribution alignment the method utilizes the cauchy schwarz divergence to measure the similarity distance between the point sets and recover the spatial transformation function needed to register them both algorithms utilize information theoretic descriptors however correntropy works at the realizations level whereas cauchy schwarz divergence works at the pdf level this allows correntropy to be less computationally expensive and for correct correspondence more accurate the two algorithms are robust against noise and outliers and perform well under varying levels of distortion they outperform several well known and state of the art methods for point set registration

maximum correntropy kalman filter abstract traditional kalman filter kf is derived under the well known minimum mean square error mmse criterion which is optimal under gaussian assumption however when the signals are non gaussian especially when the system is disturbed by some heavy tailed impulsive noises the performance of kf will deteriorate seriously to improve the robustness of kf against impulsive noises we propose in this work a new kalman filter called the maximum correntropy kalman filter mckf which adopts the robust maximum correntropy criterion mcc as the optimality criterion instead of using the mmse similar to the traditional kf the state mean vector and covariance matrix propagation equations are used to give prior estimations of the state and covariance matrix in mckf a novel fixed point algorithm is then used to update the posterior estimations a sufficient condition that guarantees the convergence of the fixed point algorithm is also given illustration examples are presented to demonstrate the effectiveness and robustness of the new algorithm

on the application of maximum entropy to the moments problem the maximum entropy approach to the solution of classical inverse problem of moments in which one seeks to reconstruct a function p x where x 0 from the values of a finite set n 1 of its moments is studied it is shown that for n 4 such a function always exists while for n 2 and n 3 the acceptable values of the moments are singled out analytically the paper extends to the general case where the results were previously bounded to the case n 2

minimum error entropy based sparse representation for robust subspace clustering this paper addresses the problem of clustering data points that are approximately drawn from multiple subspaces recently a large family of spectral clustering based methods such as sparse subspace clustering ssc and low rank representation lrr have been proposed in this paper we present a general formulation to unify many of them within a common framework based on atomic representation since mean square error mse relies heavily on the gaussianity assumption the previous mse based subspace clustering methods have the limitation of being sensitive to non gaussian noise in this paper we develop a novel subspace clustering method termed meessc by specifying the minimum error entropy mee as the loss function and the sparsity inducing atomic set we show that meessc can well overcome the above limitation the experimental results on both synthetic and real data verify the effectiveness of the proposed method

robust ellipse fitting via half quadratic and semidefinite relaxation optimization ellipse fitting is widely applied in the fields of computer vision and automatic manufacture however the introduced edge point errors especially outliers from image edge detection will cause severe performance degradation of the subsequent ellipse fitting procedure to alleviate the influence of outliers we develop a robust ellipse fitting method in this paper the main contributions of this paper are as follows first to be robust against the outliers we introduce the maximum correntropy criterion into the constrained least square cls ellipse fitting method and apply the half quadratic optimization algorithm to solve the nonlinear and nonconvex problem in an alternate manner second to ensure that the obtained solution is related to an ellipse we introduce a special quadratic equality constraint into the aforementioned cls model which results in the nonconvex quadratically constrained quadratic programming problem finally we derive the semidefinite relaxation version of the aforementioned problem in terms of the trace operator and thus determine the ellipse parameters using semidefinite programming some simulated and experimental examples are presented to illustrate the effectiveness of the proposed ellipse fitting approach

a robust regression framework with laplace kernel induced loss this work proposes a robust regression framework with nonconvex loss function two regression formulations are presented based on the laplace kernel induced loss lk loss moreover we illustrate that the lk loss function is a nice approximation for the zero norm however nonconvexity of the lk loss makes it difficult to optimize a continuous optimization method is developed to solve the proposed framework the problems are formulated as dc difference of convex functions programming the corresponding dc algorithms dcas converge linearly furthermore the proposed algorithms are applied directly to determine the hardness of licorice seeds using near infrared spectral data with noisy input experiments in eight spectral regions show that the proposed methods improve generalization compared with the traditional support vector regressions svr especially in high frequency regions experiments on several benchmark data sets demonstrate that the proposed methods achieve better results than the traditional regression methods in most of data sets we have considered

kernel methods for measuring independence we introduce two new functionals the constrained covariance and the kernel mutual information to measure the degree of independence of random variables these quantities are both based on the covariance between functions of the random variables in reproducing kernel hilbert spaces rkhss we prove that when the rkhss are universal both functionals are zero if and only if the random variables are pairwise independent we also show that the kernel mutual information is an upper bound near independence on the parzen window estimate of the mutual information analogous results apply for two correlation based dependence functionals introduced earlier we show the kernel canonical correlation and the kernel generalised variance to be independence measures for universal kernels and prove the latter to be an upper bound on the mutual information near independence the performance of the kernel dependence functionals in measuring independence is verified in the context of independent component analysis

selective ensemble of svdds with renyi entropy based diversity measure abstract in this paper a novel selective ensemble strategy for support vector data description svdd using the renyi entropy based diversity measure is proposed to deal with the problem of one class classification in order to obtain compact classification boundary the radius of ensemble is defined as the inner product of the vector of combination weights and the vector of the radii of svdds to make the center of ensemble achieve the optimal position the renyi entropy of the kernelized distances between the images of samples and the center of ensemble in the high dimensional feature space is defined as the diversity measure moreover to fulfill the selective ensemble an l 1 norm based regularization term is introduced into the objective function of the proposed ensemble the optimal combination weights can be iteratively obtained by the half quadratic optimization technique experimental results on two synthetic data sets and twenty benchmark data sets demonstrate that the proposed selective ensemble method is superior to the single svdd and the other four related ensemble approaches

robust recovery of corrupted low rankmatrix by implicit regularizers low rank matrix recovery algorithms aim to recover a corrupted low rank matrix with sparse errors however corrupted errors may not be sparse in real world problems and the relationship between l1 regularizer on noise and robust m estimators is still unknown this paper proposes a general robust framework for low rank matrix recovery via implicit regularizers of robust m estimators which are derived from convex conjugacy and can be used to model arbitrarily corrupted errors based on the additive form of half quadratic optimization proximity operators of implicit regularizers are developed such that both low rank structure and corrupted errors can be alternately recovered in particular the dual relationship between the absolute function in l1 regularizer and huber m estimator is studied which establishes a connection between robust low rank matrix recovery methods and m estimators based robust principal component analysis methods extensive experiments on synthetic and real world data sets corroborate our claims and verify the robustness of the proposed framework

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

robustness analysis of a maximum correntropy framework for linear regression abstract in this paper we formulate a solution of the robust linear regression problem in a general framework of correntropy maximization our formulation yields a unified class of estimators which includes the gaussian and laplacian kernel based correntropy estimators as special cases an analysis of the robustness properties is then provided the analysis includes a quantitative characterization of the informativity degree of the regression which is appropriate for studying the stability of the estimator using this tool a sufficient condition is expressed under which the parametric estimation error is shown to be bounded explicit expression of the bound is given and discussion on its numerical computation is supplied for illustration purpose two special cases are numerically studied

spectral fluctuations of schrodinger operators generated by critical points of the potential starting from the spectrum of schrodinger operators on mathbb r n we propose a method to detect critical points of the potential we argue semi classically on the basis of a mathematically rigorous version of gutzwiller s trace formula which expresses spectral statistics in term of classical orbits a critical point of the potential with zero momentum is an equilibrium of the flow and generates certain singularities in the spectrum via sharp spectral estimates this fluctuation indicates the presence of a critical point and allows to reconstruct partially the local shape of the potential some generalizations of this approach are also proposed

maximum entropy in the generalized moment problem the generalized moment problem in the framework of the maximum entropy approach is considered a proof for the existence conditions of the solution is provided

extraction of signals with specific temporal structure using kernel methods this work derives and evaluates a method for blind source extraction bse in a reproducing kernel hilbert space rkhs framework the a priori information about the autocorrelation function of the signal under study is translated in a linear transformation of the gram matrix of the transformed data in hilbert space our method proved to be more robust than methods presented in the literature of bse with respect to ambiguities in the available a priori information of the signal to be extracted the approach here introduced can also be seen as a generalization of kernel principal component analysis kpca to analyze autocorrelation matrices at specific time lags

joint temporal statistics of interference in decentralized wireless networks characterizing interference statistics is central to the design and analysis of both physical layer and medium access control layer techniques to mitigate interference in a wireless network the applicability of interference statistics however is limited by the assumptions adopted to derive the statistics in closed form common assumptions for a decentralized wireless network include temporally independent user locations and an unbounded pathloss function in this correspondence we derive the joint temporal statistics of interference that capture the temporal correlation in the network along with the realistic assumption of a bounded pathloss function the closed form statistics are asymptotically exact for low tail probabilities and match closely in simulations even when the tail probability is fairly high the primary contributions are to i show that joint interference statistics follow a multivariate gaussian mixture distribution under the assumption of a bounded pathloss function and ii characterize the joint tail probability decay behavior for both bounded and unbounded pathloss functions

two stage nonnegative sparse representation for large scale face recognition this paper proposes a novel nonnegative sparse representation approach called two stage sparse representation tsr for robust face recognition on a large scale database based on the divide and conquer strategy tsr decomposes the procedure of robust face recognition into outlier detection stage and recognition stage in the first stage we propose a general multisubspace framework to learn a robust metric in which noise and outliers in image pixels are detected potential loss functions including l1 l2 1 and correntropy are studied in the second stage based on the learned metric and collaborative representation we propose an efficient nonnegative sparse representation algorithm to find an approximation solution of sparse representation according to the l1 ball theory in sparse representation the approximated solution is unique and can be optimized efficiently then a filtering strategy is developed to avoid the computation of the sparse representation on the whole large scale dataset moreover theoretical analysis also gives the necessary condition for nonnegative least squares technique to find a sparse solution extensive experiments on several public databases have demonstrated that the proposed tsr approach in general achieves better classification accuracy than the state of the art sparse representation methods more importantly a significant reduction of computational costs is reached in comparison with sparse representation classifier this enables the tsr to be more suitable for robust face recognition on a large scale dataset

a maximum entropy method for particle filtering standard ensemble or particle filtering schemes do not properly represent states of low priori probability when the number of available samples is too small as is often the case in practical applications we introduce here a set of parametric resampling methods to solve this problem motivated by a general h theorem for relative entropy we construct parametric models for the filter distributions as maximum entropy minimum information models consistent with moments of the particle ensemble when the prior distributions are modeled as mixtures of gaussians our method naturally generalizes the ensemble kalman filter to systems with highly non gaussian statistics we apply the new particle filters presented here to two simple test cases a one dimensional diffusion process in a double well potential and the three dimensional chaotic dynamical system of lorenz

a semi classical trace formula for schrodinger operators in the case of a critical energy level let 2 2 v x be a schrodinger operator on rn with smooth potentialv x as x the spectrum of is discrete and one can study the asymptotic of the smoothed spectral density e k ek e as 0 here ek k nis the spectrum of and c 0 r we shall investigate the case whereeis a critical value of the symbolhof and extending the work of brummelhuis paul and uribe in 3 we will prove the existence of a full asymptotic expansion for in terms of and ln and compute the leading coefficient we will consider new weyl type estimates for the counting function nec k n ek e

semi classical spectral estimates for schrodinger operators at a critical level case of a degenerate maximum of the potential we study the semi classical trace formula at a critical energy level for a schrodinger operator on r n we assume here that the potential has a totally degenerate crit ical point associated to a local maximum the main result which establishes the contribution of the associated equilibrium in the trace formula is valid for all time in a compact subset of r and includes the singularity in t 0 for these new contributions the asymptotic expansion involves the logarithm of the parameter h depending on an explicit arithmetic condition on the dimension and the order of the critical point this logarithmic contribution can appear in the leading term

semiclassical spectral estimates for schrodinger operators at a critical energy level case of a degenerate minimum of the potential abstract we study the semi classical trace formula at a critical energy level for a schrodinger operator on r n we assume here that the potential has a totally degenerate critical point associated to a local minimum the main result which computes the contribution of this equilibrium is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point

kernel risk sensitive loss definition properties and application to robust adaptive filtering nonlinear similarity measures defined in kernel space such as correntropy can extract higher order statistics of data and offer potentially significant performance improvement over their linear counterparts especially in non gaussian signal processing and machine learning in this paper we propose a new similarity measure in kernel space called the kernel risk sensitive loss krsl and provide some important properties we apply the krsl to adaptive filtering and investigate the robustness and then develop the mkrsl algorithm and analyze the mean square convergence performance compared with correntropy the krsl can offer a more efficient performance surface thereby enabling a gradient based method to achieve faster convergence speed and higher accuracy while still maintaining the robustness to outliers theoretical analysis results and superior performance of the new algorithm are confirmed by simulation

robust gradient learning with applications this paper addresses the robust gradient learning rgl problem gradient learning models aim at learning the gradient vector of some target functions in supervised learning problems which can be further used to applications such as variable selection coordinate covariance estimation and supervised dimension reduction however existing gl models are not robust to outliers or heavy tailed noise this paper provides an rgl framework to address this problem in both regression and classification this is achieved by introducing a robust regression loss function and proposing a robust classification loss moreover our rgl algorithm works in an instance based kernelized dictionary instead of some fixed reproducing kernel hilbert space which may provide more flexibility to solve the proposed nonconvex model a simple computational algorithm based on gradient descent is provided and the convergence of the proposed method is also analyzed we then apply the proposed rgl model to applications such as nonlinear variable selection and coordinate covariance estimation the efficiency of our proposed model is verified on both synthetic and real data sets

human motion segmentation via robust kernel sparse subspace clustering studies on human motion have attracted a lot of attentions human motion capture data which much more precisely records human motion than videos do has been widely used in many areas motion segmentation is an indispensable step for many related applications but current segmentation methods for motion capture data do not effectively model some important characteristics of motion capture data such as riemannian manifold structure and containing non gaussian noise in this paper we convert the segmentation of motion capture data into a temporal subspace clustering problem under the framework of sparse subspace clustering we propose to use the geodesic exponential kernel to model the riemannian manifold structure use correntropy to measure the reconstruction error use the triangle constraint to guarantee temporal continuity in each cluster and use multi view reconstruction to extract the relations between different joints therefore exploiting some special characteristics of motion capture data we propose a new segmentation method which is robust to non gaussian noise since correntropy is a localized similarity measure we also develop an efficient optimization algorithm based on block coordinate descent method to solve the proposed model our optimization algorithm has a linear complexity while sparse subspace clustering is originally a quadratic problem extensive experiment results both on simulated noisy data set and real noisy data set demonstrate the advantage of the proposed method

correntropy based feature selection using binary projection most feature selection algorithms based on information theoretic learning itl adopt ranking process or greedy search as their searching strategies the former selects features individually so that it ignores feature interaction and dependencies the latter heavily relies on the search paths as only one path will be explored with no possible back track in addition both strategies typically lead to heuristic algorithms to cope with these problems this article proposes a novel feature selection framework based on correntropy in itl namely correntropy based feature selection using binary projection bpfs our framework selects features by projecting the original high dimensional data to a low dimensional space through a special binary projection matrix the formulated objective function aims at maximizing the correntropy between selected features and class labels and this function can be efficiently optimized via standard mathematical tools we apply the half quadratic method to optimize the objective function in an iterative manner where each iteration reduces to an assignment subproblem which can be highly efficiently solved with some off the shelf toolboxes comparative experiments on six real world datasets indicate that our framework is effective and efficient

inverse two sided laplace transform for probability density functions we present a method for the numerical inversion of two sided laplace transform of a probability density function the method assumes the knowledge of the first m derivatives at the origin of the function to be antitransformed the approximate analytical form is obtained by resorting to maximum entropy principle both entropy and l1 norm convergence are proved some numerical examples are illustrated

correntropy based evolving fuzzy neural system in this paper a correntropy based evolving fuzzy neural system cefns is proposed for approximation of nonlinear systems different from the commonly used mean square error criterion correntropy has a strong outliers rejection ability through capturing the higher moments of the error distribution considering the merits of correntropy this paper brings contributions to build evolving fuzzy neural system efns based on the correntropy concept to achieve a more stable evolution of the rule base and update of the rule parameters instead of the commonly used mean square error criterion the correntropy efns cefns begins with an empty rule base and all rules are evolved online based on the correntropy criterion the consequent part parameters are tuned based on the maximum correntropy criterion where the correntropy is used as the cost function so as to improve the non gaussian noise rejection ability the steady state convergence performance of the cefns is studied through the calculation of the steady state excess mean square error emse in two cases gaussian noise and non gaussian noise finally the cefns is validated through a benchmark system identification problem a mackey glass time series prediction problem as well as five other real world benchmark regression problems under both noise free and noisy conditions compared with other efnss the simulation results show that the proposed cefns produces better approximation accuracy using the least number of rules and training time and also owns superior non gaussian noise handling capability

maximum entropy autoregressive conditional heteroskedasticity model abstract in many applications it has been found that the autoregressive conditional heteroskedasticity arch model under the conditional normal or student s t distributions are not general enough to account for the excess kurtosis in the data moreover asymmetry in the financial data is rarely modeled in a systematic way in this paper we suggest a general density function based on the maximum entropy me approach that takes account of asymmetry excess kurtosis and also of high peakedness the me principle is based on the efficient use of available information and as is well known many of the standard family of distributions can be derived from the me approach we demonstrate how we can extract information functional from the data in the form of moment functions we also propose a test procedure for selecting appropriate moment functions our procedure is illustrated with an application to the nyse stock returns the empirical results reveal that the me approach with a fewer moment functions leads to a model that captures the stylized facts quite effectively

regularized discriminant entropy analysis in this paper we propose the regularized discriminant entropy rde which considers both class information and scatter information on original data based on the results of maximizing the rde we develop a supervised feature extraction algorithm called regularized discriminant entropy analysis rdea rdea is quite simple and requires no approximation in theoretical derivation the experiments with several publicly available data sets show the feasibility and effectiveness of the proposed algorithm with encouraging results we propose the regularized discriminant entropy rde the rde is based on the within class entropy and robust estimation we design a supervised algorithm regularized discriminant entropy analysis rdea rdea can be regarded as a framework for supervised feature extraction

robust low rank tensor recovery with regularized redescending m estimator this paper addresses the robust low rank tensor recovery problems tensor recovery aims at reconstructing a low rank tensor from some linear measurements which finds applications in image processing pattern recognition multitask learning and so on in real world applications data might be contaminated by sparse gross errors however the existing approaches may not be very robust to outliers to resolve this problem this paper proposes approaches based on the regularized redescending m estimators which have been introduced in robust statistics the robustness of the proposed approaches is achieved by the regularized redescending m estimators however the nonconvexity also leads to a computational difficulty to handle this problem we develop algorithms based on proximal and linearized block coordinate descent methods by explicitly deriving the lipschitz constant of the gradient of the data fitting risk the descent property of the algorithms is present moreover we verify that the objective functions of the proposed approaches satisfy the kurdyka ojasiewicz property which establishes the global convergence of the algorithms the numerical experiments on synthetic data as well as real data verify that our approaches are robust in the presence of outliers and still effective in the absence of outliers

fast and robust fixed point algorithms for independent component analysis independent component analysis ica is a statistical method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible we use a combination of two different approaches for linear ica comon s information theoretic approach and the projection pursuit approach using maximum entropy approximations of differential entropy we introduce a family of new contrast functions for ica these contrast functions enable both the estimation of the whole decomposition by minimizing mutual information and estimation of individual independent components as projection pursuit directions the statistical properties of the estimators based on such contrast functions are analyzed under the assumption of the linear mixture model and it is shown how to choose contrast functions that are robust and or of minimum variance finally we introduce simple fixed point algorithms for practical optimization of the contrast functions

robust level set image segmentation via a local correntropy based k means clustering it is still a challenging task to segment real world images since they are often distorted by unknown noise and intensity inhomogeneity to address these problems we propose a novel segmentation algorithm via a local correntropy based k means lck clustering due to the correntropy criterion the clustering algorithm can decrease the weights of the samples that are away from their clusters as a result lck based clustering algorithm can be robust to the outliers the proposed lck clustering algorithm is incorporated into the region based level set segmentation framework the iteratively re weighted algorithm is used to solve the lck based level set segmentation method extensive experiments on synthetic and real images are provided to evaluate our method showing significant improvements on both noise sensitivity and segmentation accuracy as compared with the state of the art approaches

learning with the maximum correntropy criterion induced losses for regression within the statistical learning framework this paper studies the regression model associated with the correntropy induced losses the correntropy as a similarity measure has been frequently employed in signal processing and pattern recognition motivated by its empirical successes this paper aims at presenting some theoretical understanding towards the maximum correntropy criterion in regression problems our focus in this paper is two fold first we are concerned with the connections between the regression model associated with the correntropy induced loss and the least squares regression model second we study its convergence property a learning theory analysis which is centered around the above two aspects is conducted from our analysis we see that the scale parameter in the loss function balances the convergence rates of the regression model and its robustness we then make some efforts to sketch a general view on robust loss functions when being applied into the learning for regression problems numerical experiments are also implemented to verify the effectiveness of the model

the mee principle in data classification a perceptron based analysis this letter focuses on the issue of whether risk functionals derived from information theoretic principles such as shannon or renyi s entropies are able to cope with the data classification problem in both the sense of attaining the risk functional minimum and implying the minimum probability of error allowed by the family of functions implemented by the classifier here denoted by min pe the analysis of this so called minimization of error entropy mee principle is carried out in a single perceptron with continuous activation functions yielding continuous error distributions in spite of the fact that the analysis is restricted to single perceptrons it reveals a large spectrum of behaviors that mee can be expected to exhibit in both theory and practice in what concerns the theoretical mee our study clarifies the role of the parameters controlling the perceptron activation function of the squashing type in often reaching the minimum probability of error our study also clarifies the role of the kernel density estimator of the error density in achieving the minimum probability of error in practice

entropy convergence in stieltjes and hamburger moment problem in the classical stieltjes and hamburger moment problem the sequence of maximum entropy approximants whose first m moments are equal to given ones is considered it is proved that whenever an infinite moment problem is determined then maximum entropy approximants converge in entropy to the function characterized by given moments entropy convergence is proved by using exclusively existence and uniqueness conditions

robust support vector machines for classification with nonconvex and smooth losses this letter addresses the robustness problem when learning a large margin classifier in the presence of label noise in our study we achieve this purpose by proposing robustified large margin support vector machines the robustness of the proposed robust support vector classifiers rsvc which is interpreted from a weighted viewpoint in this work is due to the use of nonconvex classification losses besides the robustness we also show that the proposed rscv is simultaneously smooth which again benefits from using smooth classification losses the idea of proposing rsvc comes from m estimation in statistics since the proposed robust and smooth classification losses can be taken as one sided cost functions in robust statistics its fisher consistency property and generalization ability are also investigated besides the robustness and smoothness another nice property of rsvc lies in the fact that its solution can be obtained by solving weighted squared hinge loss based support vector machine problems iteratively we further show that in each iteration it is a quadratic programming problem in its dual space and can be solved by using state of the art methods we thus propose an iteratively reweighted type algorithm and provide a constructive proof of its convergence to a stationary point effectiveness of the proposed classifiers is verified on both artificial and real data sets

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

truncated cauchy non negative matrix factorization non negative matrix factorization nmf minimizes the euclidean distance between the data matrix and its low rank approximation and it fails when applied to corrupted data because the loss function is sensitive to outliers in this paper we propose a truncated cauchynmf loss that handle outliers by truncating large errors and develop a truncated cauchynmf to robustly learn the subspace on noisy datasets contaminated by outliers we theoretically analyze the robustness of truncated cauchynmf comparing with the competing models and theoretically prove that truncated cauchynmf has a generalization bound which converges at a rate of order o sqrt ln n n where n is the sample size we evaluate truncated cauchynmf by image clustering on both simulated and real datasets the experimental results on the datasets containing gross corruptions validate the effectiveness and robustness of truncated cauchynmf for learning robust subspaces

automatic ultrasound image segmentation based on local entropy and active contour model abstract it is still a challenging task to segment ultrasound medical images automatically and accurately because of the poor quality of images to address these problems a two stage automatic segmentation scheme based on local entropy and a proposed active contour model for ultrasound images are put forward in this paper first a new region based active contour model in the level set formulation driven by global and local intensity information is established for the segmentation furthermore for automatically segmenting the ultrasound images a coarse segmentation is performed utilizing local entropy information of the ultrasound images then the coarse segmentation result is used as the initial value of the explicit segmentation based on the proposed model several experiments on real ultrasound images demonstrate that the proposed method outperforms other methods on both visual perception and objective evaluation metrics

inverse spectral problems for schrodinger operators in this article we find some explicit formulas for the semi classical wave invariants at the bottom of the well of a schrodinger operator as an application of these new formulas for the wave invariants we improve the inverse spectral results proved by guillemin and uribe in gu they proved that under some symmetry assumptions on the potential v x the taylor expansion of v x near a non degenerate global minimum can be recovered from the knowledge of the low lying eigenvalues of the associated schrodinger operator in mathbb r n we prove similar inverse spectral results using fewer symmetry assumptions we also show that in dimension 1 no symmetry assumption is needed to recover the taylor coefficients of v x

optimal state estimation for stochastic systems an information theoretic approach in this paper we examine the problem of optimal state estimation or filtering in stochastic systems using an approach based on information theoretic measures in this setting the traditional minimum mean square measure is compared with information theoretic measures kalman filtering theory is reexamined and some new interpretations are offered we show that for a linear gaussian system the kalman filter is the optimal filter not only for the mean square error measure but for several information theoretic measures which are introduced in this work for nonlinear systems these same measures generally are in conflict with each other and the feedback control policy has a dual role with regard to regulation and estimation for linear stochastic systems with general noise processes a lower bound on the achievable mutual information between the estimation error and the observation are derived the properties of an optimal probing control law and the associated optimal filter which achieve this lower bound and their relationships are investigated it is shown that for a linear stochastic system with an affine linear filter for the homogeneous system under some reachability and observability conditions zero mutual information between estimation error and observations can be achieved only when the system is gaussian

robust support vector machines based on the rescaled hinge loss function abstract the support vector machine svm is a popular classifier in machine learning but it is not robust to outliers in this paper based on the correntropy induced loss function we propose the rescaled hinge loss function which is a monotonic bounded and nonconvex loss that is robust to outliers we further show that the hinge loss is a special case of the proposed rescaled hinge loss then we develop a new robust svm based on the rescaled hinge loss after using the half quadratic optimization method we find that the new robust svm is equivalent to an iterative weighted svm which can help explain the robustness of iterative weighted svm from a loss function perspective experimental results confirm that the new robust svm not only performs better than svm and the existing robust svms on the datasets that have outliers but also presents better sparseness than svm

robust hyperspectral unmixing with correntropy based metric hyperspectral unmixing is one of the crucial steps for many hyperspectral applications the problem of hyperspectral unmixing has proved to be a difficult task in unsupervised work settings where the endmembers and abundances are both unknown in addition this task becomes more challenging in the case that the spectral bands are degraded by noise this paper presents a robust model for unsupervised hyperspectral unmixing specifically our model is developed with the correntropy based metric where the nonnegative constraints on both endmembers and abundances are imposed to keep physical significance besides a sparsity prior is explicitly formulated to constrain the distribution of the abundances of each endmember to solve our model a half quadratic optimization technique is developed to convert the original complex optimization problem into an iteratively reweighted nonnegative matrix factorization with sparsity constraints as a result the optimization of our model can adaptively assign small weights to noisy bands and put more emphasis on noise free bands in addition with sparsity constraints our model can naturally generate sparse abundances experiments on synthetic and real data demonstrate the effectiveness of our model in comparison to the related state of the art unmixing models

sparse modal additive model sparse additive models have been successfully applied to high dimensional data analysis due to the flexibility and interpretability of their representation however the existing methods are often formulated using the least squares loss with learning the conditional mean which is sensitive to data with the non gaussian noises e g skewed noise heavy tailed noise and outliers to tackle this problem we propose a new robust regression method called as sparse modal additive model spmam by integrating the modal regression metric the data dependent hypothesis space and the weighted lq 1 norm regularizer q 1 into the additive models specifically the modal regression metric assures the model robustness to complex noises via learning the conditional mode the data dependent hypothesis space offers the model adaptivity via sample based presentation and the lq 1 norm regularizer addresses the algorithmic interpretability via sparse variable selection in theory the proposed spmam enjoys statistical guarantees on asymptotic consistency for regression estimation and variable selection simultaneously experimental results on both synthetic and real world benchmark data sets validate the effectiveness and robustness of the proposed model

quantized minimum error entropy criterion comparing with traditional learning criteria such as mean square error the minimum error entropy mee criterion is superior in nonlinear and non gaussian signal processing and machine learning the argument of the logarithm in renyi s entropy estimator called information potential ip is a popular mee cost in information theoretic learning the computational complexity of ip is however quadratic in terms of sample number due to double summation this creates the computational bottlenecks especially for large scale data sets to address this problem in this paper we propose an efficient quantization approach to reduce the computational burden of ip which decreases the complexity from o n 2 to o mn with m ll n the new learning criterion is called the quantized mee qmee some basic properties of qmee are presented illustrative examples with linear in parameter models are provided to verify the excellent performance of qmee

the correntropy mace filter the minimum average correlation energy mace filter is well known for object recognition this paper proposes a nonlinear extension to the mace filter using the recently introduced correntropy function correntropy is a positive definite function that generalizes the concept of correlation by utilizing second and higher order moments of the signal statistics because of its positive definite nature correntropy induces a new reproducing kernel hilbert space rkhs taking advantage of the linear structure of the rkhs it is possible to formulate the mace filter equations in the rkhs induced by correntropy and obtained an approximate solution due to the nonlinear relation between the feature space and the input space the correntropy mace cmace can potentially improve upon the mace performance while preserving the shift invariant property additional computation for all shifts will be required in the cmace to alleviate the computation complexity of the solution this paper also presents the fast cmace using the fast gauss transform fgt we apply the cmace filter to the mstar public release synthetic aperture radar sar data set as well as pie database of human faces and show that the proposed method exhibits better distortion tolerance and outperforms the linear mace in both generalization and rejection abilities

a semi classical trace formula at a totally degenerate critical level we study the semi classical trace formula at a critical energy level for an h pseudo differential operator on open image in new window n whose principal symbol has a totally degenerate critical point for that energy this problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point

sharp semiclassical estimates for the number of eigenvalues below a totally degenerate critical level we consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than e for elliptic operators in l2 rd we describe a method of obtaining remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the intervals e e h where e is close to e if the volume of this region is o h then we obtain remainder estimates o h1 d with no assumptions on the hessian of the principal symbol at the critical level moreover we do not assume that the coefficients are smooth all results hold if second order derivatives of coefficients are holder continuous

maximum entropy in the finite stieltjes and hamburger moment problem a necessary and sufficient condition for the existence of the maximum entropy me function defined in an infinite or semiinfinite interval is provided the conclusions reached show that except in a few particular cases the necessary and sufficient conditions for the existence of maximum entropy function are identical to the conditions for the solution of the moment problem when the first m 1 moments are assigned even if the conclusions reached are very similar to the hausdorff case the specificity of the hamburger and stieltjes cases demands a different handling a sufficient condition for the entropy convergence of the resulting sequence of maximum entropy estimators to the entropy of the recovering function is also provided

robust principal component analysis based on maximum correntropy criterion principal component analysis pca minimizes the mean square error mse and is sensitive to outliers in this paper we present a new rotational invariant pca based on maximum correntropy criterion mcc a half quadratic optimization algorithm is adopted to compute the correntropy objective at each iteration the complex optimization problem is reduced to a quadratic problem that can be efficiently solved by a standard optimization method the proposed method exhibits the following benefits 1 it is robust to outliers through the mechanism of mcc which can be more theoretically solid than a heuristic rule based on mse 2 it requires no assumption about the zero mean of data for processing and can estimate data mean during optimization and 3 its optimal solution consists of principal eigenvectors of a robust covariance matrix corresponding to the largest eigenvalues in addition kernel techniques are further introduced in the proposed method to deal with nonlinearly distributed data numerical results demonstrate that the proposed method can outperform robust rotational invariant pcas based on l1 norm when outliers occur

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure