hamiltonians on discrete structures jumps of the integrated density of states and uniform convergence we study equivariant families of discrete hamiltonians on amenable geometries and their integrated density of states ids we prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support the size of a jump of the ids is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions from this we deduce uniform convergence w r t the spectral parameter of the finite volume approximants of the ids our framework includes quasiperiodic operators on delone sets periodic and random operators on quasi transitive graphs and operators on percolation graphs

mixing and non mixing local minima of the entropy contrast for blind source separation in this paper both non mixing and mixing local minima of the entropy are analyzed from the viewpoint of blind source separation bss they correspond respectively to acceptable and spurious solutions of the bss problem the contribution of this work is twofold first a taylor development is used to show that the exact output entropy cost function has a non mixing minimum when this output is proportional to any of the non gaussian sources and not only when the output is proportional to the lowest entropic source second in order to prove that mixing entropy minima exist when the source densities are strongly multimodal an entropy approximator is proposed the latter has the major advantage that an error bound can be provided even if this approximator and the associated bound is used here in the bss context it can be applied for estimating the entropy of any random variable with multimodal density

localization for quasiperiodic operators with unbounded monotone potentials abstract we establish non perturbative anderson localization for a wide class of 1d quasiperiodic operators with unbounded monotone potentials extending the classical results on maryland model and perturbative results for analytic potentials by bellissard lima and scoppola

extended gaussianization method for blind separation of post nonlinear mixtures the linear mixture model has been investigated in most articles tackling the problem of blind source separation recently several articles have addressed a more complex model blind source separation bss of postnonlinear pnl mixtures these mixtures are assumed to be generated by applying an unknown invertible nonlinear distortion to linear instantaneous mixtures of some independent sources the gaussianization technique for bss of pnl mixtures emerged based on the assumption that the distribution of the linear mixture of independent sources is gaussian in this letter we review the gaussianization method and then extend it to apply to pnl mixture in which the linear mixture is close to gaussian our proposed method approximates the linear mixture using the cornishfisher expansion we choose the mutual information as the independence measurement to develop a learning algorithm to separate pnl mixtures this method provides better applicability and accuracy we then discuss the sufficient condition for the method to be valid the characteristics of the nonlinearity do not affect the performance of this method with only a few parameters to tune our algorithm has a comparatively low computation finally we present experiments to illustrate the efficiency of our method

holder continuity of the spectral measures for one dimensional schrodinger operator in exponential regime avila and jitomirskaya prove that the spectral measure v xf of quasi periodic schrodinger operator is 1 2 holder continuous with appropriate initial vector f if satisfies diophantine condition and is small in the present paper the conclusion is extended to that for all with the spectral measure v xf is 1 2 holder continuous with small if v is real analytic in a neighbor of x c where c is a large absolute constant in particular the spectral measure xf of almost mathieu operator is 1 2 holder continuous if e c with c a large absolute constant

blind separation of instantaneous mixture of sources via an independent component analysis in this paper we introduce a procedure for separating a multivariate distribution into nearly independent components based on minimizing a criterion defined in terms of the kullback leibner distance by replacing the unknown density with a kernel estimate we derive useful forms of this criterion when only a sample from that distribution is available we also compute the gradient and hessian of our criteria for use in an iterative minimization setting this gradient to zero yields a set of separating functions similar to the ones considered in the source separation problem except that here these functions are adapted to the observed data finally some simulations are given illustrating the good performance of the method

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

on the risk of using renyi s entropy for blind source separation recently some researchers have suggested renyi s entropy in its general form as a blind source separation bss objective function this was motivated by two arguments 1 shannon s entropy which is known to be a suitable criterion for bss is a particular case of renyi s entropy and 2 some practical advantages can be obtained by choosing another specific value for the renyi exponent yielding to e g quadratic entropy unfortunately by doing so there is no longer guarantee that optimizing this generalized criterion would lead to recovering the original sources in this paper we show that renyi s entropy in its exact form i e out of any consideration about its practical estimation or computation might lead to not recovering the sources depending on the source densities and on renyi s exponent value this is illustrated on specific examples we also compare our conclusions with previous works involving renyi s entropies for blind deconvolution

convergence properties and data efficiency of the minimum error entropy criterion in adaline training recently we have proposed the minimum error entropy mee criterion as an information theoretic alternative to the widely used mean square error criterion in supervised adaptive system training for this purpose we have formulated a nonparametric estimator for renyi s entropy that employs parzen windowing mathematical investigation of the proposed entropy estimator revealed interesting insights about the process of information theoretical learning this new estimator and the associated criteria have been applied to the supervised and unsupervised training of adaptive systems in a wide range of problems successfully in this paper we analyze the structure of the mee performance surface around the optimal solution and we derive the upper bound for the step size in adaptive linear neuron adaline training with the steepest descent algorithm using mee in addition the effects of the entropy order and the kernel size in parzen windowing on the shape of the performance surface and the eigenvalues of the hessian at and around the optimal solution are investigated conclusions from the theoretical analyses are illustrated through numerical examples

fast algorithms for mutual information based independent component analysis this paper provides fast algorithms to perform independent component analysis based on the mutual information criterion the main ingredient is the binning technique and the use of cardinal splines which allows the fast computation of the density estimator over a regular grid using a discretized form of the entropy the criterion can be evaluated quickly together with its gradient which can be expressed in terms of the score functions both offline and online separation algorithms have been developed our density entropy and score estimators also have their own interest

absence of localisation in the almost mathieu equation the author considers the discrete schrodinger operator on z with the potential lambda cos 2 pi alpha n theta this one dimensional model occurs in the study of an electron in a two dimensional periodic potential with a uniform magnetic field first it is proved that for every alpha and for lambda 2 this operator has no eigenvalue i e localised state furthermore at lambda 2 the eigenvalues if they exist belong to the set where the lyapunov exponent vanishes and the associated eigenvectors at in l2 z but are not summable

a generalized contrast function and stability analysis for overdetermined blind separation of instantaneous mixtures in this letter the problem of blind separation of n independent sources from their m linear instantaneous mixtures is considered first a generalized contrast function is defined as a valuable extension of the existing classical and nonsymmetrical contrast functions it is applicable to the overdetermined blind separation m n with an unknown number of sources because not only independent components but also redundant ones are allowed in the outputs of a separation system second a natural gradient learning algorithm developed primarily for the complete case m n is shown to work as well with an n m or m m separating matrix for each optimizes a certain mutual information contrast function finally we present stability analysis for a newly proposed generalized orthogonal natural gradient algorithm which can perform the overdetermined blind separation when n is unknown obtaining an expectable result that its local stability conditions are slightly stricter than those of the conventional natural gradient algorithm using an invertible mixing matrix m n

sharp holder continuity of the lyapunov exponent of finitely differentiable quasi periodic cocycles we show that if the base frequency is diophantine then the lyapunov exponent of a c k quasi periodic sl 2 mathbb r cocycle is 1 2 holder continuous in the almost reducible regime as a consequence we show that if the frequency is diophantine and the potential is small then the integrated density of states of the corresponding quasi periodic schrodinger operator is 1 2 holder continuous

on the estimation of differential entropy from data located on embedded manifolds estimation of the differential entropy from observations of a random variable is of great importance for a wide range of signal processing applications such as source coding pattern recognition hypothesis testing and blind source separation in this paper we present a method for estimation of the shannon differential entropy that accounts for embedded manifolds the method is based on high rate quantization theory and forms an extension of the classical nearest neighbor entropy estimator the estimator is consistent in the mean square sense and an upper bound on the rate of convergence of the estimator is given because of the close connection between compression and shannon entropy the proposed method has an advantage over methods estimating the renyi entropy through experiments on uniformly distributed data on known manifolds and real world speech data we show the accuracy and usefulness of our proposed method

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

random schrodinger operators arising from lattice gauge fields i existence and examples we consider new models of ergodic schrodinger operators whose existence relies on a cohomological theorem of feldman and moore in ergodic theory these operators generalize the harper operator which describes the case of a constant magnetic field an example is the case when the magnetic field is given by independent random variables attached to the lattice plaquettes a generalization of the feldman moore theorem by lind to non abelian groups also allows us to consider schrodinger operators obtained from non abelian lattice gauge fields the existence result extends to more general graphs like to operators on tilings and to higher dimensions we compute some moment expansions for the density of states for example for independent identically and uniformly distributed magnetic fields a model which has been studied at least since 1970 and whose existence can also be seen without involving the above mentioned existence theorem we show that the nth moment is the number of closed paths in the two dimensional lattice starting at the origin for which the winding number vanishes at each plaquette point this goes beyond the brinkman rice self retracing path approximation other examples are a higher dimensional example a one dimensional anderson model which can be treated in this framework as well as the hofstadter model with constant magnetic field where one averages over all possible magnetic fields we also reprove a result of jitomirskaya mandelshtam stating that the deterministic aharonov bohm model is a compact perturbation of the free laplacian

on entropy rate for the complex domain and its application to i i d sampling we derive the entropy rate formula for a complex gaussian random process by using a widely linear model the resulting expression is general and applicable to both circular and noncircular gaussian processes since any second order stationary process can be modeled as the output of a widely linear system driven by a circular white noise furthermore we demonstrate application of the derived formula to an order selection problem we extend a scheme for independent and identically distributed i i d sampling to the complex domain to improve the estimation performance of information theoretic criteria when samples are correlated we show the effectiveness of the approach for order selection for simulated and actual functional magnetic resonance imaging fmri data that are inherently complex valued

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

globally convergent deflationary instantaneous blind source separation algorithm for digital communication signals recently an instantaneous blind source separation bss approach that exploits the bounded magnitude structure of digital communications signals has been introduced in this paper we introduce a deflationary adaptive algorithm based on this criterion and provide its convergence analysis we show that the resulting algorithm is convergent to one of the globally optimal points that correspond to perfect separation the simulation examples related to the separation of digital communication signals are provided to illustrate the convergence and the performance of the algorithm

exact wave functions of an electron on a quasiperiodic lattice definition of an infinite dimensional riemann theta function a scheme for obtaining the exact wave functions of an electron on a quasiperiodic lattice is presented it is shown that the trace map plays a very important role for construction of the infinite dimensional riemann theta function in terms of which the wave functions can be represented

stability analyses of information theoretic blind separation algorithms in the case where the sources are nonlinear processes a basic approach to blind source separation is to define an index representing the statistical dependency among the output signals of the separator and minimize it with respect to the separator s parameters the most natural index might be mutual information among the output signals of the separator in the case of a convolutive mixture however since the signals must be treated as a time series it becomes very complicated to concretely express the mutual information as a function of the parameters to cope with this difficulty in most of the conventional methods the source signals are assumed to be independent identically distributed i i d or linear based on this assumption some simpler indices are defined and their minimization is made by such an iterative calculation as the gradient method in actual applications however the sources are often not linear processes this paper discusses what will happen when those algorithms postulating the linearity of the sources are applied to the case of nonlinear sources an analysis of local stability derives a couple of conditions guaranteeing that the separator stably tends toward a desired one with iteration the obtained results reveal that those methods which are based on the minimization of some indices related to the mutual information do not work well when the sources signals are far from linear

computing the spectral function for singular sturm liouville problems algorithms for computing sturm liouville spectral density functions are developed based on several mathematical characterizations convergence and error bounds are derived and methods are tested on several examples the results are compared with those from the existing software package sledge

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

mutual information approach to blind separation of stationary sources this paper presents a unified approach to the problem of blind separation of sources based on the concept of mutual information this concept is applied to the whole source sequences as stationary processes and thus provides a universal contrast applicable to both the instantaneous and convolutive mixture cases for practical implementation we introduce several degraded forms of this contrast computable from a finite dimensional distribution of the reconstructed source processes only from them we derive several sets of estimating equations generalizing those considered earlier

an expectation maximization method for spatio temporal blind source separation using an ar mog source model in this paper we develop a maximum likelihood ml spatio temporal blind source separation bss algorithm where the temporal dependencies are explained by assuming that each source is an autoregressive ar process and the distribution of the associated independent identically distributed i i d innovations process is described using a mixture of gaussians unlike most ml methods the proposed algorithm takes into account both spatial and temporal information optimization is performed using the expectation maximization em method the source model is adapted to maximize the likelihood and the update equations have a simple analytical form the proposed method which we refer to as autoregressive mixture of gaussians ar mog outperforms nine other methods for artificial mixtures of real audio we also show results for using ar mog to extract the fetal cardiac signal from real magnetocardiographic mcg data

blind separation of mixture of independent sources through a quasi maximum likelihood approach we propose two methods for separating mixture of independent sources without any precise knowledge of their probability distribution they are obtained by considering a maximum likelihood ml solution corresponding to some given distributions of the sources and relaxing this assumption afterward the first method is specially adapted to temporally independent non gaussian sources and is based on the use of nonlinear separating functions the second method is specially adapted to correlated sources with distinct spectra and is based on the use of linear separating filters a theoretical analysis of the performance of the methods has been made a simple procedure for optimally choosing the separating functions is proposed further in the second method a simple implementation based on the simultaneous diagonalization of two symmetric matrices is provided finally some numerical and simulation results are given illustrating the performance of the method and the good agreement between the experiments and the theory

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

beyond independent components trees and clusters we present a generalization of independent component analysis ica where instead of looking for a linear transform that makes the data components independent we look for a transform that makes the data components well fit by a tree structured graphical model this tree dependent component analysis tca provides a tractable and flexible approach to weakening the assumption of independence in ica in particular tca allows the underlying graph to have multiple connected components and thus the method is able to find clusters of components such that components are dependent within a cluster and independent between clusters finally we make use of a notion of graphical models for time series due to brillinger 1996 to extend these ideas to the temporal setting in particular we are able to fit models that incorporate tree structured dependencies among multiple time series

a simple geometric blind source separation method for bounded magnitude sources a novel blind source separation approach and the corresponding adaptive algorithm is presented it is assumed that the observation mixture is obtained through an unknown memoryless linear mapping of independent and bounded magnitude sources we further assume an initial adaptive prewhitening of the original observation vector which transforms it into a white vector with the same dimension as the original source vector our approach is centered around the basic geometric fact that under a certain boundedness assumption the unitary mapping which transforms the whitening output vector into an independent vector has the minimum value of maximum real component magnitude output over the ensemble of all output components therefore the related criterion is the minimization of the infinity norm of the real component of the unitary separator s output over all possible output combinations for the minimization of the corresponding nondifferentiable cost function we propose the use of subgradient optimization methods to obtain a low complexity iterative adaptive solution the resulting algorithm is fairly intuitive and simple and provides a low complexity solution especially to a class of multiuser digital communications problems we provide examples at the end of this paper to illustrate the performance of our algorithm

markovian source separation a maximum likelihood ml approach is used to separate the instantaneous mixtures of temporally correlated independent sources with neither preliminary transformation nor a priori assumption about the probability distribution of the sources a markov model is used to represent the joint probability density of successive samples of each source the joint probability density functions are estimated from the observations using a kernel method for the special case of autoregressive models the theoretical performance of the algorithm is computed and compared with the performance of second order algorithms and i i d based separation algorithms

multichannel blind deconvolution of nonminimum phase systems using filter decomposition in this paper we present a new filter decomposition method for multichannel blind deconvolution of nonminimum phase systems with this approach we decompose a doubly finite impulse response filter into a cascade form of two filters a causal finite impulse response fir filter and an anticausal fir filter after introducing a lie group to the manifold of fir filters we discuss geometric properties of the fir filter manifold using the nonholonomic transform we derive the natural gradient on the fir manifold by simplifying the mutual information rate we present a very simple cost function for blind deconvolution of nonminimum phase systems subsequently the natural gradient algorithms are developed both for the causal fir filter and for the anticausal fir filter simulations are presented to illustrate the validity and favorable learning performance of the proposed algorithms

a note on stone s conjecture of blind signal separation stone s method is one of the novel approaches to the blind source separation bss problem and is based on stone s conjecture however this conjecture has not been proved we present a simple simulation to demonstrate that stone s conjecture is incorrect we then modify stone s conjecture and prove this modified conjecture as a theorem which can be used a basis for bss algorithms

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

global solution for a generalized boussinesq equation this paper considers the existence and uniqueness of the solution to the cauchy problem for a class of boussinesq equation proves the global existence and finite time blowup of the solution to the problem by the potential well method

holder continuity of absolutely continuous spectral measure for multi frequency schrodinger operators abstract we establish sharp results on the modulus of continuity of the distribution of the spectral measure for multi frequency schrodinger operators with diphantine frequencies and small analytic potentials

uniqueness analysis of non unitary matrix joint diagonalization matrix joint diagonalization mjd is a powerful approach for solving the blind source separation bss problem it relies on the construction of matrices which are diagonalized by the unknown demixing matrix their joint diagonalizer serves as a correct estimate of this demixing matrix only if it is uniquely determined thus a critical question is under what conditions is a joint diagonalizer unique in the present work we fully answer this question about the identifiability of mjd based bss approaches and provide a general result on uniqueness conditions of matrix joint diagonalization it unifies all existing results which exploit the concepts of non circularity non stationarity non whiteness and non gaussianity as a corollary we propose a solution for complex bss which can be formulated in closed form in terms of an eigen and a singular value decomposition of two matrices