testing for stochastic independence application to blind source separation in this paper we address the issue of testing for stochastic independence and its application as a guide to selecting the standard independent component analysis ica algorithms when solving blind source separation bss problems our investigation focuses on the problem of establishing tests for the quality of separation among recovered sources obtained by ica algorithms in an unsupervised environment we review existing tests and propose two contingency table based algorithms the first procedure is based on the measure of goodness of fit of the observed signals to the model of independence provided by the power divergence pd family of test statistics we provide conditions that guarantee the validity of the independence test when the individual sources are nonstationary when the sources exhibit significant time dependence we show how to adopt hotelling s t sup 2 test statistic for zero mean to create an accurate test of independence experimental results obtained from a variety of synthetic and real life benchmark data sets confirm the success of the pd based test when the individual source samples preserve the so called constant cell probability assumption as well as the validity of the t sup 2 based test for sources with significant time dependence

on the finite amplitudes for open graphs in abelian dynamical colored boulatov ooguri models in the work ben geloun and bonzom 2011 int j theor phys 50 2819 it has been proved that the radiative corrections of the 2 point function in the su 2 boulatov tensor model generates a marginal in the renormalization group sense contribution of the form of a laplacian such a term which was missing in the initial boulatov model action should be added in that action before discussing the renormalization analysis of this model in this work by linearizing the group manifold we prove that the amplitudes associated with feynman graphs with external legs of the colored boulatov model over u 1 3 endowed with a laplacian dynamics are all convergent we conjecture that the same feature happens for the corresponding boulatov model over su 2 higher rank models are also discussed

norm estimations for the moore penrose inverse of multiplicative perturbations of matrices abstract let b be a multiplicative perturbation of a c m n given by b d 1 a d 2 where d 1 c m m and d 2 c n n are both nonsingular new norm upper bounds for b a f and b a 2 are derived where a and b are the moore penrose inverses of a and b respectively the main results of meng and zheng 2015 10 are improved in the cases of the frobenius norm and the spectral norm

on the perturbation of the moore penrose inverse of a matrix abstract the moore penrose inverse of a matrix has been extensively investigated and widely applied in many fields over the past decades one reason for the interest is that the moore penrose inverse can succinctly express some important geometric constructions in finite dimensional spaces such as the orthogonal projection onto a subspace and the linear least squares problem in this paper we establish new perturbation bounds for the moore penrose inverse under the frobenius norm some of which are sharper than the existing ones

ward takahashi identities for the colored boulatov model ward takahashi wt identities of the colored boulatov model are derived using a generic unitary field transformation in a specific instance this generic transformation turns out to be a symmetry of the interaction so that particular classes of reduced wt identities for that symmetry are consequently identified and interpreted

perturbation bounds of generalized inverses let complex matrices a and b have the same sizes we characterize the generalized inverse matrix b 1 i called an 1 i inverse of b for each i 3 and 4 such that the distance between a given 1 i inverse of a matrix a and the set of all 1 i inverses of the matrix b reaches minimum under 2 norm spectral norm and frobenius norm similar problems are also studied for 1 2 i inverse in practice the matrix b is often considered as the perturbed matrix of a and hence based on the previous results the additive perturbation bounds for the 1 i and 1 2 i inverses and multiplicative perturbation bounds for the 1 1 i and 1 2 i inverses are proposed numerical examples show that these multiplicative perturbation bounds can be achieved respective under 2 norm and frobenius norm

generalised conservation laws in non local field theories we propose a geometrical treatment of symmetries in non local field theories where the non locality is due to a lack of identification of field arguments in the action we show that the existence of a symmetry of the action leads to a generalized conservation law in which the usual conserved current acquires an additional non local correction term obtaining a generalization of the standard noether theorem we illustrate the general formalism by discussing the specific physical example of complex scalar field theory of the type describing the hydrodynamic approximation of bose einstein condensates we expect our analysis and results to be of particular interest for the group field theory formulation of quantum gravity

a new relative perturbation theorem for singular subspaces abstract this note addresses the sensitivity of singular subspaces of a matrix under relative perturbations it employs a new technique of separating a multiplicative perturbation d into two components one is the distance of a scalar multiple of d to the nearest unitary matrix q and the other is the distance of q to the identity consequently the new bounds reflect the intrinsic differences in how left and right multiplicative perturbations affect left and right singular subspaces

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

renormalizable models in rank d geq 2 tensorial group field theory classes of renormalizable models in the tensorial group field theory framework are investigated the rank d tensor fields are defined over d copies of a group manifold g d u 1 d or g d su 2 d with no symmetry and no gauge invariance assumed on the fields in particular we explore the space of renormalizable models endowed with a kinetic term corresponding to a sum of momenta of the form p 2a a in 0 1 this study is tailored for models equipped with laplacian dynamics on gd case a 1 but also for more exotic nonlocal models in quantum topology case 0 a 1 a generic model can be written dim g d phi k d a where k is the maximal valence of its interactions using a multi scale analysis for the generic situation we identify several classes of renormalizable actions including matrix model actions in this specific instance we find a tower of renormalizable matrix models parametrized by k geq 4 in a second part of this work we study the uv behavior of the models up to maximal valence of interaction k 6 all rank d geq 3 tensor models proved renormalizable are asymptotically free in the uv all matrix models with k 4 have a vanishing function at one loop and very likely reproduce the same feature of the grosse wulkenhaar model commun math phys 256 305 2005

additive and multiplicative perturbation bounds for the moore penrose inverse abstract in this paper we obtain the additive and multiplicative perturbation bounds for the moore penrose inverse under the unitarily invariant norm and the q norm which improve the corresponding ones in p a wedin perturbation theory for pseudo inverses bit 13 1973 217 232

blind separation of instantaneous mixtures of nonstationary sources most source separation algorithms are based on a model of stationary sources however it is a simple matter to take advantage of possible nonstationarities of the sources to achieve separation this paper develops novel approaches in this direction based on the principles of maximum likelihood and minimum mutual information these principles are exploited by efficient algorithms in both the off line case via a new joint diagonalization procedure and in the on line case via a newton like procedure some experiments showing the good performance of our algorithms and evidencing an interesting feature of our methods are presented their ability to achieve a kind of super efficiency the paper concludes with a discussion contrasting separating methods for non gaussian and nonstationary models and emphasizing that as a matter of fact what makes the algorithms work is strictly speaking not the nonstationarity itself but rather the property that each realization of the source signals has a time varying envelope

on computing accurate singular values and eigenvalues of matrices with acyclic graphs abstract it is known that small relative perturbations in the entries of a bidiagonal matrix only cause small relative perturbations in its singular values independentof the values of the matrix entries in this paper we show that a matrix has this property if and only if its associated bipartite graph is acyclic we also show how to compute the singular values of such a matrix to high relative accuracy the same algorithm can compute eigenvalues of symmetric matrices with acyclic graphs with tiny componentwise relative backward error this class includes tridiagonal matrices arrow matrices and exponentially many others

the optimal perturbation bounds of the moore penrose inverse under the frobenius norm abstract we obtain the optimal perturbation bounds of the moore penrose inverse under the frobenius norm by using singular value decomposition which improved the results in the earlier paper p a wedin perturbation theory for pseudo inverses bit 13 1973 217 232 in addition a perturbation bound of the moore penrose inverse under the frobenius norm in the case of the multiplicative perturbation model is also given

computing the singular value decomposition with high relative accuracy we analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy this means that each computed singular value is guaranteed to have some correct digits even if the singular values have widely varying magnitudes this is in contrast to the absolute accuracy provided by conventional backward stable algorithms which in general only guarantee correct digits in the singular values with large enough magniturds it is of interest to compute the tiniest singular values with several correct digits because in some cases such as finite element problems and quantum mechanics it is the smallest singular values that have physical meaning and should be determined accurately by the data many recent papers have identified special classes of matrices where high relative accuracy is possible since it is not possible in general the perturbation theory and algorithms for these matrix classes have been quite different motivating us to seek a common perturbation theory and common algorithm we provide these in this paper and show that high relative accuracy is possible in many new cases as well the briefest way to describe our results is that we can compute the svd to high relative accuracy provided we can compute a high accuracy pivoted ldu decomposition we provide many examples of matrix classes permitting such an ldu decomposition

a note on the perturbation bounds of eigenspaces for hermitian matrices let h be a hermitian matrix and h be its perturbed matrix in this paper both additive and multiplicative perturbations of eigenspace are studied and some absolute and relative bounds are presented which are better than the existing bounds in some sense

renormalization of tensorial group field theories abelian u 1 models in four dimensions we tackle the issue of renormalizability for tensorial group field theories tgft including gauge invariance conditions with the rigorous tool of multi scale analysis to prepare the ground for applications to quantum gravity models in the process we define the appropriate generalization of some key qft notions including connectedness locality and contraction of high subgraphs we also define a new notion of wick ordering corresponding to the subtraction of maximal melonic tadpoles we then consider the simplest examples of dynamical 4 dimensional tgft with gauge invariance conditions for the abelian u 1 case we prove that they are super renormalizable for any polynomial interaction

relative perturbation theory iv sin 2θ theorems abstract the double angle theorems of davis and kahan bound the change in an invariant subspace when a hermitian matrix a is subject to an additive perturbation a a a a this paper supplies analogous results when a is subject to a congruential or multiplicative perturbation a a d ad the relative gaps that appear in the bounds involve the spectrum of only one matrix either a or a in contrast to the gaps that appear in the single angle bounds the double angle theorems do not directly bound the difference between the old invariant subspace s and the new one s but instead bound the difference between s and its reflection j s where the mirror is s and j reverses s the orthogonal complement of s the double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix d def d 1 jdj note that d is invariant under the transformation d d for 0 whereas the single angle theorems give bounds proportional to d s departure from the identity and from orthogonality the corresponding results for the singular value problem when a nonsquare matrix b is perturbed to b d 1 bd 2 are also presented

relative perturbation theory iii more bounds on eigenvalue variation abstract classically the relative error in ga 1 as an approximation to is measured by relative error in ga a the quantity log10 is usually used for the number of correct decimal digits in numerical results although this measure is clearly not a metric since it lacks symmetry between and ga in part i of this series two other kinds of relative distances which have much better mathematical properties have been introduced and employed to establish theories it is shown that these different measurements are topologically equivalent however the measure is more convenient to use in practice in this part we established relative perturbation bounds directly using the classical measure the new bounds for diagonalizable matrices are cleaner than the corresponding ones in part i and yield nice bounds for hermitian matrices too but when applied to nonnegative definite hermitian matrices the new bounds are weaker than those in part i

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

relative perturbation techniques for singular value problems a technique is presented for deriving bounds on the relative change in the singular values of a real matrix or the eigenvalues of a real symmetric matrix due to a perturbation as well as bounds on the angles between the unperturbed and perturbed singular vectors or eigenvectors the class of perturbations considered consists of all delta b for which b delta b d l bd r for some nonsingular matrices d l and d r this class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off diagonal block in a block triangular matrix many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations also some new relative perturbation and deflation results for the singular values and vectors of biacyclic triangular and shifted triangular matrices are presented

methods for the analysis of contingency tables with large and small cell counts abstract the traditional practice has been to analyze contingency tables using asymptotic 2 approximations for the tail probability of certain test statistics even when the approximation is known to be poor for two dimensional tables pearson s x 2 statistic is most commonly used whereas for multidimensional tables the likelihood ratio test is preferred because it makes it easy to test series of nested hypotheses when asymptotic approximations are not adequate exact tests are preferred recent advances in algorithms have broadened the range of two dimensional tables for which satisfactory answers can be obtained baglivo olivier and pagano 1985 mehta and patel 1983 1986 pagano and taylor halvorsen 1981 in this article we propose algorithms that use a mixture of exact and asymptotic techniques to calculate tail probabilities our methods can be used with several different test statistics for two dimensional and multidimensional contingency table models and hence further broaden the set of tables

a bayesian independence test for small datasets we propose a bayesian test for independence among signals where only a small dataset is available traditional frequentist approaches often fail in this case due to inaccurate estimation of either the source statistical models or the threshold used by the test statistics in addition these frequentist methods cannot incorporate prior information into the computation of the test statistics our procedure renders parametric the nonparametric problem of testing for independence by quantizing the observed data samples into a table of cell counts the test statistic is based on the likelihood of the observed cell counts under the independence hypothesis where the marginal cell probabilities are modeled by independent symmetric dirichlet priors we apply our bayesian test to validate the solutions to the problem of blind source separation with small datasets using both synthetic and real life benchmark data the experimental results indicate that our approach can overcome the scarcity of data samples and significantly outperform the standard frequentist parametric methods with a proper selection of the prior parameters

an implementation of the dqds algorithm positive case abstract the dqds algorithm was introduced in 1994 to compute singular values of bidiagonal matrices to high relative accuracy but it may also be used to compute eigenvalues of tridiagonal matrices this paper discusses in detail the issues that have to be faced when the algorithm is to be realized on a computer criteria for accepting a value for splitting the matrix and for choosing a shift to reduce the number of iterations as well as the relative advantages of using ieee arithmetic when available ways to avoid unnecessary over underflows are described in addition some new formulae are developed to approximate the smallest eigenvalue from a twisted factorization of a matrix the results of extensive testing are presented at the end the list of contents is a valuable guide to the reader interested in specific features of the algorithm

implicit standard jacobi gives high relative accuracy we prove that the jacobi algorithm applied implicitly on a decomposition a xdx t of the symmetric matrix a where d is diagonal and x is well conditioned computes all eigenvalues of a to high relative accuracy the relative error in every eigenvalue is bounded by o epsilon kappa x where epsilon is the machine precision and kappa x equiv x 2 cdot x 1 2 is the spectral condition number of x the eigenvectors are also computed accurately in the appropriate sense we believe that this is the first algorithm to compute accurate eigenvalues of symmetric indefinite matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations

on perturbations of matrix pencils with real spectra perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil a ab with real spectrum are developed it is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices the applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of hermitian matrices such as the weyl lidskii theorem and the hoffman wielandt theorem and those for generalized eigenspaces are counterparts of the celebrated davis kahan sin 0 sin 20 theorems for the eigenspace variations of hermitian matrices the paper consists of two parts part i is for generalized eigenvalue perturbations while part ii deals with generalized eigenspace perturbations

multiplicative perturbation bounds for spectral and singular value decompositions let h be a hermitian matrix and email protected d hd be its perturbed matrix in this paper the multiplicative perturbations for both spectral decompositions and singular value decompositions are studied and some new perturbation bounds for these decompositions are presented our results improve some existing bounds

a robust approach to independent component analysis of signals with high level noise measurements we propose a robust approach for independent component analysis ica of signals where observations are contaminated with high level additive noise and or outliers the source signals may contain mixtures of both sub gaussian and super gaussian components and the number of sources is unknown our robust approach includes two procedures in the first procedure a robust prewhitening technique is used to reduce the power of additive noise the dimensionality and the correlation among sources a cross validation technique is introduced to estimate the number of sources in this first procedure in the second procedure a nonlinear function is derived using the parameterized t distribution density model this nonlinear function is robust against the undue influence of outliers fundamentally moreover the stability of the proposed algorithm and the robust property of misestimating the parameters kurtosis have been studied by combining the t distribution model with a family of light tailed distributions sub gaussian model we can separate the mixture of sub gaussian and super gaussian source components through the analysis of artificially synthesized data and real world magnetoencephalographic meg data we illustrate the efficacy of this robust approach

relative perturbation bounds for the subunitary polar factor under unitarily invariant norms let a be an m n complex matrix a decomposition a qh is said to be a generalized polar decomposition of a if q is an m n subunitary matrix and h is an n n hermitian positive semidefinite matrix in this paper some relative perturbation bounds of the subunitary polar factor under unitarily invariant norms are presented which are tighter in some sense compared with the absolute perturbation bounds obtained by li c r and so on

the perturbation bounds for eigenspaces of a definite matrix pair leta andb be hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b equiv mathop min limits left x right 1 x h a ib x 0 the perturbation bounds for eigenspaces of a definite pair on every unitary invariant matrix norm were obtained by imposing additional restrictions on the location of the generalized eigenvalues thus it gives a positive answer for an open question proposed by stewart 7 the famous davis kahan sin theorems and sin 2 theorem 2 can also be deduced from the present results

perturbation bounds for the generalized inverse at s 2 with respect to the frobenius norm abstract in this paper we present the perturbation bounds and the relative error bound of the generalized inverse a t s 2 with respect to the frobenius norm and apply the results to the relative errors of the solution of the general restricted linear equation

the lidskii mirsky wielandt theorem additive and multiplicative versions we use a simple matrix splitting technique to give an elementary new proof of the lidskii mirsky wielandt theorem and to obtain a multiplicative analog of the lidskii mirsky wielandt theorem which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of hermitian matrices and singular values of general matrices we apply our bound to obtain numerous bounds on the matching distance between the eigenvalues and singular values of matrices our results strengthen and generalize those in the literature

continuous point symmetries in group field theories we discuss the notion of symmetries in non local field theories characterized by integro differential equations of motion from a geometric perspective we then focus on group field theory gft models of quantum gravity and provide a general analysis of their continuous point symmetry transformations including the generalized conservation laws following from them

residual bounds for some or all singular values abstract for a given real or complex matrix we derive perturbation bounds for some or all singular values the bounds are of the same quality as corresponding residual bounds for the hermitian eigenvalue problem they are suitable for numerical computation

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

singularity wielandt s lemma and singular values in this study some upper and lower bounds for singular values of a general complex matrix are investigated according to singularity and wielandt s lemma of matrices especially some relationships between the singular values of the matrix a and its block norm matrix are established based on these relationships one may obtain the effective estimates for the singular values of large matrices by using the lower dimension norm matrices in addition a small error in piazza 2002 g piazza t politi an upper bound for the condition number of a matrix in spectral norm j comput appl math 143 1 2002 141 144 is also corrected some numerical experiments on saddle point problems show that these results are simple and sharp under suitable conditions