cramer rao induced bounds for candecomp parafac tensor decomposition this paper presents a cramer rao lower bound crlb on the variance of unbiased estimates of factor matrices in canonical polyadic cp or candecomp parafac cp decompositions of a tensor from noisy observations i e the tensor plus a random gaussian i i d tensor a novel expression is derived for a bound on the mean square angular error of factors along a selected dimension of a tensor of an arbitrary dimension the expression needs less operations for computing the bound o nr6 than the best existing state of the art algorithm o n3r6 operations where n and r are the tensor order and the tensor rank insightful expressions are derived for tensors of rank 1 and rank 2 of arbitrary dimension and for tensors of arbitrary dimension and rank where two factor matrices have orthogonal columns the results can be used as a gauge of performance of different approximate cp decomposition algorithms prediction of their accuracy and for checking stability of a given decomposition of a tensor condition whether the crlb is finite or not a novel expression is derived for a hessian matrix needed in popular damped gauss newton method for solving the cp decomposition of tensors with missing elements beside computing the crlb for these tensors the expression may serve for design of damped gauss newton algorithm for the decomposition

the analysis of multitrait multimethod matrices via constrained components analysis multitrait multimethod mtmm matrices are often analyzed by means of confirmatory factor analysis cfa however fitting mtmm models often leads to improper solutions or non convergence in an attempt to overcome these problems various alternative cfa models have been proposed but with none of these the problem of finding improper solutions was solved completely in the present paper an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed the approach is based on constrained variants of components analysis ca besides the fact that these methods do not give improper solutions they have the advantage that they provide component scores which can later on be used to relate the components to external variables the new methods are illustrated by means of simulated data as well as empirical data sets

the comparative efficacy of imputation methods for missing data in structural equation modeling abstract missing data is a problem that permeates much of the research being done today traditional techniques for replacing missing values may have serious limitations recent developments in computing allow more sophisticated techniques to be used this paper compares the efficacy of five current and promising methods that can be used to deal with missing data this efficacy will be judged by examining the percent of bias in estimating parameters the focus of this paper is on structural equation modeling sem a popular statistical technique which subsumes many of the traditional statistical procedures to make the comparison this paper examines a full structural equation model that is generated by simulation in accord with previous research the five techniques used for comparison are expectation maximization em full information maximum likelihood fiml mean substitution mean multiple imputation mi and regression imputation regression all of these techniques other than fiml impute missing data and result in a complete dataset that can be used by researchers for other research fiml on the other hand can still estimate the parameters of the model the study involves two levels of sample size 100 and 500 and seven levels of incomplete data 2 4 8 12 16 24 and 32 missing completely at random after extensive bootstrapping and simulation the results indicate that fiml is a superior method in the estimation of most different types of parameters in a sem format furthermore mi is found to be superior in the estimation of standard errors multiple imputation mi also is an excellent estimator with the exception of datasets with over 24 missing information considering the fact that fiml is a direct method and does not actually impute the missing data whereas mi does and can yield a complete set of data for the researcher to analyze we conclude that mi because of its theoretical and distributional underpinnings is probably most promising for future applications in this field

the heteroscedastic graded response model with a skewed latent trait testing statistical and substantive hypotheses related to skewed item category functions the graded response model grm samejima estimation of ability using a response pattern of graded scores psychometric monograph no 17 richmond va the psychometric society 1969 can be derived by assuming a linear regression of a continuous variable z on the trait to underlie the ordinal item scores takane de leeuw in psychometrika 52 393 408 1987 traditionally a normal distribution is specified for z implying homoscedastic error variances and a normally distributed in this paper we present the heteroscedastic grm with skewed latent trait which extends the traditional grm by incorporation of heteroscedastic error variances and a skew normal latent trait an appealing property of the extended grm is that it includes the traditional grm as a special case this enables specific tests on the normality assumption of z we show how violations of normality in z can lead to asymmetrical category response functions the ability to test this normality assumption is beneficial from both a statistical and substantive perspective in a simulation study we show the viability of the model and investigate the specificity of the effects we apply the model to a dataset on affect and a dataset on alexithymia

blind multiuser detection from moe to subspace methods the minimum output energy moe multiuser receiver has been shown to approach the minimum mean square error mmse receiver at high signal to noise ratio snr however performance degradation is incurred by noise induced channel estimation error in this paper we propose a power of r por technique to significantly improve the performance of the moe receiver it is shown that the new receiver asymptotically converges to the mmse receiver without performance penalty the convergence is established either under high snr with large exponent raised in the power of the covariance matrix or with sufficiently large number of data samples connection between our por method and a widely studied subspace method is investigated from the respective optimization criteria asymptotic equivalence between these two methods is also established extensive simulations based on finite data samples show that the proposed method significantly outperforms the subspace method in systems with medium to heavy loading severe multipath distortion or smaller processing gain moreover adaptive implementation of the proposed method exhibits very robust performance in a dynamic loading environment

kruskal s uniqueness inequality is sharp abstract kruskal proved that a tensor in v 1 v 2 v m of rank r has a unique decomposition as a sum of r pure tensors if a certain inequality is satisfied we will show the uniqueness fails if the inequality is weakened we give two different constructions for counterexamples

simplicity transformations for three way arrays with symmetric slices and applications to tucker 3 models with sparse core arrays abstract tucker three way pca and candecomp parafac are two well known methods of generalizing principal component analysis to three way data candecomp parafac yields component matrices a e g for subjects or objects b e g for variables and c e g for occasions that are typically unique up to jointly permuting and rescaling columns tucker 3 analysis on the other hand has full transformational freedom that is the fit does not change when a b and c are postmultiplied by nonsingular transformation matrices provided that the inverse transformations are applied to the so called core array g this freedom of transformation can be used to create a simple structure in a b c and or in g this paper deals with the latter possibility exclusively it revolves around the question of how a core array or in fact any three way array can be transformed to have a maximum number of zero elements direct applications are in tucker 3 analysis where simplicity of the core may facilitate the interpretation of a tucker 3 solution and in constrained tucker 3 analysis where hypotheses involving sparse cores are taken into account in the latter cases it is important to know what degree of sparseness can be attained as a tautology by using the transformational freedom in addition simplicity transformations have proven useful as a mathematical tool to examine rank and generic or typical rank of three way arrays so far a number of simplicity results have been attained pertaining to arrays sampled randomly from continuous distributions these results do not apply to three way arrays with symmetric slices in one direction the present paper offers a number of simplicity results for arrays with symmetric slices of order 2 2 3 3 and 4 4 some generalizations to higher orders are also discussed as a mathematical application the problem of determining the typical rank of 4 3 3 and 5 3 3 arrays with symmetric slices will be revisited using a sparse form with only 8 out of 36 elements nonzero for the former case and 10 out of 45 elements nonzero for the latter one that can be attained almost surely for such arrays the issue of maximal simplicity of the targets to be presented will be addressed either by formal proofs or by relying on simulation results

new als methods with extrapolating search directions and optimal step size for complex valued tensor decompositions in signal processing data analysis and scientific computing one often encounters the problem of decomposing a tensor into a sum of contributions to solve such problems both the search direction and the step size are two crucial elements in numerical algorithms such as alternating least squares algorithm als owing to the nonlinearity of the problem the often used linear search direction is not always powerful enough in this paper we propose two higher order search directions the first one geometric search direction is constructed via a combination of two successive linear directions the second one algebraic search direction is constructed via a quadratic approximation of three successive iterates then in an enhanced line search along these directions the optimal complex step size contains two arguments modulus and phase a current strategy is elscs that finds these two arguments alternately so it may suffer from a local optimum we broach a direct method which determines these two arguments simultaneously so as to obtain the global optimum finally numerical comparisons on various search direction and step size schemes are reported in the context of blind separation equalization of convolutive ds cdma mixtures the results show that the new search directions have greatly improve the efficiency of als and the new step size strategy is competitive

a multitrait multimethod model with minimal assumptions a new model of confirmatory factor analysis cfa for multitrait multimethod mtmm data sets is presented it is shown that this model can be defined by only three assumptions in the framework of classical psychometric test theory ctt all other properties of the model particularly the uncorrelated ness of the trait with the method factors are logical consequences of the definition of the model in the model proposed there are as many trait factors as different traits considered but the number of method factors is one fewer than the number of methods included in an mtmm study the covariance structure implied by this model is derived and it is shown that this model is identified even under conditions under which other cfa mtmm models are not the model is illustrated by two empirical applications furthermore its advantages and limitations are discussed with respect to previously developed cfa models for mtmm data

deterministic and probabilistic conditions for finite completability of low tucker rank tensor we investigate the fundamental conditions on the sampling pattern i e locations of the sampled entries for finite completability of a low rank tensor given some components of its tucker rank in order to find the deterministic necessary and sufficient conditions we propose an algebraic geometric analysis on the tucker manifold which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the grassmannian manifold this analysis characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern which is closely related to finite completability of the sampled tensor where finite completability simply means that the number of possible completions of the sampled tensor is finite probabilistic conditions are then studied and a lower bound on the sampling probability is given which guarantees that the proposed deterministic conditions on the sampling patterns for finite completability hold with high probability furthermore using the proposed geometric approach for finite completability we propose a sufficient condition on the sampling pattern that ensures there exists exactly one completion of the sampled tensor

imprimitive permutations in primitive groups abstract the goal of this paper is to study primitive groups that are contained in the union of maximal in the symmetric group imprimitive groups the study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups however this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult fortunately the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem in this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact in addition we introduce some algorithms to handle permutations provide the corresponding gap implementation solve some open problems and propose a large list of open problems

two generalizations of homogeneity in groups with applications to regular semigroups let x be a finite set such that x n and let i 6 j 6 n a group g 6 sn is said to be i j homogeneous if for every i j x such that i i and j j there exists g g such that ig j clearly i i homogeneity is i homogeneity in the usual sense a group g 6 sn is said to have the k universal transversal property if given any set i x with i k and any partition p of x into k blocks there exists g g such that ig is a section for p that is the orbit of each k subset of x contains a section for each k partition of x in this paper we classify the groups with the k universal transversal prop erty with the exception of two classes of 2 homogeneous groups and the k 1 k homogeneous groups for 2 k 6 n 1 as a corollary of the clas sification we prove that a k 1 k homogeneous group is also k 2 k 1 homogeneous with two exceptions and similarly but with no exceptions groups having the k universal transversal property have the k 1 universal transversal property a corollary of all the previous results is a classification of the groups that together with any rank k transformation on x generate a regular semigroup for 1 6 k 6 n 1 the paper ends with a number of challenges for experts in number theory group and or semigroup theory linear algebra and matrix theory

generalized structured component analysis we propose an alternative method to partial least squares for path analysis with components called generalized structured component analysis the proposed method replaces factors by exact linear combinations of observed variables it employs a well defined least squares criterion to estimate model parameters as a result the proposed method avoids the principal limitation of partial least squares i e the lack of a global optimization procedure while fully retaining all the advantages of partial least squares e g less restricted distributional assumptions and no improper solutions the method is also versatile enough to capture complex relationships among variables including higher order components and multi group comparisons a straightforward estimation algorithm is developed to minimize the criterion

candecomp parafac decomposition of high order tensors through tensor reshaping in general algorithms for order 3 candecomp parafac cp also coined canonical polyadic decomposition cpd are easy to implement and can be extended to higher order cpd unfortunately the algorithms become computationally demanding and they are often not applicable to higher order and relatively large scale tensors in this paper by exploiting the uniqueness of cpd and the relation of a tensor in kruskal form and its unfolded tensor we propose a fast approach to deal with this problem instead of directly factorizing the high order data tensor the method decomposes an unfolded tensor with lower order e g order 3 tensor on the basis of the order 3 estimated tensor a structured kruskal tensor of the same dimension as the data tensor is then generated and decomposed to find the final solution using fast algorithms for the structured cpd in addition strategies to unfold tensors are suggested and practically verified in the paper

component analysis with different sets of constraints on different dimensions many of the classical multivariate data analysis and multidimensional scaling techniques call for approximations by lower dimensional configurations a model is proposed in which different sets of linear constraints are imposed on different dimensions in component analysis and classical multidimensional scaling frameworks a simple efficient and monotonically convergent algorithm is presented for fitting the model to the data by least squares the basic algorithm is extended to cover across dimension constraints imposed in addition to the dimensionwise constraints and to the case of a symmetric data matrix examples are given to demonstrate the use of the method

degeneracy in candecomp parafac explained for p p 2 arrays of rank p 1 or higher the candecomp parafac cp model decomposes a three way array into a prespecified number r of rank 1 arrays and a residual array in which the sum of squares of the residual array is minimized the practical use of cp is sometimes complicated by the occurrence of so called degenerate solutions in which some components are highly correlated in all three modes and the elements of these components become arbitrarily large we consider the real valued cp model in which p p 2 arrays of rank p 1 or higher are decomposed into p rank 1 arrays and a residual array it is shown that the cp objective function does not have a minimum in these cases but an infimum moreover any sequence of cp approximations of which the objective value approaches the infimum will become degenerate this result extends ten berge kiers de leeuw 1988 who consider a particular 2 2 2 array of rank 3

generalized structured component analysis with latent interactions generalized structured component analysis gsca is a component based approach to structural equation modeling in practice researchers may often be interested in examining the interaction effects of latent variables however gsca has been geared only for the specification and testing of the main effects of variables thus an extension of gsca is proposed to effectively deal with various types of interactions among latent variables in the proposed method a latent interaction is defined as a product of interacting latent variables as a result this method does not require the construction of additional indicators for latent interactions moreover it can easily accommodate both exogenous and endogenous latent interactions an alternating least squares algorithm is developed to minimize a single optimization criterion for parameter estimation a monte carlo simulation study is conducted to investigate the parameter recovery capability of the proposed method an application is also presented to demonstrate the empirical usefulness of the proposed method

on the non existence of optimal solutions and the occurrence of degeneracy in the candecomp parafac model the candecomp parafac cp model decomposes a three way array into a prespecified number of r factors and a residual array by minimizing the sum of squares of the latter it is well known that an optimal solution for cp need not exist we show that if an optimal cp solution does not exist then any sequence of cp factors monotonically decreasing the cp criterion value to its infimum will exhibit the features of a so called degeneracy that is the parameter matrices become nearly rank deficient and the euclidean norm of some factors tends to infinity we also show that the cp criterion function does attain its infimum if one of the parameter matrices is constrained to be column wise orthonormal

tensor space time frequency coding with semi blind receivers for mimo wireless communication systems in this paper we first introduce two new classes of constrained tensor models that we call generalized paratuck n1 n and tucker n1 n models a new tensor space time frequency tstf coding structure is then proposed for mimo ofdm cdma wireless communication systems two semi blind receivers relying on the new generalized paratuck model are derived for solving the problem of joint channel and symbol estimation one is iterative and based on a two step alternating least squares als algorithm the other one is a closed form and low complexity solution which consists of the kronecker product least squares kpls estimation of the symbol matrix and a matrix unfolding of the channel tensor uniqueness of the underlying tensor model is discussed and system design requirements are derived for applicability of the als and kpls receivers we also show that the so called tstf system can be viewed as an extension of three existing tensor based st tst stf coding systems that are described in a unified framework computer simulation results illustrate the good performance of the tstf system which outperforms the considered existing tensor based systems both in terms of symbol estimation quality and allocation flexibility

rank regularization and bayesian inference for tensor completion and extrapolation a novel regularizer of the parafac decomposition factors capturing the tensor s rank is proposed in this paper as the key enabler for completion of three way data arrays with missing entries set in a bayesian framework the tensor completion method incorporates prior information to enhance its smoothing and prediction capabilities this probabilistic approach can naturally accommodate general models for the data distribution lending itself to various fitting criteria that yield optimum estimates in the maximum a posteriori sense in particular two algorithms are devised for gaussian and poisson distributed data that minimize the rank regularized least squares error and kullback leibler divergence respectively the proposed technique is able to recover the ground truth tensor rank when tested on synthetic data and to complete brain imaging and yeast gene expression datasets with 50 and 15 of missing entries respectively resulting in recovery errors at 11 db and 15 db

a block component model based blind ds cdma receiver in this paper we consider the problem of blind multiuser separation equalization in the uplink of a wideband ds cdma system in a multipath propagation environment with intersymbol interference isi to solve this problem we propose a multilinear algebraic receiver that relies on a new third order tensor decomposition and generalizes the parallel factor parafac model our method is deterministic and exploits the temporal spatial and spectral diversities to collect the received data in a third order tensor the specific algebraic structure of this tensor is then used to decompose it in a sum of user s contributions the so called block component model bcm receiver does not require knowledge of the spreading codes the propagation parameters nor statistical independence of the sources but relies instead on a fundamental uniqueness condition of the decomposition that guarantees identifiability of every user s contribution the development of fast and reliable techniques to calculate this decomposition is important we propose a blind receiver based either on an alternating least squares als algorithm or on a levenberg marquardt lm algorithm simulations illustrate the performance of the algorithms

a concise proof of kruskal s theorem on tensor decomposition a theorem of j kruskal from 1977 motivated by a latent class statistical model established that under certain explicit conditions the expression of a third order tensor as the sum of rank 1 tensors is essentially unique we give a new proof of this fundamental result which is substantially shorter than both the original one and recent versions along the original lines

functional parallel factor analysis for functions of one and two dimensional arguments parallel factor analysis parafac is a useful multivariate method for decomposing three way data that consist of three different types of entities simultaneously this method estimates trilinear components each of which is a low dimensional representation of a set of entities often called a mode to explain the maximum variance of the data functional parafac permits the entities in different modes to be smooth functions or curves varying over a continuum rather than a collection of unconnected responses the existing functional parafac methods handle functions of a one dimensional argument e g time only in this paper we propose a new extension of functional parafac for handling three way data whose responses are sequenced along both a two dimensional domain e g a plane with x and y axis coordinates and a one dimensional argument technically the proposed method combines parafac with basis function expansion approximations using a set of piecewise quadratic finite element basis functions for estimating two dimensional smooth functions and a set of one dimensional basis functions for estimating one dimensional smooth functions in a simulation study the proposed method appeared to outperform the conventional parafac we apply the method to eeg data to demonstrate its empirical usefulness

a mortar finite element approach to lubricated contact problems abstract in this paper a new numerical approach based on mortar discretization of interface fields is proposed to solve lubricated contact problems between deformable solid bodies the reynolds approximation to the thin film fluid equations is used to describe the fluid behavior on the interface while the equations of small strain elasticity to which finite elements are applied are assumed to govern the solid body behavior the result of this approach is a monolithic scheme in which the fluid and solid mechanics are solved simultaneously without the need for iterative or staggered approaches between the fluid and solid phases key features of the approach are that the fluid film thickness directly related to the deformation of the solid phase is computed from a least squares projection based on dual basis functions as pioneered by wohlmuth b i wohlmuth discretization methods and iterative solvers based on domain decomposition springer verlag heidelberg 2001 the free boundary problem for the fluid phase corresponding to the boundary between lubricated and cavitated regions is regularized with a penalty method and thus resolved as part of the solution several numerical examples are given to demonstrate the approach

khatri rao space time codes space time st coding techniques exploit the spatial diversity afforded by multiple transmit and receive antennas to achieve reliable transmission in scattering rich environments st block codes are capable of realizing full diversity and spatial coding gains at relatively low rates st trellis codes can achieve better rate diversity tradeoffs at the cost of high complexity on the other hand v blast supports high rates but has no built in spatial coding and does not work well with fewer receive than transmit antennas we propose a novel linear block coding scheme based on the khatri rao matrix product the proposed scheme offers flexibility for achieving full rate or full diversity or a desired rate diversity tradeoff and it can handle any transmit receive antenna configuration or signal constellation the proposed codes are shown to have numerous desirable properties including guaranteed unique linear decodability built in blind channel identifiability and efficient near maximum likelihood decoding

decomposition of homogeneous polynomials with low rank let f be a homogeneous polynomial of degree d in m 1 variables defined over an algebraically closed field of characteristic 0 and suppose that f belongs to the s th secant variety of the d uple veronese embedding of mathbb p m into mathbb p tiny left begin array c rm m d rm d end array right 1 but that its minimal decomposition as a sum of d th powers of linear forms m1 mr is f m 1 d cdots m r d with r s we show that if s r 2d 1 then such a decomposition of f can be split in two parts one of them is made by linear forms that can be written using only two variables the other part is uniquely determined once one has fixed the first part we also obtain a uniqueness theorem for the minimal decomposition of f if r is at most d and a mild condition is satisfied

on uniqueness in candecomp parafac one of the basic issues in the analysis of three way arrays by candecomp parafac cp has been the question of uniqueness of the decomposition kruskal 1977 has proved that uniqueness is guaranteed when the sum of thek ranks of the three component matrices involved is at least twice the rank of the solution plus 2 since then little has been achieved that might further qualify kruskal s sufficient condition attempts to prove that it is also necessary for uniqueness except for rank 1 or 2 have failed but counterexamples to necessity have not been detected the present paper gives a method for generating the class of all solutions or at least a subset of that class given a cp solution that satisfies certain conditions this offers the possibility to examine uniqueness for a great variety of specific cp solutions it will be shown that kruskal s condition is necessary and sufficient when the rank of the solution is three but that uniqueness may hold even if the condition is not satisfied when the rank is four or higher

an efficient algorithm for tuckals3 on data with large numbers of observation units a modification of the tuckals3 algorithm is proposed that handles three way arrays of order i j k for any i when i is much larger than jk the modified algorithm needs less work space to store the data during the iterative part of the algorithm than does the original algorithm because of this and the additional feature that execution speed is higher the modified algorithm is highly suitable for use on personal computers

blind parafac receivers for ds cdma systems this paper links the direct sequence code division multiple access ds cdma multiuser separation equalization detection problem to the parallel factor parafac model which is an analysis tool rooted in psychometrics and chemometrics exploiting this link it derives a deterministic blind parafac ds cdma receiver with performance close to non blind minimum mean squared error mmse the proposed parafac receiver capitalizes on code spatial and temporal diversity combining thereby supporting small sample sizes more users than sensors and or less spreading than users interestingly parafac does not require knowledge of spreading codes the specifics of multipath interchip interference doa calibration information finite alphabet constant modulus or statistical independence whiteness to recover the information bearing signals instead parafac relies on a fundamental result regarding the uniqueness of low rank three way array decomposition due to kruskal 1977 1988 and generalized herein to the complex valued case that guarantees identifiability of all relevant signals and propagation parameters these and other issues are also demonstrated in pertinent simulation experiments

uni mode uniqueness conditions for candecomp parafac decomposition of n way arrays with linearly dependent loadings abstract guo miron brie and stegeman x guo s miron d brie a stegeman uni mode and partial uniqueness conditions for candecomp parafac of three way arrays with linearly dependent loadings siam j matrix anal appl 33 2012 111 129 give three sufficient conditions for the three way candecomp parafac cp model which ensure uniqueness in one of the three modes uni mode uniqueness in this paper we generalize these uniqueness conditions to n 3 based on these conditions a partial uniqueness condition is given which allows collinear loadings in only one mode

tensor decomposition for signal processing and machine learning tensors or multiway arrays are functions of three or more indices i j k ldots similar to matrices two way arrays which are functions of two indices r c for row column tensors have a rich history stretching over almost a century and touching upon numerous disciplines but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing statistics data mining and machine learning this overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors as such it focuses on fundamentals and motivation using various application examples aiming to strike an appropriate balance of breadth and depth that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and or developing tensor algorithms and software some background in applied optimization is useful but not strictly required the material covered includes tensor rank and rank decomposition basic tensor factorization models and their relationships and properties including fairly good coverage of identifiability broad coverage of algorithms ranging from alternating optimization to stochastic gradient statistical performance analysis and applications ranging from source separation to collaborative filtering mixture and topic modeling classification and multilinear subspace learning

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

kruskal s permutation lemma and the identification of candecomp parafac and bilinear models with constant modulus constraints candecomp parafac cp analysis is an extension of low rank matrix decomposition to higher way arrays which are also referred to as tensors cp extends and unifies several array signal processing tools and has found applications ranging from multidimensional harmonic retrieval and angle carrier estimation to blind multiuser detection the uniqueness of cp decomposition is not fully understood yet despite its theoretical and practical significance toward this end we first revisit kruskal s permutation lemma which is a cornerstone result in the area using an accessible basic linear algebra and induction approach the new proof highlights the nature and limits of the identification process we then derive two equivalent necessary and sufficient uniqueness conditions for the case where one of the component matrices involved in the decomposition is full column rank these new conditions explain a curious example provided recently in a previous paper by sidiropoulos who showed that kruskal s condition is in general sufficient but not necessary for uniqueness and that uniqueness depends on the particular joint pattern of zeros in the possibly pretransformed component matrices as another interesting application of the permutation lemma we derive a similar necessary and sufficient condition for unique bilinear factorization under constant modulus cm constraints thus providing an interesting link to and unification with cp

representation of measurement error in marketing variables review of approaches and extension to three facet designs this paper explores approaches for modeling measurement error in marketing research including random method and measure specific sources of error the following approaches are considered classic confirmatory factor analysis second order models panel models additive trait method models correlated uniqueness models covariance components analysis additive trait method measure specific error models and the direct product model where traits and methods interact finally a three facet multiplicative model is addressed wherein latent variables underlying a phenomenon under investigation are shown to interact with multiple methods and occasions of measurement the three facet model is illustrated on a study of consumer attitudes toward losing weight explicitly conducted for this paper

high order tensor estimation via trains of coupled third order cp and tucker decompositions abstract in this work equivalence relations between a tensor train tt decomposition and the canonical polyadic decomposition cpd tucker decomposition td are investigated it is shown that a q order tensor following a cpd td with q 3 can be written using the graph based formalism as a train of q tensors of order at most 3 following the same decomposition as the initial q order tensor this means that for any practical problem of interest involving the cpd td there exists an equivalent tt based formulation this equivalence allows us to overcome the curse of dimensionality when dealing with the big data tensors in this paper it is shown that the native difficult optimization problems for cpd td of q order tensors can be efficiently solved using the tt decomposition according to flexible strategies that involve q 2 optimization problems with 3 order tensors this methodology hence involves a number of free parameters linear with q and thus allows to mitigate the exponential growth of parameters for q order tensors then by capitalizing on the tt decomposition we also formulate several robust and fast algorithms to accomplish joint dimensionality reduction and factors retrieval jirafe for the cpd td in particular based on the tt svd algorithm we show how to exploit the existing coupling between two successive tt cores in the graph based formalism the advantages of the proposed solutions in terms of storage cost computational complexity and factor estimation accuracy are also discussed

three mode factor analysis by means of candecomp parafac a three mode covariance matrix contains covariances of n observations e g subject scores on j variables for k different occasions or conditions we model such an jk jk covariance matrix as the sum of a common covariance matrix having candecomp parafac form and a diagonal matrix of unique variances the candecomp parafac form is a generalization of the two mode case under the assumption of parallel factors we estimate the unique variances by minimum rank factor analysis the factors can be chosen oblique or orthogonal our approach yields a model that is easy to estimate and easy to interpret moreover the unique variances the factor covariance matrix and the communalities are guaranteed to be proper a percentage of explained common variance can be obtained for each variable condition combination and the estimated model is rotationally unique under mild conditions we apply our model to several datasets in the literature and demonstrate our estimation procedure in a simulation study

degeneracy in candecomp parafac and indscal explained for several three sliced arrays with a two valued typical rank the candecomp parafac cp method decomposes a three way array into a prespecified number r of rank 1 arrays by minimizing the sum of squares of the residual array the practical use of cp is sometimes complicated by the occurrence of so called degenerate sequences of solutions in which several rank 1 arrays become highly correlated in all three modes and some elements of the rank 1 arrays become arbitrarily large we consider the real valued cp decomposition of all known three sliced arrays i e of size p q 3 with a two valued typical rank these are the 5 3 3 and 8 4 3 arrays and the 3 3 4 and 3 3 5 arrays with symmetric 3 3 slices in the latter two cases cp is equivalent to the indscal model for a typical rank of m m 1 we consider the cp decomposition with r m of an array of rank m 1 we show that in most cases the cp objective function does not have a minimum but an infimum moreover any sequence of feasible cp solutions in which the objective value approaches the infimum will become degenerate we use the tools developed in stegeman 2006 who considers p p 2 arrays and present a framework of analysis which is of use to the future study of cp degeneracy related to a two valued typical rank moreover our examples show that cp uniqueness is not necessary for degenerate solutions to occur

blind identification of underdetermined mixtures by simultaneous matrix diagonalization in this paper we study simultaneous matrix diagonalization based techniques for the estimation of the mixing matrix in underdetermined independent component analysis ica this includes a generalization to underdetermined mixtures of the well known sobi algorithm the problem is reformulated in terms of the parallel factor decomposition parafac of a higher order tensor we present conditions under which the mixing matrix is unique and discuss several algorithms for its computation

accelerated canonical polyadic decomposition using mode reduction canonical polyadic decomposition candecomp cpd also known as parallel factor analysis parafac is widely applied to nth order n 3 tensor analysis existing cpd methods mainly use alternating least squares iterations and hence need to unfold tensors to each of their n modes frequently which is one major performance bottleneck for large scale data especially when the order n is large to overcome this problem in this paper we propose a new cpd method in which the cpd of a high order tensor i e is realized by applying cpd to a mode reduced one typically third order tensor followed by a khatri rao product projection procedure this way is not only quite efficient as frequently unfolding to n modes is avoided but also promising to conquer the bottleneck problem caused by high collinearity of components we show that under mild conditions any nth order cpd can be converted to an equivalent third order one but without destroying essential uniqueness and theoretically they simply give consistent results besides once the cpd of any unfolded lower order tensor is essentially unique it is also true for the cpd of the original higher order tensor error bounds of truncated cpd are also analyzed in the presence of noise simulations show that compared with state of the art cpd methods the proposed method is more efficient and is able to escape from local solutions more easily

synchronizing groups and automata pin showed that every p state automaton p a prime containing a cyclic permutation and a non permutation has a synchronizing word of length at most p 1 2 in this paper we consider permutation automata with the property that adding any nonpermutation will lead to a synchronizing word and establish bounds on the lengths of such synchronizing words in particular we show that permutation groups whose permutation character over the rationals splits into a sum of only two irreducible characters have the desired property

sufficient conditions for uniqueness in candecomp parafac and indscal with random component matrices a key feature of the analysis of three way arrays by candecomp parafac is the essential uniqueness of the trilinear decomposition we examine the uniqueness of the candecomp parafac and indscal decompositions in the latter the array to be decomposed has symmetric slices we consider the case where two component matrices are randomly sampled from a continuous distribution and the third component matrix has full column rank in this context we obtain almost sure sufficient uniqueness conditions for the candecomp parafac and indscal models separately involving only the order of the three way array and the number of components in the decomposition both uniqueness conditions are closer to necessity than the classical uniqueness condition by kruskal

methods for mediation analysis with missing data despite wide applications of both mediation models and missing data techniques formal discussion of mediation analysis with missing data is still rare we introduce and compare four approaches to dealing with missing data in mediation analysis including listwise deletion pairwise deletion multiple imputation mi and a two stage maximum likelihood ts ml method an r package bmem is developed to implement the four methods for mediation analysis with missing data in the structural equation modeling framework and two real examples are used to illustrate the application of the four methods the four methods are evaluated and compared under mcar mar and mnar missing data mechanisms through simulation studies both mi and ts ml perform well for mcar and mar data regardless of the inclusion of auxiliary variables and for av mnar data with auxiliary variables although listwise deletion and pairwise deletion have low power and large parameter estimation bias in many studied conditions they may provide useful information for exploring missing mechanisms

on insensitivity of the chi square model test to nonlinear misspecification in structural equation models in this paper we show that for some structural equation models sem the classical chi square goodness of fit test is unable to detect the presence of nonlinear terms in the model as an example we consider a regression model with latent variables and interactions terms not only the model test has zero power against that type of misspecifications but even the theoretical chi square distribution of the test is not distorted when severe interaction term misspecification is present in the postulated model we explain this phenomenon by exploiting results on asymptotic robustness in structural equation models the importance of this paper is to warn against the conclusion that if a proposed linear model fits the data well according to the chi quare goodness of fit test then the underlying model is linear indeed it will be shown that the underlying model may in fact be severely nonlinear in addition the present paper shows that such insensitivity to nonlinear terms is only a particular instance of a more general problem namely the incapacity of the classical chi square goodness of fit test to detect deviations from zero correlation among exogenous regressors either being them observable or latent when the structural part of the model is just saturated

adaptive algorithms to track the parafac decomposition of a third order tensor the parafac decomposition of a higher order tensor is a powerful multilinear algebra tool that becomes more and more popular in a number of disciplines existing parafac algorithms are computationally demanding and operate in batch mode both serious drawbacks for on line applications when the data are serially acquired or the underlying model changes with time adaptive parafac algorithms that can track the sought decomposition at low complexity would be highly desirable this is a challenging task that has not been addressed in the literature and the topic of this paper given an estimate of the parafac decomposition of a tensor at instant t we propose two adaptive algorithms to update the decomposition at instant t 1 the new tensor being obtained from the old one after appending a new slice in the time dimension the proposed algorithms can yield estimation performance that is very close to that obtained via repeated application of state of art batch algorithms at orders of magnitude lower complexity the effectiveness of the proposed algorithms is illustrated using a mimo radar application tracking of directions of arrival and directions of departure as an example

constrained tensor modeling approach to blind multiple antenna cdma schemes in this paper we consider an uplink multiple antenna code division multiple access cdma system linking several multiple antenna mobile users to one multiple antenna base station for this system a constrained third order tensor decomposition is introduced for modeling the multiple antenna transmitter as well as the received signal the constrained structure of the proposed tensor decomposition is characterized by two constraint matrices that have a meaningful physical interpretation in our context they can be viewed as canonical allocation matrices that define the allocation of users data streams and spreading codes to the transmit antennas the distinguishing features of the proposed tensor modeling with respect to the already existing tensor based cdma models are i it copes with multiple transmit antennas and spreading codes per user and ii it models several spatial spreading multiplexing schemes with multiple spreading codes by controlling the constrained structure of the tensor signal model a systematic design procedure for the canonical allocation matrices leading to a unique blind symbol or joint blind symbol code recovery is proposed which allows us to derive a finite set of multiple antenna schemes for a fixed number of transmit antennas identifiability of the proposed tensor model is discussed and a blind multiuser detection receiver based on the alternating least squares algorithm is considered for performance evaluation of several multiple antenna cdma schemes derived from the constrained modeling approach

finite primitive permutation groups containing a permutation having at most four cycles we classify the finite primitive groups containing a permutation with at most four cycles including fixed points in its disjoint cycle representation

identification with deficient rank loading matrices in confirmatory factor analysis multitrait multimethod models this paper presents some results on identification in multitrait multimethod mtmm confirmatory factor analysis cfa models some mtmm models are not identified when the factorial patterned loadings matrix is of deficient column rank for at least one other mtmm model identification does exist despite such deficiency it is also shown that for some mtmm cfa models howe s 1955 conditions sufficient for rotational uniqueness can fail yet the model may well be identified and rotationally unique implications of these results for cfa models in general are discussed

the direct product model for the mtmm matrix parameterized as a second order factor analysis model the composite direct product model for the multitrait multimethod matrix is reparameterized as a second order factor analysis model this facilitates the use of widely available computer programs such as lisrel and liscomp for fitting the model

moment testing for interaction terms in structural equation modeling starting with kenny and judd psychol bull 96 201 210 1984 several methods have been introduced for analyzing models with interaction terms in all these methods more information from the data than just means and covariances is required in this paper we also use more than just first and second order moments however we are aiming to adding just a selection of the third order moments the key issue in this paper is to develop theoretical results that will allow practitioners to evaluate the strength of different third order moments in assessing interaction terms of the model to select the third order moments we propose to be guided by the power of the goodness of fit test of a model with no interactions which varies with each selection of third order moments a theorem is presented that relates the power of the usual goodness of fit test of the model with the power of a moment test for the significance of third order moments the latter has the advantage that it can be computed without fitting a model the main conclusion is that the selection of third order moments can be based on the power of a moment test thus assessing the relevance in the analysis of different sets of third order moments can be computationally simple the paper gives an illustration of the method and argues for the need of refraining from adding into the analysis an excess of higher order moments

cramer rao lower bounds for low rank decomposition of multidimensional arrays unlike low rank matrix decomposition which is generically nonunique for rank greater than one low rank three and higher dimensional array decomposition is unique provided that the array rank is lower than a certain bound and the correct number of components equal to array rank is sought in the decomposition parallel factor parafac analysis is a common name for low rank decomposition of higher dimensional arrays this paper develops cramer rao bound crb results for low rank decomposition of three and four dimensional 3 d and 4 d arrays illustrates the behavior of the resulting bounds and compares alternating least squares algorithms that are commonly used to compute such decompositions with the respective crbs simple to check necessary conditions for a unique low rank decomposition are also provided

on the asymptotic relative efficiency of planned missingness designs in planned missingness pm designs certain data are set a priori to be missing pm designs can increase validity and reduce cost however little is known about the loss of efficiency that accompanies these designs the present paper compares pm designs to reduced sample rn designs that have the same total number of data points concentrated in fewer participants in 4 studies we consider models for both observed and latent variables designs that do or do not include an x set of variables with complete data and a full range of between and within set correlation values all results are obtained using asymptotic relative efficiency formulas and thus no data are generated this novel approach allows us to examine whether pm designs have theoretical advantages over rn designs removing the impact of sampling error our primary findings are that a in manifest variable regression models estimates of regression coefficients have much lower relative efficiency in pm designs as compared to rn designs b relative efficiency of factor correlation or latent regression coefficient estimates is maximized when the indicators of each latent variable come from different sets and c the addition of an x set improves efficiency in manifest variable regression models only for the parameters that directly involve the x set variables but it substantially improves efficiency of most parameters in latent variable models we conclude that pm designs can be beneficial when the model of interest is a latent variable model recommendations are made for how to optimize such a design

factor uniqueness of the structural parafac model factor analysis is a well known method for describing the covariance structure among a set of manifest variables through a limited number of unobserved factors when the observed variables are collected at various occasions on the same statistical units the data have a three way structure and standard factor analysis may fail to overcome these limitations three way models such as the parafac model can be adopted it is often seen as an extension of principal component analysis able to discover unique latent components the structural version i e as a reparameterization of the covariance matrix has been also formulated but rarely investigated in this article such a formulation is studied by discussing under what conditions factor uniqueness is preserved it is shown that under mild conditions such a property holds even if the specific factors are assumed to be within variable or within occasion correlated and the model is modified to become scale invariant

on kruskal s uniqueness condition for the candecomp parafac decomposition let x be a real valued three way array the candecomp parafac cp decomposition is written as x under bar y under bar 1 y under bar r e where y r are rank 1 arrays and e under bar is a rest term each rank 1 array is defined by the outer product of three vectors a r b r and c r i e y ijk r a i r b j r c k t these vectors make up the r columns of the component matrices a b and c if 2r 2 is less than or equal to the sum of the k ranks of a b and c then the fitted part of the decomposition is unique up to a change in the order of the rank 1 arrays and rescaling counterscaling of each triplet of vectors a r b r c r forming a rank 1 array this classical result was shown by kruskal his proof is however rather inaccessible and does not seem intuitive in order to contribute to a better understanding of cp uniqueness this paper provides an accessible and intuitive proof of kruskal s condition the proof is both self contained and compact and can easily be adapted for the complex valued cp decomposition c 2006 elsevier inc all rights reserved

the special sign indeterminacy of the direct fitting parafac2 model some implications cautions and recommendations for simultaneous component analysis parafac2 is the most flexible simultaneous component analysis sca model that produces an essentially unique solution in this paper we discuss how parafac2 s special sign indeterminacy affects applications of sca and we reveal how an external criterion variable can be used to ensure that estimated parafac2 weights are meaningfully signed across the levels of the nesting mode we present an example with real data from clinical psychology that illustrates the importance of parafac2 s special sign indeterminacy as well as the effectiveness of our proposed solution we also discuss the implications of our results for general applications of sca

design and analysis of incomplete multitrait multimethod studies from a multiplicative perspective the composite direct product cdp model is a multiplicative model for multitrait multimethod mtmm designs it is extended to incomplete mtmm correlation matrices where some trait method combinations are not available rules for omitting trait method combinations without resulting in an indeterminate model are also suggested maximum likelihood estimation and the log absolute correlation procedure are used to fit the model and are found to yield similar results the balanced incomplete mtmm design tends to yield more accurate estimates than the randomly missing design

blind multichannel deconvolution and convolutive extensions of canonical polyadic and block term decompositions tensor decompositions such as the canonical polyadic decomposition cpd or the block term decomposition btd are basic tools for blind signal separation most of the literature concerns instantaneous mixtures memoryless channels in this paper we focus on convolutive extensions more precisely we present a connection between convolutive cpd btd models and coupled but instantaneous cpd btd we derive a new identifiability condition dedicated to convolutive low rank factorization problems we explain that under this condition the convolutive extension of cpd btd can be computed by means of an algebraic method guaranteeing perfect source separation in the noiseless case in the inexact case the algorithm can be used as a cheap initialization for an optimization based method we explain that in contrast to the memoryless case convolutive signal separation is in certain cases possible despite only two way diversities e g space times time

simplicity and typical rank results for three way arrays matrices can be diagonalized by singular vectors or when they are symmetric by eigenvectors pairs of square matrices often admit simultaneous diagonalization and always admit block wise simultaneous diagonalization generalizing these possibilities to more than two non square matrices leads to methods of simplifying three way arrays by nonsingular transformations such transformations have direct applications in tucker pca for three way arrays where transforming the core array to simplicity is allowed without loss of fit simplifying arrays also facilitates the study of array rank the typical rank of a three way array is the smallest number of rank one arrays that have the array as their sum when the array is generated by random sampling from a continuous distribution in some applications the core array of tucker pca is constrained to have a vast majority of zero elements both simplicity and typical rank results can be applied to distinguish constrained tucker pca models from tautologies an update of typical rank results over the real number field is given in the form of two tables

a note on the identification of restricted factor loading matrices it is shown that problems of rotational equivalence of restricted factor loading matrices in orthogonal factor analysis are equivalent to problems of identification in simultaneous equations systems with covariance restrictions a necessary under a regularity assumption and sufficient condition for local uniqueness is given and a counterexample is provided to a theorem by j algina concerning necessary and sufficient conditions for global uniqueness

confac decomposition approach to blind identification of underdetermined mixtures based on generating function derivatives this work proposes a new tensor based approach to solve the problem of blind identification of underdetermined mixtures of complex valued sources exploiting the cumulant generating function cgf of the observations we show that a collection of second order derivatives of the cgf of the observations can be stored in a third order tensor following a constrained factor confac decomposition with known constrained structure in order to increase the diversity we combine three derivative types into an extended third order confac decomposition a detailed uniqueness study of this decomposition is provided from which easy to check sufficient conditions ensuring the essential uniqueness of the mixing matrix are obtained from an algorithmic viewpoint we develop a confac based enhanced line search confac els method to be used with an alternating least squares estimation procedure for accelerated convergence and also analyze the numerical complexities of two confac based algorithms namely confac als and confac els in comparison with the levenberg marquardt lm based algorithm recently derived to solve the same problem simulation results compare the proposed approach with some higher order methods our results also corroborate the advantages of the confac based approach over the competing lm based approach in terms of performance and computational complexity

semi blind receivers for joint symbol and channel estimation in space time frequency mimo ofdm systems in wireless communications increased spectral efficiency and low error rates can be achieved by means of space time frequency coded mimo ofdm systems in this work we consider a mimo ofdm transmit signal design combining space frequency modulation with a time varying linear precoding technique which allows spreading and multiplexing the transmitted symbols in both space time and frequency domains for this system we propose two closed form semi blind receivers that exploit differently the multilinear structure of the received signal which is formulated as a nested parallel factor parafac model first we devise a least squares khatri rao factorization ls krf based receiver for joint channel and symbol estimation by making an efficient use of a short frame of pilot symbols the ls krf receiver provides the same performance at a lower computational complexity compared to the alternating least squares als based receiver for further reducing pilot overhead we develop a simplified closed form parafac s cfp receiver coupled with a pairing algorithm that yields an unambiguous estimation of the transmitted symbols without the need of a pilot frame the uniqueness conditions spectral efficiency and computational complexity of the ls krf and s cfp with pairing receivers are analyzed and compared with the als receiver it is shown that the s cfp with pairing receiver has the same order of computational complexity as the als receiver meanwhile simulation results show that our s cfp with pairing receiver achieves the same or very similar performance of the competing receivers with extra pilot overhead at sufficiently high signal to noise ratio snr conditions on the other hand it is slightly inferior to them in terms of channel estimation accuracy and bit error rate at lower snrs

a constrained factor decomposition with application to mimo antenna systems in this paper we formulate a new tensor decomposition herein called constrained factor confac decomposition it consists in decomposing a third order tensor into a triple sum of rank one tensor factors where interactions involving the components of different tensor factors are allowed the interaction pattern is controlled by three constraint matrices the columns of which are canonical vectors each constraint matrix is associated with a given dimension or mode of the tensor the explicit use of these constraint matrices provides degrees of freedom to the confac decomposition for modeling tensor signals with constrained structures which cannot be handled with the standard parallel factor parafac decomposition the uniqueness of this decomposition is discussed and an application to multiple input multiple output mimo antenna systems is presented a new transmission structure is proposed the core of which consists of a precoder tensor decomposed as a function of the confac constraint matrices by adjusting the precoder constraint matrices we can control the allocation of data streams and spreading codes to transmit antennas based on a confac model of the received signal blind symbol code channel recovery is possible using the alternating least squares algorithm for illustrating this application we evaluate the bit error rate ber performance for some configurations of the precoder constraint matrices

a numerical approach to the approximate and the exact minimum rank of a covariance matrix a concept of approximate minimum rank for a covariance matrix is defined which contains the exact minimum rank as a special case a computational procedure to evaluate the approximate minimum rank is offered the procedure yields those proper communalities for which the unexplained common variance ignored in low rank factor analysis is minimized the procedure also permits a numerical determination of the exact minimum rank of a covariance matrix within limits of computational accuracy a set of 180 covariance matrices with known or bounded minimum rank was analyzed the procedure was successful throughout in recovering the desired rank

blind signal separation via tensor decomposition with vandermonde factor canonical polyadic decomposition several problems in signal processing have been formulated in terms of the canonical polyadic decomposition of a higher order tensor with one or more vandermonde constrained factor matrices we first propose new relaxed uniqueness conditions we explain that under these conditions the number of components may simply be estimated as the rank of a matrix we propose an efficient algorithm for the computation of the factors that only resorts to basic linear algebra we demonstrate the use of the results for various applications in wireless communication and array processing

space time frequency stf mimo communication systems with blind receiver based on a generalized paratuck2 model in this paper we first propose a generalized fourth order paratuck2 tensor model for multiple input multiple output mimo communication systems with space time frequency stf spreading multiplexing the core of the proposed paratuck2 model is composed of two third order interaction tensors that define a joint time and frequency allocation of the data streams to the transmit antennas thus allowing to adjust the multiplexing degree and spreading redundancy in three domains space transmit antennas time blocks and frequency subcarriers then we investigate the identifiability of the paratuck2 stf mimo system by deriving sufficient conditions which are translated into design recommendations for the stf allocation structure in particular essential uniqueness is discussed by interpreting the generalized fourth order paratuck2 model as an equivalent third order constrained factor confac model with two fixed constraint matrices and one variable constraint matrix that depends on the stream to antenna allocation structure we also present a blind receiver using the levenberg marquardt lm algorithm based on the generalized fourth order paratuck2 model numerical results are provided for a bit error rate performance evaluation and a comparison with some competing algorithms

blind deconvolution of ds cdma signals by means of decomposition in rank 1 l l terms in this paper we present a powerful technique for the blind extraction of direct sequence code division multiple access ds cdma signals from convolutive mixtures received by an antenna array the technique is based on a generalization of the canonical or parallel factor decomposition candecomp parafac in multilinear algebra we present a bound on the number of users under which blind separation and deconvolution is guaranteed the solution is computed by means of an alternating least squares als algorithm the excellent performance is illustrated by means of a number of simulations we include an explicit expression of the cramer rao bound crb of the transmitted symbols

subtracting a best rank 1 approximation may increase tensor rank it has been shown that a best rank r approximation of an order k tensor may not exist when r 2 and k 3 this poses a serious problem to data analysts using tensor decompositions it has been observed numerically that generally this issue cannot be solved by consecutively computing and subtracting best rank 1 approximations the reason for this is that subtracting a best rank 1 approximation generally does not decrease tensor rank in this paper we provide a mathematical treatment of this property for real valued 2 2 2 tensors with symmetric tensors as a special case regardless of the symmetry we show that for generic 2 2 2 tensors which have rank 2 or 3 subtracting a best rank 1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank 2 and rank 3 sets hence for a typical tensor of rank 2 subtracting a best rank 1 approximation increases the tensor rank

on uniqueness conditions for candecomp parafac and indscal with full column rank in one mode abstract in the candecomp parafac cp model a three way array x is written as the sum of r outer vector product arrays and a residual array the former comprise the columns of the component matrices a b and c for fixed residuals a b c is unique up to trivial ambiguities if 2 r 2 is less than or equal to the sum of the k ranks of a b and c this classical result was shown by kruskal in 1977 in this paper we consider the case where one of a b c has full column rank and show that in this case kruskal s uniqueness condition implies a recently obtained uniqueness condition moreover we obtain kruskal type uniqueness conditions that are weaker than kruskal s condition itself also for a b c with rank a r 1 and c full column rank we obtain easy to check necessary and sufficient uniqueness conditions we extend our results to the indscal decomposition in which the array x has symmetric slices and a b is imposed we consider the real valued cp and indscal decompositions but our results are also valid for their complex valued counterparts

maximal rank of m x n x mn k tensors it is shown that the maximal rank of m times n times mn k tensors with k leqq min m 1 2 2 n 1 2 2 is greater than mn 4 sqrt 2k o 1

a novel method for integrating first and second order differential equations in elastohydrodynamic lubrication for the solution of smooth isothermal line contact problems this paper contains details of recent developments in the analysis of elastohydrodynamic lubrication problems using the finite element method a steady state isothermal finite element formulation of the smooth line contact problem with newtonian lubricant behaviour is presented containing both first and second order formulations of the hydrodynamic equation previous problems with the limited range of applicability of both first and second order finite difference solutions have been overcome by summing both the first and second order equations weighted contributions application of the method to a range of problems unattainable by either single first or second order formulations is presented copyright 1999 john wiley sons ltd

finite element solution of a reynolds koiter coupled problem for the elastic journal bearing abstract the aim of this work is to present a numerical algorithm for solving a new elastohydrodynamic free boundary problem which models the lubricant pressure behaviour by means of reynolds equation jointly with the free boundary model of elrod adams for cavitation and the elastic deformation of the bearing with the koiter shell model a non local constraint on the pressure is imposed the proposed algorithm uncouples the hydrodynamic and elastic parts to achieve a fixed point for the two dimensional problem a combination of finite elements transport diffusion and duality methods is performed for the three dimensional elastic problem a finite element discretization is applied to a mixed formulation for the koiter model

primitive groups graph endomorphisms and synchronization the third author has been partially supported by the fundacao para a ciencia e a tecnologia through the project cemat ciencias uid multi 04621 2013

tensor decompositions for identifying directed graph topologies and tracking dynamic networks directed networks are pervasive both in nature and engineered systems often underlying the complex behavior observed in biological systems microblogs and social interactions over the web as well as global financial markets since their structures are often unobservable in order to facilitate network analytics one generally resorts to approaches capitalizing on measurable nodal processes to infer the unknown topology structural equation models sems are capable of incorporating exogenous inputs to resolve inherent directional ambiguities however conventional sems assume full knowledge of exogenous inputs which may not be readily available in some practical settings this paper advocates a novel sem based topology inference approach that entails factorization of a three way tensor constructed from the observed nodal data using the well known parallel factor parafac decomposition it turns out that second order piecewise stationary statistics of exogenous variables suffice to identify the hidden topology capitalizing on the uniqueness properties inherent to high order tensor factorizations it is shown that topology identification is possible under reasonably mild conditions in addition to facilitate real time operation and inference of time varying networks an adaptive parafac tensor decomposition scheme that tracks the topology revealing tensor factors is developed extensive tests on simulated and real stock quote data demonstrate the merits of the novel tensor based approach

advanced structural equation modeling issues and techniques contents g a marcoulides r e schumacker introduction w wothke models for multitrait multimethod matrix analysis k g joreskog f yang nonlinear structural equation models the kenny judd model with interaction effects j j mcardle f hamagami multilevel models from a multiple group structural equation perspective j b willett a g sayer cross domain analyses of change over time combining growth modeling and covariance structure analysis s l hershberger p c m molenaar s e corneal a hierarchy of univariate and multivariate structural times series models y f yung p m bentler bootstrapping techniques in analysis of mean and covariance structures k a bollen a limited information estimator for lisrel models with or without heteroscedastic errors j l arbuckle full information estimation in the presence of incomplete data l j williams h bozdogan l aiman smith inference problems with equivalent models h w marsh j r balla k t hau an evaluation of incremental fit indices a clarification of mathematical and empirical properties