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

vector sparse representation of color image using quaternion matrix analysis traditional sparse image models treat color image pixel as a scalar which represents color channels separately or concatenate color channels as a monochrome image in this paper we propose a vector sparse representation model for color images using quaternion matrix analysis as a new tool for color image representation its potential applications in several image processing tasks are presented including color image reconstruction denoising inpainting and super resolution the proposed model represents the color image as a quaternion matrix where a quaternion based dictionary learning algorithm is presented using the k quaternion singular value decomposition qsvd generalized k means clustering for qsvd method it conducts the sparse basis selection in quaternion space which uniformly transforms the channel images to an orthogonal color space in this new color space it is significant that the inherent color structures can be completely preserved during vector reconstruction moreover the proposed sparse model is more efficient comparing with the current sparse models for image restoration tasks due to lower redundancy between the atoms of different color channels the experimental results demonstrate that the proposed sparse image model avoids the hue bias issue successfully and shows its potential as a general and powerful tool in color image analysis and processing domain

stable coexistence states in the volterra lotka competition model with diffusion we study the existence uniqueness and stability of coexistence states in the lotka volterra model with diffusion for two competing species we assume that the parameters describing the interactio

some additional results on principal components analysis of three mode data by means of alternating least squares algorithms kroonenberg and de leeuw 1980 have developed an alternating least squares method tuckals 3 as a solution for tucker s three way principal components model the present paper offers some additional features of their method starting from a reanalysis of tucker s problem in terms of a rank constrained regression problem it is shown that the fitted sum of squares in tuckals 3 can be partitioned according to elements of each mode of the three way data matrix an upper bound to the total fitted sum of squares is derived finally a special case of tuckals 3 is related to the carroll harshman candecomp parafac model

parameter ranges for the existence of solutions whose state components have specified nodal structure in coupled multiparameter systems of nonlinear sturm liouville boundary value problems

uniqueness proof for a family of models sharing features of tucker s three mode factor analysis and parafac candecomp some existing three way factor analysis and mds models incorporate cattell s principle of parallel proportional profiles these models can with appropriate data empirically determine a unique best fitting axis orientation without the need for a separate factor rotation stage but they have not been general enough to deal with what tucker has called interactions among dimensions this article presents a proof of unique axis orientation for a considerably more general parallel profiles model which incorporates interacting dimensions the model xk aadk hbdk b does not assume symmetry in the data or in the interactions among factors a second proof is presented for the symmetrically weighted case i e whereadk bdk the generality of these models allows one to impose successive restrictions to obtain several useful special cases including parafac2 and three way dedicom

constrained candecomp parafac via the lasso the candecomp parafac cp model is a well known tool for summarizing a three way array by extracting a limited number of components unfortunately in some cases the model suffers from the so called degeneracy that is a solution with diverging and uninterpretable components to avoid degeneracy orthogonality constraints are usually applied to one of the component matrices this solves the problem only from a technical point of view because the existence of orthogonal components underlying the data is not guaranteed for this purpose we consider some variants of the cp model where the orthogonality constraints are relaxed either by constraining only a pair or a subset of components or by stimulating the cp solution to be possibly orthogonal we theoretically clarify that only the latter approach based on the least absolute shrinkage and selection operator and named the cp lasso is helpful in solving the degeneracy problem the results of the application of cp lasso on simulated and real life data show its effectiveness

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

an approach to n mode components analysis as an extension of lastovicka s four mode components analysis ann mode components analysis is developed using a convenient notation both a canonical and a least squares solution are derived the relation between both solutions and their computational aspects are discussed

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

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

on the minimal ranks of matrix pencils and the existence of a best approximate block term tensor decomposition abstract under the action of the general linear group with tensor structure the ranks of matrices a and b forming an m n pencil a b can change but in a restricted manner specifically with every pencil one can associate a pair of minimal ranks which is unique up to a permutation this notion can be defined for matrix pencils and more generally also for matrix polynomials of arbitrary degree in this paper we provide a formal definition of the minimal ranks discuss its properties and the natural hierarchy it induces in a pencil space then we show how the minimal ranks of a pencil can be determined from its kronecker canonical form for illustration we classify the orbits according to their minimal ranks under the action of the general linear group in the case of real pencils with m n 4 subsequently we show that real regular 2 k 2 k pencils having only complex valued eigenvalues which form an open positive volume set do not admit a best approximation in the norm topology on the set of real pencils whose minimal ranks are bounded by 2 k 1 our results can be interpreted from a tensor viewpoint where the minimal ranks of a degree d 1 matrix polynomial characterize the minimal ranks of matrices constituting a block term decomposition of an m n d tensor into a sum of matrix vector tensor products

bifurcation of steady state solutions in predator prey and competition systems we discuss steady state solutions of systems of semilinear reaction diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region it is easy to find solutions to the systems such that either u or v is identically zero such solutions correspond to the case where one of the species is extinct by using decoupling and global bifurcation theory techniques we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co exist

a simplification of a result by zellini on the maximal rank of symmetric three way arrays zellini 1979 theorem 3 1 has shown how to decompose an arbitrary symmetric matrix of ordern n as a linear combination of 1 2n n 1 fixed rank one matrices thus constructing an explicit tensor basis for the set of symmetricn n matrices zellini s decomposition is based on properties of persymmetric matrices in the present paper a simplified tensor basis is given by showing that a symmetric matrix can also be decomposed in terms of 1 2n n 1 fixed binary matrices of rank one the decomposition implies that ann n p array consisting ofp symmetricn n slabs has maximal rank 1 2n n 1 likewise an unconstrained indscal symmetric candecomp parafac decomposition of such an array will yield a perfect fit in 1 2n n 1 dimensions when the fitting only pertains to the off diagonal elements of the symmetric matrices as is the case in a version of parafac where communalities are involved the maximal number of dimensions can be further reduced to 1 2n n 1 however when the saliences in indscal are constrained to be nonnegative the tensor basis result does not apply in fact it is shown that in this case the number of dimensions needed can be as large asp the number of matrices analyzed

a fast approach for dimensionality reduction with image data an important objective in image analysis is dimensionality reduction the most often used data exploratory technique with this objective is principal component analysis which performs a singular value decomposition on a data matrix of vectorized images when considering an array data or tensor instead of a matrix the high order generalization of pca for computing principal components offers multiple ways to decompose tensors orthogonally as an alternative we propose a new method based on the projection of the images as matrices and show that it leads to a better reconstruction of images than previous approaches

on the distortion of knots on embedded surfaces our main result is a nontrivial lower bound for the distortion of some specic knots in particular we show that the distortion of the torus knot tp q satises tp q 1 160 min p q this answers a 1983 question of gromov

simple structure in component analysis techniques for mixtures of qualitative and quantitative variables several methods have been developed for the analysis of a mixture of qualitative and quantitative variables and one called pcamix includes ordinary principal component analysis pca and multiple correspondence analysis mca as special cases the present paper proposes several techniques for simple structure rotation of a pcamix solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for pca in addition a recently developed technique for the analysis of mixtures of qualitative and quantitative variables called indomix is shown to construct component scores without rotational freedom maximizing the quartimax criterion over all possible sets of component scores a numerical example is used to illustrate the implication that when used for qualitative variables indomix provides axes that discriminate between the observation units better than do those generated from mca

some clarifications of the tuckals2 algorithm applied to the idioscal problem kroonenberg and de leeuw have suggested fitting the idioscal model by the tuckals2 algorithm for three way components analysis in theory this is problematic because tuckals2 produces two possibly different coordinate matrices that are useless for idioscal unless they are equal kroonenberg has claimed that when idioscal is fitted by tuckals2 the resulting coordinate matrices will be identical in the present paper this claim is proven valid when the data matrices are semidefinite however counterexamples for indefinite matrices are also constructed by examining the global minimum in the case where the data matrices have the same eigenvectors similar counterexamples have been considered by ten berge and kiers in the related context of candecomp parafac to fit the indscal model

shift invariant multilinear decomposition of neuroimaging data we present an algorithm for multilinear decomposition that allows for arbitrary shifts along one modality the method is applied to neural activity arranged in the three modalities space time and trial thus the algorithm models neural activity as a linear superposition of components with a fixed time course that may vary across either trials or space in its overall intensity and latency its utility is demonstrated on simulated data as well as actual eeg and fmri data we show how shift invariant multilinear decompositions of multiway data can successfully cope with variable latencies in data derived from neural activity a problem that has caused degenerate solutions especially in modeling neuroimaging data with instantaneous multilinear decompositions our algorithm is available for download at www erpwavelab org

multicomponent wave separation using hosvd unimodal ica subspace method multicomponent sensor arrays now are commonly used in seismic acquisition to record polarized waves in this article we use a three mode model polarization mode distance mode and temporal mode to take into account the specific structure of signals that are recorded with these arrays providing a data structure preserving processing with the suggested model we propose a multilinear decomposition named higher order singular value decomposition and unimodal independent component analysis hosvd unimodal ica to split the recorded threemode data into two orthogonal subspaces the signal and noise subspaces this decomposition allows the separation and identification of polarized waves with infinite apparent horizontal propagation velocity the hosvd leads to a definition of a subspace method that is the counterpart of the well known subspace method for matrices that is driven by singular value decompositionsvd aclassictoolinmonocomponentarrayprocessing the proposed three mode subspace decomposition provides a multicomponentwave separationalgorithm toenhancetheseparation result when the signal to noise ratio is low or when orthogonality constraints are not well adapted to the recorded waves a unimodal ica step is included on the temporal mode doingthisreplacestheclassicorthogonalityconstraintsbetween estimated waves with independence constraints that might allow better recovery of recorded seismic waves asimulation on realistic two component 2c geophysical data shows qualitative and quantitative improvements for the wavefield separation results the relative mean square errors between the original and estimated signal subspaces are respectively 52 for svd applied on each component separately 27 4 for hosvd based technique applied to the whole three mode dataset and 7 3 for hosvd unimodal ica technique the efficiency of the threemodesubspacedecompositionsalsoisshownonrealthree component 3c geophysical data these results emphasize the potentialofthehosvd unimodal icasubspacemethodformulticomponentseismic waveseparation

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

typical ranks for 3 tensors nonsingular bilinear maps and determinantal ideals abstract let m n 3 m 1 n 1 2 p m n and u m n p the set r u n m of all real tensors with size u n m is one to one corresponding to the set of bilinear maps r m r n r u we show that r m n p has plural typical ranks p and p 1 if and only if there exists a nonsingular bilinear map r m r n r u we show that there is a dense open subset o of r u n m such that for any y o the ideal of maximal minors of a matrix defined by y in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal further we show that there is a dense open subset t of r n p m and continuous surjective open maps o r u p and t r u p where r u p is the set of u p matrices with entries in r such that if y t then rank t p if and only if the ideal of maximal minors of the matrix defined by y is a real prime ideal

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

operator norm inequalities between tensor unfoldings on the partition lattice interest in higher order tensors has recently surged in data intensive fields with a wide range of applications including image processing blind source separation community detection and feature extraction a common paradigm in tensor related algorithms advocates unfolding or flattening the tensor into a matrix and applying classical methods developed for matrices despite the popularity of such techniques how the functional properties of a tensor changes upon unfolding is currently not well understood in contrast to the body of existing work which has focused almost exclusively on matricizations we here consider all possible unfoldings of an order k tensor which are in one to one correspondence with the set of partitions of 1 k we derive general inequalities between the lp norms of arbitrary unfoldings defined on the partition lattice in particular we demonstrate how the spectral norm p 2 of a tensor is bounded by that of its unfoldings and obtain an improved upper bound on the ratio of the frobenius norm to the spectral norm of an arbitrary tensor for specially structured tensors satisfying a generalized definition of orthogonal decomposability we prove that the spectral norm remains invariant under specific subsets of unfolding operations

convergence rate analysis for the higher order power method in best rank one approximations of tensors a popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method hopm it is known in the literature that the iterative sequence generated by hopm converges globally while the convergence rate can be superlinear linear or sublinear in this paper we examine the convergence rate of hopm in solving the best rank one approximation problem of real tensors we first show that the iterative sequence of hopm always converges globally and provide an explicit eventual sublinear convergence rate the sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space then we examine the concept of nondegenerate singular vector tuples and show that if the sequence of hopm converges to a nondegenerate singular vector tuple then the global convergence rate is r linear we show that for almost all tensors in the sense of lebesgue measure all the singular vector tuples are nondegenerate and so the hopm typically exhibits global r linear convergence rate moreover without any regularity assumption we establish that the sequence generated by hopm always converges globally and r linearly for orthogonally decomposable tensors with order at least 3 we achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate

the carroll and chang conjecture of equal indscal components when candecomp parafac gives perfect fit abstract the candecomp parafac algorithm approximates a set of matrices x 1 x i by products of the form ac i b with c i diagonal i 1 i carroll and chang have conjectured that when the matrices are symmetric the resulting a and b will be column wise proportional for cases of perfect fit ten berge et al have shown that the conjecture holds true in a variety of cases but may fail when there is no unique solution in such cases obtaining proportionality by changing part of the solution seems possible the present paper extends and further clarifies their results in particular where ten berge et al solved all i 2 2 cases now all i 3 3 cases and also the i 4 4 cases for i 2 8 and 9 are clarified in a number of cases a and b necessarily have column wise proportionality when candecomp parafac is run to convergence in other cases proportionality can be obtained by using specific methods no cases were found that seem to resist proportionality

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

explicit candecomp parafac solutions for a contrived 2 2 2 array of rank three kruskal harshman and lundy have contrived a special 2 2 2 array to examine formal properties of degenerate candecomp parafac solutions it is shown that for this array the candecomp parafac loss has an infimum of 1 in addition the array will be used to challenge the tradition of fitting indscal and related models by means of the candecomp parafac process

on positive solutions of some pairs of differential equations ii on etudie l existence de solutions strictement positives de systemes predateur proie on demontre l unicite de la solution strictement positive dans le systeme de conway smoller

higher order tensor based method for delayed exponential fitting we present subspace based schemes for the estimation of the poles angular frequencies and damping factors of a sum of damped and delayed sinusoids in our model each component is supported over a different time frame depending on the delay parameter classical subspace based methods are not suited to handle signals with varying time supports in this contribution we propose solutions based on the approximation of a partially structured hankel type tensor on which the data are mapped we show by means of several examples that the approach based on the best rank r1 r2 r3 approximation of the data tensor outperforms the current tensor and matrix based techniques in terms of the accuracy of the angular frequency and damping factor parameter estimates especially in the context of difficult scenarios as in the low signal to noise ratio regime and for closely spaced sinusoids

a case of extreme simplicity of the core matrix in three mode principal components analysis in three mode principal components analysis thep q r core matrixg can be transformed to simple structure before it is interpreted it is well known that whenp qr g can be transformed to the identity matrix which implies that all elements become equal to values specified a priori in the present paper it is shown that whenp qr 1 g can be transformed to have nearly all elements equal to values spectified a priori a cllsed form solution for this transformation is offered theoretical and practical implications of this simple structure transformation ofg are discussed

blind spatial signature estimation via time varying user power loading and parallel factor analysis in this paper the problem of blind spatial signature estimation using the parallel factor parafac analysis model is addressed in application to wireless communications a time varying user power loading in the uplink mode is proposed to make the model identifiable and to enable application of parafac analysis then identifiability issues are studied in detail and closed form expressions for the corresponding modified crame spl acute r rao bound crb are obtained furthermore two blind spatial signature estimation algorithms are developed the first technique is based on the parafac fitting trilinear alternating least squares tals regression procedure whereas the second one makes use of the joint approximate diagonalization algorithm these techniques do not require any knowledge of the propagation channel and or sensor array manifold and are applicable to a more general class of scenarios than earlier approaches to blind spatial signature estimation

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

the typical rank of tall three way arrays the rank of a three way array refers to the smallest number of rank one arrays outer products of three vectors that generate the array as their sum it is also the number of components required for a full decomposition of a three way array by candecomp parafac the typical rank of a three way array refers to the rank a three way array has almost surely the present paper deals with typical rank and generalizes existing results on the typical rank ofi j k arrays withk 2 to a particular class of arrays withk 2 it is shown that the typical rank isi when the array is tall in the sense thatjk j i jk in addition typical rank results are given for the case wherei equalsjk j

three way metric unfolding via alternating weighted least squares three way unfolding was developed by desarbo 1978 and reported in desarbo and carroll 1980 1981 as a new model to accommodate the analysis of two mode three way data e g nonsymmetric proximities for stimulus objects collected over time and three mode three way data e g subjects rendering preference judgments for various stimuli in different usage occasions or situations this paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis we begin with an introduction of the problem and a review of three way unfolding the three way unfolding model weighted objective function and new algorithm are presented monte carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance finally three applications of the methodology are reported illustrating the flexibility and robustness of the procedure

typical rank and indscal dimensionality for symmetric three way arrays of order i 2 2 or i 3 3 a peculiar property of three way arrays is that the rank they typically have does not necessarily coincide with the maximum possible rank given their order typical tensorial rank has much been studied over algebraically closed fields however very few results have been found pertaining to the typical rank of three way arrays over the real field these results refer to arrays sampled randomly from continuous distributions arrays that consist of symmetric slices do not fit into this sampling scheme the present paper offers typical rank results over the real field for arrays containing symmetric slices of order 2 2a nd 3 3 symmetric arrays often appear to have lower typical ranks than their asymmetric counterparts this paper also examines whether or not the rank of a symmetric array coincides with the smallest number of dimensions that allow a perfect fit of indscal for all cases considered this is indeed true thus a full indscal solution may require fewer dimensions for a symmetric array than a full cp decomposition applied to an asymmetric array of the same size the reverse situation does not seem to arise next we examine in which cases cp solutions inevitably are indscal solutions finally the rank reducing impact of double standardizing the slices is discussed

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

matrices with special reference to applications in psychometrics multidimensional scaling item response theory and factor analysis may be considered three major contributions of psychometricians to statistics matrix theory played an important role in early developments of these techniques unfortunately nonlinear models are currently very prevalent in these areas still one can identify several areas of psychometrics where matrix algebra plays a prominent role they include analysis of asymmetric square tables multiway data analysis reduced rank regression analysis and multiple set t set canonical correlation analysis among others in this article we review some of the important matrix results in these areas and suggest future studies

simplicity of core arrays in three way principal component analysis and the typical rank of p q 2 arrays abstract interpreting the solution of a principal component analysis of a three way array is greatly simplified when the core array has a large number of zero elements the possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand and by mathematical analysis on the other in the present paper it is shown that a p q 2 array with p q 2 can almost surely be transformed to have all but 2 q elements zero it is also shown that arrays of that form have three way rank p at most this has direct implications for the typical rank of p q 2 arrays also when p q when p 2q the typical rank is 2 q when q it is p and when p q the rank is typically almost surely p or p 1 these typical rank results pertain to the decomposition of real valued three way arrays in terms of real valued rank one arrays and do not apply in the complex setting where the typical rank of p q 2 arrays is also min p 2q when p q but it is p when p q

monotonically convergent algorithms for symmetric tensor approximation abstract reduced rank approximations to symmetric tensors find use in data compaction and in multi user blind source separation we derive iterative algorithms which feature monotonic convergence to a minimum of a frobenius norm approximation criterion for a certain rank r tucker product version of the approximation problem the approach exploits the gradient inequality for convex functions to establish monotonic convergence while sparing the cumbersome step size analysis required from a manifold gradient approach it likewise overcomes some limitations of symmetric versions of alternating least squares the computational load per iteration amounts to computing an unfolded matrix and a qr decomposition

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

correspondence analysis used complementary to loglinear analysis loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables in this paper we will show that there are cases in which these two techniques can be used complementary to each other more specifically we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices each following a specific loglinear model therefore in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models a generalization of correspondence analysis recently proposed by escofier will also be discussed with this decomposition which includes classical correspondence analysis as a special case it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis in this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models

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

some clarifications of the candecomp algorithm applied to indscal carroll and chang have claimed that candecomp applied to symmetric matrices yields equivalent coordinate matrices as needed for indscal although this claim has appeared to be valid for all practical purposes it has gone without a rigorous mathematical footing the purpose of the present paper is to clarify candecomp in this respect it is shown that equivalent coordinate matrices are not granted at global minima when the symmetric matrices are not gramian or when these matrices are gramian but the solution not globally optimal

an efficient algorithm for parafac of three way data with large numbers of observation units the candecomp algorithm for the parafac analysis ofn m p three way arrays is adapted to handle arrays in whichn mp more efficiently for such arrays the adapted algorithm needs less memory space to store the data during the iterations and uses less computation time than the original candecomp algorithm the size of the arrays that can be handled by the new algorithm is in no way limited by the number of observation units n in the data

global higher bifurcations in coupled systems of nonlinear eigenvalue problems on etablit des conditions pour l existence d une bifurcation n aire globale de dimension elevee dans des systemes generaux de problemes aux valeurs propres non lineaires multiparametre qui conservent la structure de couplage des modeles de lotka volterra diffusifs etat stationnaire

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

structure seeking multilinear methods for the analysis of fmri data in comprehensive fmri studies of brain function the data structures often contain higher order ways such as trial task condition subject and group in addition to the intrinsic dimensions of time and space while multivariate bilinear methods such as principal component analysis pca have been used successfully for extracting information about spatial and temporal features in data from a single fmri run the need to unfold higher order data sets into bilinear arrays has led to decompositions that are nonunique and to the loss of multiway linkages and interactions present in the data these additional dimensions or ways can be retained in multilinear models to produce structures that are unique and which admit interpretations that are neurophysiologically meaningful multiway analysis of fmri data from multiple runs of a bilateral finger tapping paradigm was performed using the parallel factor parafac model a trilinear model was fitted to a data cube of dimensions voxels by time by run similarly a quadrilinear model was fitted to a higher way structure of dimensions voxels by time by trial by run the spatial and temporal response components were extracted and validated by comparison to results from traditional svd pca analyses based on scenarios of unfolding into lower order bilinear structures

a tensor approximation approach to dimensionality reduction dimensionality reduction has recently been extensively studied for computer vision applications we present a novel multilinear algebra based approach to reduced dimensionality representation of multidimensional data such as image ensembles video sequences and volume data before reducing the dimensionality we do not convert it into a vector as is done by traditional dimensionality reduction techniques like pca our approach works directly on the multidimensional form of the data matrix in 2d and tensor in higher dimensions to yield what we call a datum as is representation this helps exploit spatio temporal redundancies with less information loss than image as vector methods an efficient rank r tensor approximation algorithm is presented to approximate higher order tensors we show that rank r tensor approximation using datum as is representation generalizes many existing approaches that use image as matrix representation such as generalized low rank approximation of matrices glram ye y in mach learn 61 167 191 2005 rank one decomposition of matrices rodm shashua a levin a in cvpr 01 proceedings of the 2001 ieee computer society conference on computer vision and pattern recognition p 42 2001 and rank one decomposition of tensors rodt wang h ahuja n in icpr 04 icpr 04 proceedings of the 17th international conference on pattern recognition icpr 04 vol 1 pp 44 47 2004 our approach yields the most compact data representation among all known image as matrix methods in addition we propose another rank r tensor approximation algorithm based on slice projection of third order tensors which needs fewer iterations for convergence for the important special case of 2d image ensembles e g video we evaluated the performance of our approach vs other approaches on a number of datasets with the following two main results first for a fixed compression ratio the proposed algorithm yields the best representation of image ensembles visually as well as in the least squares sense second proposed representation gives the best performance for object classification

kruskal s polynomial for 2 2 2 arrays and a generalization to 2 n n arrays a remarkable difference between the concept of rank for matrices and that for three way arrays has to do with the occurrence of non maximal rank the set ofn n matrices that have a rank less thann has zero volume kruskal pointed out that a 2 2 2 array has rank three or less and that the subsets of those 2 2 2 arrays for which the rank is two or three both have positive volume these subsets can be distinguished by the roots of a certain polynomial the present paper generalizes kruskal s results to 2 n n arrays incidentally it is shown that twon n matrices can be diagonalized simultaneously with positive probability

on the typical rank of real bivariate polynomials abstract here we study the typical rank for real bivariate homogeneous polynomials of degree d 6 the case d 5 being settled by p comon and g ottaviani we prove that d 1 is a typical rank and that if d is odd then d 3 2 is a typical rank the comon ottaviani conjecture was later completely solved by g blekherman

optimization based algorithms for tensor decompositions canonical polyadic decomposition decomposition in rank l_r l_r 1 terms and a new generalization the canonical polyadic and rank l r l r 1 block term decomposition cpd and btd respectively are two closely related tensor decompositions the cpd and recently btd are important tools in psychometrics chemometrics neuroscience and signal processing we present a decomposition that generalizes these two and develop algorithms for its computation among these algorithms are alternating least squares schemes several general unconstrained optimization techniques and matrix free nonlinear least squares methods in the latter we exploit the structure of the jacobian s gramian to reduce computational and memory cost combined with an effective preconditioner numerical experiments confirm that these methods are among the most efficient and robust currently available for computing the cpd rank l r l r 1 btd and their generalized decomposition

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

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

on the generic and typical ranks of 3 tensors abstract we study the generic and typical ranks of 3 tensors of dimension l m n using results from matrices and algebraic geometry we state a conjecture about the exact values of the generic rank of 3 tensors over the complex numbers which is verified numerically for l m n 14 we also discuss the typical ranks over the real numbers and give an example of an infinite family of 3 tensors of the form l m n m 1 2 1 m 3 4 which have at least two typical ranks

exact probabilities for typical ranks of 2 2 2 and 3 3 2 tensors we show that the probability to be of rank 2 for a 2 2 2 tensor with elements from a standard normal distribution is 4 and that the probability to be of rank 3 for a 3 3 2 tensor is 1 2 in the

transforming three way arrays to maximal simplicity transforming the core array in tucker three way component analysis to simplicity is an intriguing way of revealing structures in between standard tucker three way pca where the core array is unconstrained and candecomp parafac where the core array has a generalized diagonal form for certain classes of arrays transformations to simplicity that is transformations that produce a large number of zeros can be obtained explicitly by solving sets of linear equations the present paper extends these results first a method is offered to simplifyj j 2 arrays next it is shown that the transformation that simplifies ani j k array can be used to also simplify the complementary arrays of order jk i j k of orderi ik j k and of orderi j ij k finally the question of what constitutes the maximal simplicity for arrays the maximal number of zero elements will be considered it is shown that cases of extreme simplicity considered in the past are in fact cases of maximal simplicity

symmetry transformations for square sliced three way arrays with applications to their typical rank the typical 3 tensorial rank has been much studied over algebraically closed fields but very little has been achieved in the way of results pertaining to the real field the present paper examines the typical 3 tensorial rank over the real field when the slices of the array involved are square matrices the typical rank of 3 3 3 arrays is shown to be five the typical rank of p q q arrays is shown to be larger than q 1 unless there are only two slices p 2 or there are three slices of order 2 2 p 3 and q 2 the key result is that when the rank is q 1 there usually exists a rank preserving transformation of the array to one with symmetric slices

typical ranks for m n m 1 n tensors with m n abstract in various application fields tensor type data are used recently and then a typical rank is important there may be more than one typical ranks over the real number field it is well known that the set of 2 n n tensors has two typical ranks n n 1 for n 2 that the set of 3 4 8 tensors has two typical ranks 8 9 and that the set of 4 4 12 tensors has two typical ranks 12 13 in this paper we show that the set of m n m 1 n tensors with m n has two typical ranks m 1 n m 1 n 1 if m n where is the hurwitz radon function defined as n 2 b 8 c for nonnegative integers a b c such that n 2 a 1 2 b 4 c and 0 b 4

a modified newton s method for best rank one approximation to tensors in this paper a modified newton s method for the best rank one approximation problem to tensor is proposed we combine the iterative matrix of jacobi gauss newton jgn algorithm or alternating least squares als algorithm with the iterative matrix of grq newton method and present a modified version of grq newton algorithm a line search along the projective direction is employed to obtain the global convergence preliminary numerical experiments and numerical comparison show that our algorithm is efficient

the k indscal model for heterogeneous three way dissimilarity data a weighted euclidean distance model for analyzing three way dissimilarity data stimuli by stimuli by subjects for heterogeneous subjects is proposed first it is shown that indscal may fail to identify a common space representative of the observed data structure in presence of heterogeneity a new model that removes the rotational invariance of the classical multidimensional scaling problem and specifies k common homogeneous spaces is proposed the model called mixture indscal in k classes or briefly k indscal still includes individual saliencies however the large number of parameters in k indscal may produce instability of the estimates and therefore a parsimonious model will also be discussed the parameters of the model are estimated in a least squares fitting context and an efficient coordinate descent algorithm is given the usefulness of k indscal is demonstrated by both artificial and real data analyses

generic and typical ranks of multi way arrays abstract the concept of tensor rank was introduced in the 20s in the 70s when methods of component analysis on arrays with more than two indices became popular tensor rank became a much studied topic the generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor we explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions based on terracini s lemma and compare it with the algebraic results already known in the real or complex fields in particular we examine the cases of symmetric tensors tensors with symmetric matrix slices complex tensors enjoying hermitian symmetries or merely tensors with free entries

dimensionality reduction in higher order signal processing and rank r1 r2 rn reduction in multilinear algebra abstract in this paper we review a multilinear generalization of the singular value decomposition and the best rank r 1 r 2 r n approximation of higher order tensors we show that they are important tools for dimensionality reduction in higher order signal processing we discuss applications in independent component analysis simultaneous matrix diagonalization and subspace variants of algorithms based on higher order statistics

a comparison of different notions of ranks of symmetric tensors abstract we introduce various notions of rank for a high order symmetric tensor taking values over the complex numbers namely rank border rank catalecticant rank generalized rank scheme length border scheme length extension rank and smoothable rank we analyze the stratification induced by these ranks the mutual relations between these stratifications allow us to describe the hierarchy among all the ranks we show that strict inequalities are possible between rank border rank extension rank and catalecticant rank moreover we show that scheme length generalized rank and extension rank coincide

an alternating least squares algorithm for fitting the two and three way dedicom model and the idioscal model the dedicom model is a model for representing asymmetric relations among a set of objects by means of a set of coordinates for the objects on a limited number of dimensions the present paper offers an alternating least squares algorithm for fitting the dedicom model the model can be generalized to represent any number of sets of relations among the same set of objects an algorithm for fitting this three way dedicom model is provided as well based on the algorithm for the three way dedicom model an algorithm is developed for fitting the idioscal model in the least squares sense