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

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

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

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

parallel factor analysis in sensor array processing this paper links multiple invariance sensor array processing mi sap to parallel factor parafac analysis which is a tool rooted in psychometrics and chemometrics parafac is a common name for low rank decomposition of three and higher way arrays this link facilitates the derivation of powerful identifiability results for mi sap shows that the uniqueness of single and multiple invariance esprit stems from uniqueness of low rank decomposition of three way arrays and allows tapping on the available expertise for fitting the parafac model the results are applicable to both data domain and subspace mi sap formulations the paper also includes a constructive uniqueness proof for a special parafac model

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

supervisory control of timed discrete event systems under partial observation based on activity models and eligible time bounds to avoid the state space explosion by including tick events in timed discrete event systems dess under partial observation a notion of eligible time bounds is introduced and based on the notion controllability and observability conditions of languages are presented in particular this paper shows that these controllability and observability conditions are necessary and sufficient for the existence of a supervisor to achieve the given language specification 2005 elsevier b v all rights reserved

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

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

multidimensional harmonic retrieval via coupled canonical polyadic decomposition part i model and identifiability multidimensional harmonic retrieval mhr is a fundamental problem in signal processing we make a connection with coupled canonical polyadic decomposition cpd which allows us to better exploit the rich mhr structure than existing approaches in the derivation of uniqueness results we discuss both deterministic and generic conditions we obtain a deterministic condition that is both necessary and sufficient but which may be difficult to check in practice we derive mild deterministic relaxations that are easy to verify we also discuss the variant in which the generators have unit norm we narrow the transition zone between generic uniqueness and generic non uniqueness to two values of the number of harmonics we explain differences with one dimensional hr

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

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

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

data driven power control for state estimation we consider sensor transmission power control for state estimation using a bayesian inference approach a sensor node sends its local state estimate to a remote estimator over an unreliable wireless communication channel with random data packet drops as related to packet dropout rate transmission power is chosen by the sensor based on the relative importance of the local state estimate the proposed power controller is proved to preserve gaussianity of local estimate innovation which enables us to obtain a closed form solution of the expected state estimation error covariance comparisons with alternative non data driven controllers demonstrate performance improvement using our approach

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

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

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

a candecomp parafac perspective on uniqueness of doa estimation using a vector sensor array we address the uniqueness problem in estimating the directions of arrival doas of multiple narrowband and fully polarized signals impinging on a passive sensor array composed of identical vector sensors the data recorded on such an array present the so called multiple invariances which can be linked to the candecomp parafac cp model cp refers to a family of low rank decompositions of three way or higher way mutidimensional data arrays where each dimension is termed as a mode a sufficient condition is derived for uniqueness of the cp decomposition of a three way three mode array in the particular case where one of the three loading matrices each associated to one mode involved in the decomposition has full column rank based on this upper bounds on the maximal number of identifiable doas are deduced for the two typical cases i e the general case of uncorrelated or partially correlated sources and the case where the sources are coherent

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

tensor deflation for candecomp parafac part i alternating subspace update algorithm candecomp parafac cp approximates multiway data by sum of rank 1 tensors unlike matrix decomposition the procedure which estimates the best rank r tensor approximation through r sequential best rank 1 approximations does not work for tensors because the deflation does not always reduce the tensor rank in this paper we propose a novel deflation method for the problem when one factor matrix of a rank r cp decomposition is of full column rank the decomposition can be performed through r 1 rank 1 reductions at each deflation stage the residue tensor is constrained to have a reduced multilinear rank for decomposition of order 3 tensors of size r times r times r and rank r estimation of one rank 1 tensor has a computational cost of cal o r 3 per iteration which is lower than the cost cal o r 4 of the als algorithm for the overall cp decomposition the method can be extended to tracking one or a few rank one tensors of slow changes or inspect variations of common patterns in individual datasets

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

supervisory control of timed discrete event systems under partial observation this paper extends the authors previous work on observability of discrete event systems by taking time into consideration in a timed discrete event system events must occur within their respective lower and upper time bounds a supervisor can disable enable or force some events to achieve a given control objective the authors assume that the supervisor does not observe all events which is often the case in practice the authors generalize the concept of observability to timed discrete event systems and show that it characterizes the existence condition for a supervisor the authors also generalize normality a stronger version of observability to timed discrete event systems which has nice properties that are absent in observability the authors then derive conditions under which observability and normality are equivalent the authors propose two methods to synthesize a supervisor a direct approach and an indirect approach an example is given to illustrate the results

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