uniqueness of three mode factor models with sparse cores the 3 x 3 x 3 case three mode factor analysis 3mfa and parafac are methods to describe three way data both methods employ models with components for the three modes of a three way array the 3mfa model also uses a three way core array for linking all components to each other the use of the core array makes the 3mfa model more general than the parafac model thus allowing a better fit but also more complicated moreover in the 3mfa model the components are not uniquely determined and it seems hard to choose among all possible solutions a particularly interesting feature of the parafac model is that it does give unique components the present paper introduces a class of 3mfa models in between 3mfa and parafac that share the good properties of the 3mfa model and the parafac model they fit almost as well as the 3mfa model they are relatively simple and they have the same uniqueness properties as the parafac model

estimating correlation dimension from a chaotic time series when does plateau onset occur abstract suppose that a dynamical system has a chaotic attractor a with a correlation dimension d 2 a common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m dimensional space using the delay coordinate technique the estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau on which the dimension estimate is relatively constant for a range of large enough m values the plateaued dimension value is then assumed to be an estimate of d 2 for the attractor in the original full phase space in this paper we first present rigorous results which state that for a long enough data string with low enough noise the plateau onset occurs at m ceil d 2 where ceil d 2 standing for ceiling of d 2 is the smallest integer greater than or equal to d 2 we then numerical examples illustrating the theoretical prediction in addition we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m ceil d 2

a synthetic approach to markov kernels conditional independence and theorems on sufficient statistics abstract we develop markov categories as a framework for synthetic probability and statistics following work of golubtsov as well as cho and jacobs this means that we treat the following concepts in purely abstract categorical terms conditioning and disintegration various versions of conditional independence and its standard properties conditional products almost surely sufficient statistics versions of theorems on sufficient statistics due to fisher neyman basu and bahadur besides the conceptual clarity offered by our categorical setup its main advantage is that it provides a uniform treatment of various types of probability theory including discrete probability theory measure theoretic probability with general measurable spaces gaussian probability stochastic processes of either of these kinds and many others

parafac based blind estimation of possibly underdetermined convolutive mimo systems in this paper we consider the problem of blind identification of a convolutive multiple input multiple output mimo system with no outputs and ni inputs while many methods have been proposed to blindly identify convolutive mimo systems with no ges ni overdetermined very scarce results exist for the case of underdetermined all of which refer to systems that either have some special structure or special and values in this paper we show that as long as independent of whether the system is overdetermined or underdetermined we can always find the appropriate order of statistics that guarantees identifiability of the system response within trivial ambiguities we also propose an algorithm to reach the solution that consists of parallel factorization parafac of a way tensor containing th order statistics of the system outputs followed by an iterative scheme for a certain order of statistics we provide the description of the class of identifiable mimo systems we also show that this class can be expanded by applying parafac decomposition to a pair of tensors instead of one tensor the proposed approach constitutes a novel scheme for estimation of underdetermined systems and improves over existing approaches for overdetermined systems

permanental partition models and markovian gibbs structures we study both time invariant and time varying gibbs distributions for configurations of particles into disjoint clusters specifically we introduce and give some fundamental properties for a class of partition models called permanental partition models whose distributions are expressed in terms of the permanent of a similarity matrix parameter we show that in the time invariant case the permanental partition model is a refinement of the celebrated pitman ewens distribution whereas in the time varying case the permanental model refines the ewens cut and paste markov chains j appl probab 43 3 778 791 2011 by a special property of the permanent the partition function can be computed exactly allowing us to make several precise statements about this general model including a characterization of exchangeable and consistent permanental models

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

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

generalized ewens pitman model for bayesian clustering we propose a bayesian method for clustering from discrete data structures that commonly arise in genetics and other applications this method is equivariant with respect to relabelling units unsampled units do not interfere with sampled data and missing data do not hinder inference cluster inference using the posterior mode performs well on simulated and real datasets and the posterior predictive distribution enables supervised learning based on a partial clustering of the sample

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

on the unidentifiability of the fixed effects 3pl model the paper offers a general review of the basic concepts of both statistical model and parameter identification and revisits the conceptual relationships between parameter identification and both parameter interpretability and properties of parameter estimates all these issues are then exemplified for the 1pl 2pl and 1pl g fixed effects models for the 3pl model however we provide a theorem proving that the item parameters are not identified do not have an empirical interpretation and that it is not possible to obtain consistent and unbiased estimates of them

a theory of statistical models for monte carlo integration summary the task of estimating an integral by monte carlo methods is formulated as a statistical model using simulated observations as data the difficulty in this exercise is that we ordinarily have at our disposal all of the information required to compute integrals exactly by calculus or numerical integration but we choose to ignore some of the information for simplicity or computational feasibility our proposal is to use a semiparametric statistical model that makes explicit what information is ignored and what information is retained the parameter space in this model is a set of measures on the sample space which is ordinarily an infinite dimensional object none the less from simulated data the base line measure can be estimated by maximum likelihood and the required integrals computed by a simple formula previously derived by vardi and by lindsay in a closely related model for biased sampling the same formula was also suggested by geyer and by meng and wong using entirely different arguments by contrast with geyer s retrospective likelihood a correct estimate of simulation error is available directly from the fisher information the principal advantage of the semiparametric model is that variance reduction techniques are associated with submodels in which the maximum likelihood estimator in the submodel may have substantially smaller variance than the traditional estimator the method is applicable to markov chain and more general monte carlo sampling schemes with multiple samplers

three mode orthomax rotation factor analysis and principal components analysis pca are often followed by an orthomax rotation to rotate a loading matrix to simple structure the simple structure is usually defined in terms of the simplicity of the columns of the loading matrix in three mode pca rotational freedom of the so called core a three way array relating components for the three different modes can be used similarly to find a simple structure of the core simple structure of the core can be defined with respect to all three modes simultaneously possibly with different emphases on the different modes the present paper provides a fully flexible approach for orthomax rotation of the core to simple structure with respect to three modes simultaneously computationally this approach relies on repeated two way orthomax applied to supermatrices containing the frontal lateral or horizontal slabs respectively the procedure is illustrated by means of a number of exemplary analyses as a by product application of the three mode orthomax procedures to two way arrays is shown to reveal interesting relations with and interpretations of existing two way simple structure rotation techniques

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

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

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

randomness and degrees of irregularity the fundamental question are sequential data random arises in myriad contexts often with severe data length constraints furthermore there is frequently a critical need to delineate nonrandom sequences in terms of closeness to randomness e g to evaluate the efficacy of therapy in medicine we address both these issues from a computable framework via a quantification of regularity apen approximate entropy defining maximal randomness for sequences of arbitrary length indicating the applicability to sequences as short as n 5 points an infinite sequence formulation of randomness is introduced that retains the operational and computable features of the finite case in the infinite sequence setting we indicate how the foundational definition of independence in probability theory and the definition of normality in number theory reduce to limit theorems without rates of convergence from which we utilize apen to address rates of convergence of a deficit from maximal randomness refining the aforementioned concepts in a computationally essential manner representative applications among many are indicated to assess i random number generation output ii well shuffled arrangements and iii the quality of bootstrap replicates

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

identifying gene gene interactions using penalized tensor regression gene gene g g interactions have been shown to be critical for the fundamental mechanisms and development of complex diseases beyond main genetic effects the commonly adopted marginal analysis is limited by considering only a small number of g factors at a time with the main effects interactions hierarchical constraint many of the existing joint analysis methods suffer from prohibitively high computational cost in this study we propose a new method for identifying important g g interactions under joint modeling the proposed method adopts tensor regression to accommodate high data dimensionality and the penalization technique for selection it naturally accommodates the strong hierarchical structure without imposing additional constraints making optimization much simpler and faster than in the existing studies it outperforms multiple alternatives in simulation the analysis of the cancer genome atlas tcga data on lung cancer and melanoma demonstrates that it can identify markers with important implications and better prediction performance

measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

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

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

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

local versus global association for bivariate ordered responses summary local and global parameters modelling association are examined for contingency tables with both margins ordered responses local association parameters are crossratios of 2 x 2 subtables of adjacent cell probabilities global association parameters are cross ratios of quadrant probabilities the interpretation and various properties of these measures of association are compared and illustrated

on quantum statistical inference interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially in step with dramatic new developments in experimental techniques for studying small quantum systems furthermore developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory the present paper reviews this field and proposes and interrelates some new concepts for an extension of classical statistical inference to the quantum context

an analysis of transformations revisited rebutted abstract in a paper written in 1964 box and cox described a method for estimating transformations and showed how in suitable cases valuable increases in simplicity and efficiency were possible since that time this technique has enjoyed wide practical use and considerable success however a recent theoretical paper by bickel and doksum 1981 seems to suggest that serious dangers are associated with the employment of this method and speaks of instability and cost of estimation of the transformation these difficulties seem to be associated with 1 examples which common sense would rule out namely situations where the effect of transformation on the data is almost linear so that it is a matter of indifference which transformation is used 2 the idea that it makes sense to state conclusions in terms of a number measured on an arbitrary scale 3 failure to take proper account of the jacobian of the transformation

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

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

blind mimo system estimation based on parafac decomposition of higher order output tensors we present a novel framework for the identification of a multiple input multiple output mimo system driven by white mutually independent unobservable inputs samples of the system frequency response are obtained based on parallel factorization parafac of three or four way tensors constructed based on respectively third or fourth order cross spectra of the system outputs the main difficulties in frequency domain methods are frequency dependent permutation and filtering ambiguities we show that the information available in the higher order spectra allows for the ambiguities to be resolved up to a constant scaling and permutation ambiguities and a linear phase ambiguity important features of the proposed approach are that it does not require channel length information needs no phase unwrapping and unlike the majority of existing methods needs no prewhitening of the system outputs

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

blind identification of underdetermined mixtures based on the characteristic function the complex case blind identification of underdetermined mixtures can be addressed efficiently by using the second characteristic function caf of the observations our contribution is twofold first we propose the use of a levenberg marquardt algorithm herein called lemacaf as an alternative to an alternating least squares algorithm known as alescaf which has been used recently in the case of real mixtures of real sources second we extend the caf approach to the case of complex sources for which the previous algorithms are not suitable we show that the complex case involves an appropriate tensor stowage which is linked to a particular tensor decomposition an extension of the lemacaf algorithm called then proposed to blindly estimate the mixing matrix by exploiting this tensor decomposition in our simulation results we first provide performance comparisons between third and fourth order versions of alescaf and lemacaf in various situations involving bpsk sources then a performance study of is carried out considering 4 qam sources these results show that the proposed algorithm provides satisfying estimations especially in the case of a large underdeterminacy level

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

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

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

sequential unfolding svd for tensors with applications in array signal processing this paper contributes to the field of higher order n 2 tensor decompositions in signal processing a novel paratree tensor model is introduced accompanied with sequential unfolding svd susvd algorithm susvd as the name indicates applies a matrix singular value decomposition sequentially on the unfolded tensor reshaped from the right hand basis vectors of the svd of the previous mode the consequent paratree model is related to the well known family of parafac tensor decomposition models both of them describe a tensor as a sum of rank 1 tensors but paratree has several advantages over parafac when it is applied as a lower rank approximation technique paratree is orthogonal due to susvd fast and reliable to compute and the order or rank of the decomposition can be adaptively adjusted the low rank paratree approximation can be applied for e g reducing computational complexity in inverse problems measurement noise suppression as well as data compression the benefits of the proposed algorithm are illustrated through application examples in signal processing in comparison to parafac and hosvd

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

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

a coordinate descent algorithm for complex joint diagonalization under hermitian and transpose congruences abstract this paper deals with the problem of joint complexmatrix diagonalization by considering both the hermitian andtranspose congruences we address the general case where thesearched diagonalizing matrix is apriori nonunitary based onthe minimization of a quadratic criterion we propose a ex ible algorithm in the sense that it allows to directly consider arectangular diagonalizing matrix and to take into considerationboth the hermitian and transpose congruences within the sameframework the proposed algorithm is a coordinate descentalgorithm that is based on an approximate criterion leading to theanalytical derivation of the minima arguments computer simu lations are drawn to illustrate the usefulness and performancesof the algorithm and a comparison to state of the art algorithmsis presented finally an application to independent componentanalysis based on fourth order statistics is also presented index terms blind source separation complex symmetric ma trices hermitian matrices independent component analysis jointdiagonalization

the focused information criterion a variety of model selection criteria have been developed of general and specific types most of these aim at selecting a single model with good overall properties for example formulated via average prediction quality or shortest estimated overall distance to the true model the akaike the bayesian and the deviance information criteria along with many suitable variations are examples of such methods these methods are not concerned however with the actual use of the selected model which varies with context and application the present article takes the view that the model selector should instead focus on the parameter singled out for interest in particular a model that gives good precision for one estimand may be worse when used for inference for another estimand we develop a method that for a given focus parameter estimates the precision of any submodel based estimator the framework is that of large sample likelihood inference using an unbiased estimate of limiting risk we propose a foc

maximum likelihood estimation in semiparametric regression models with censored data summary semiparametric regression models play a central role in formulating the effects of covariates on potentially censored failure times and in the joint modelling of incomplete repeated measures and failure times in longitudinal studies the presence of infinite dimensional parameters poses considerable theoretical and computational challenges in the statistical analysis of such models we present several classes of semiparametric regression models which extend the existing models in important directions we construct appropriate likelihood functions involving both finite dimensional and infinite dimensional parameters the maximum likelihood estimators are consistent and asymptotically normal with efficient variances we develop simple and stable numerical techniques to implement the corresponding inference procedures extensive simulation experiments demonstrate that the inferential and computational methods proposed perform well in practical settings applications to three medical studies yield important new insights we conclude that there is no reason theoretical or numerical not to use maximum likelihood estimation for semiparametric regression models we discuss several areas that need further research

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

mobius transformation and cauchy parameter estimation some properties of the ordinary two parameter cauchy family the circular or wrapped cauchy family and their connection via mobius transformation are discussed a key simplification is achieved by taking the parameter i to be a point in the complex plane rather than the real plane maximum likelihood estimation is studied in some detail it is shown that the density of any equivariant estimator is harmonic on the upper half plane in consequence the maximum likelihood estimator is unbiased for n 3 and every harmonic or analytic function of the maximum likelihood estimator is unbiased if its expectation is finite the joint density of the maximum likelihood estimator is obtained in exact closed form for samples of size n 4 and in approximate form for n 5 various marginal distributions including that of student s pivotal ratio are also obtained most results obtained in the context of the real cauchy family also apply to the wrapped cauchy family by mobius transformation

an h likelihood method for spatial mixed linear models based on intrinsic auto regressions summary we consider sparse spatial mixed linear models particularly those described by besag and higdon and develop an h likelihood method for their statistical inference the method proposed allows for singular precision matrices as it produces estimates that coincide with those from the residual maximum likelihood based on appropriate differencing of the data and has a novel approach to estimating precision parameters by a gamma linear model furthermore we generalize the h likelihood method to include continuum spatial variations by making explicit use of scaling limit connections between gaussian intrinsic markov random fields on regular arrays and the de wijs process keeping various applications of spatial mixed linear models in mind we devise a novel sparse conjugate gradient algorithm that allows us to achieve fast matrix free statistical computations we provide two applications the first is an extensive analysis of an agricultural variety trial that brings forward various new aspects of nearest neighbour adjustment such as effects on statistical analyses to changes of scale and use of implicit continuum spatial formulation the second application concerns an analysis of a large cotton field which gives a focus to matrix free computations the paper closes with some further considerations such as applications to irregularly spaced data use of the parametric bootstrap and some generalizations to the gaussian matern mixed effect models

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

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

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

maximum approximate entropy and r threshold a new approach for regularity changes detection abstract approximate entropy apen has been widely used as an estimator of regularity in many scientific fields it has proved to be a useful tool because of its ability to distinguish different system s dynamics when there is only available short length noisy data incorrect parameter selection embedding dimension m threshold r and data length n and the presence of noise in the signal can undermine the apen discrimination capacity in this work we show that r m a x a p e n m r m a x n a p e n m a x can also be used as a feature to discern between dynamics moreover the combined use of a p e n m a x and r m a x allows a better discrimination capacity to be accomplished even in the presence of noise we conducted our studies using real physiological time series and simulated signals corresponding to both low and high dimensional systems when a p e n m a x is incapable of discerning between different dynamics because of the noise presence our results suggest that r m a x provides additional information that can be useful for classification purposes based on cross validation tests we conclude that for short length noisy signals the joint use of a p e n m a x and r m a x can significantly decrease the misclassification rate of a linear classifier in comparison with their isolated use

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

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

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

on the relation between correlation dimension approximate entropy and sample entropy parameters and a fast algorithm for their calculation abstract we explore the relation between correlation dimension approximate entropy and sample entropy parameters which are commonly used in nonlinear systems analysis using theoretical considerations we identify the points which are shared by all these complexity algorithms and show explicitly that the above parameters are intimately connected and mutually interdependent a new geometrical interpretation of sample entropy and correlation dimension is provided and the consequences for the interpretation of sample entropy its relative consistency and some of the algorithms for parameter selection for this quantity are discussed to get an exact algorithmic relation between the three parameters we construct a very fast algorithm for simultaneous calculations of the above which uses the full time series as the source of templates rather than the usual 10 this algorithm can be used in medical applications of complexity theory as it can calculate all three parameters for a realistic recording of 10 4 points within minutes with the use of an average notebook computer

what is a statistical model this paper addresses two closely related questions what is a statistical model and what is a parameter the notions that a model must make sense and that a parameter must have a well defined meaning are deeply ingrained in applied statistical work reasonably well understood at an instinctive level but absent from most formal theories of modelling and inference in this paper these concepts are defined in algebraic terms using morphisms functors and natural transformations it is argued that inference on the basis of a model is not possible unless the model admits a natural extension that includes the domain for which inference is required for example prediction requires that the domain include all future units subjects or time points although it is usually not made explicit every sensible statistical model admits such an extension examples are given to show why such an extension is necessary and why a formal theory is required in the definition of a subparameter it is shown that certain parameter functions are natural and others are not inference is meaningful only for natural parameters this distinction has important consequences for the construction of prior distributions and also helps to resolve a controversy concerning the box cox model

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

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

exploiting efficient representations in large scale tensor decompositions decomposing tensors into simple terms is often an essential step toward discovering and understanding underlying processes or toward compressing data however storing the tensor and computing its

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

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

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

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

on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

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

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

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

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

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

not all possibly random sequences are created equal the need to assess the randomness of a single sequence especially a finite sequence is ubiquitous yet is unaddressed by axiomatic probability theory here we assess randomness via approximate entropy apen a computable measure of sequential irregularity applicable to single sequences of both even very short finite and infinite length we indicate the novelty and facility of the multidimensional viewpoint taken by apen in contrast to classical measures furthermore and notably for finite length finite state sequences one can identify maximally irregular sequences and then apply apen to quantify the extent to which given sequences differ from maximal irregularity via a set of deficit defm functions the utility of these defm functions which we show allows one to considerably refine the notions of probabilistic independence and normality is featured in several studies including i digits of e 2 and 3 both in base 2 and in base 10 and ii sequences given by fractional parts of multiples of irrationals we prove companion analytic results which also feature in a discussion of the role and validity of the almost sure properties from axiomatic probability theory insofar as they apply to specified sequences and sets of sequences in the physical world we conclude by relating the present results and perspective to both previous and subsequent studies

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

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

evidence for conformal invariance of crop yields the aim of this paper is to study the nature of spatial correlation of yields of agricultural crops the focus is primarily on natural or non anthropogenic spatial variation patterns that cannot be explained by topography by variety or treatment effects or by agricultural practices conformal invariance implies stationarity and isotropy and also determines the rate of decay of spatial correlations the resulting gaussian model is studied empirically to see whether it describes satisfactorily the pattern of spatial correlations observed in field trials of various crops by embedding the law in a larger

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

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

tensor decompositions with banded matrix factors the computation of the model parameters of a canonical polyadic decomposition cpd also known as the parallel factor parafac or canonical decomposition candecomp or cp decomposition is typically done by resorting to iterative algorithms e g either iterative alternating least squares type or descent methods in many practical problems involving tensor decompositions such as signal processing some of the matrix factors are banded first we develop methods for the computation of cpds with one banded matrix factor it results in best rank 1 tensor approximation problems second we propose methods to compute cpds with more than one banded matrix factor third we extend the developed methods to also handle banded and structured matrix factors such as hankel or toeplitz computer results are also reported

conditional inference and cauchy models summary many computations associated with the two parameter cauchy model are shown to be greatly simplified if the parameter space is represented by the complex plane rather than the real plane with this convention we show that the family is closed under m6bius transformation of the sample space there is a similar induced transformation on the parameter space the chief raison d etre of the paper however is that the two parameter cauchy model provides an example of a nonunique configuration ancillary in the sense of fisher 1934 such that the maximum likelihood estimate together with either ancillary is minimal sufficient some consequences for bayesian inference and non bayesian conditional inference are explored in particular it is shown that conditional coverage probability assessments depend on the choice of ancillary for moderate deviations the effect occurs in the op n 1 term for large deviations the relative effect is op n2

scaling multidimensional inference for structured gaussian processes exact gaussian process gp regression has cal o n 3 runtime for data size n making it intractable for large n many algorithms for improving gp scaling approximate the covariance with lower rank matrices other work has exploited structure inherent in particular covariance functions including gps with implied markov structure and inputs on a lattice both enable cal o n or cal o n log n runtime however these gp advances have not been well extended to the multidimensional input setting despite the preponderance of multidimensional applications this paper introduces and tests three novel extensions of structured gps to multidimensional inputs for models with additive and multiplicative kernels first we present a new method for inference in additive gps showing a novel connection between the classic backfitting method and the bayesian framework we extend this model using two advances a variant of projection pursuit regression and a laplace approximation for non gaussian observations lastly for multiplicative kernel structure we present a novel method for gps with inputs on a multidimensional grid we illustrate the power of these three advances on several data sets achieving performance equal to or very close to the naive gp at orders of magnitude less cost

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

computing the unique candecomp parafac decomposition of unbalanced tensors by homotopy method abstract the candecomp parafac decomposition cpd of the tensor whose maximal dimension is greater than its rank is considered we derive the upper bound of rank under which computing the cpd is equivalent to solving a structured polynomial system that is determined by the full rank factorization of the matricization of the tensor under the generic uniqueness conditions the cpd solutions of the system are isolated so that these solutions can be achieved by homotopy method

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

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