heterogeneous tensor decomposition for clustering via manifold optimization tensor clustering is an important tool that exploits intrinsically rich structures in real world multiarray or tensor datasets often in dealing with those datasets standard practice is to use subspace clustering that is based on vectorizing multiarray data however vectorization of tensorial data does not exploit complete structure information in this paper we propose a subspace clustering algorithm without adopting any vectorization process our approach is based on a novel heterogeneous tucker decomposition model taking into account cluster membership information we propose a new clustering algorithm that alternates between different modes of the proposed heterogeneous tensor model all but the last mode have closed form updates updating the last mode reduces to optimizing over the multinomial manifold for which we investigate second order riemannian geometry and propose a trust region algorithm numerical experiments show that our proposed algorithm compete effectively with state of the art clustering algorithms that are based on tensor factorization

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

weighted least squares fitting using ordinary least squares algorithms a general approach for fitting a model to a data matrix by weighted least squares wls is studied this approach consists of iteratively performing steps of existing algorithms for ordinary least squares ols fitting of the same model the approach is based on minimizing a function that majorizes the wls loss function the generality of the approach implies that for every model for which an ols fitting algorithm is available the present approach yields a wls fitting algorithm in the special case where the wls weight matrix is binary the approach reduces to missing data imputation

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

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

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

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

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

reduction of multivariate mixtures and its applications abstract we consider fast deterministic algorithms to identify the best linearly independent terms in multivariate mixtures and use them to compute up to a user selected accuracy an equivalent representation with fewer terms importantly the multivariate mixtures do not have to be a separated representation of a function one algorithm employs a pivoted cholesky decomposition of the gram matrix constructed from the terms of the mixture to select what we call skeleton terms and the other uses orthogonalization for the same purpose both algorithms require o r 2 n p d r n operations where n is the initial number of terms in the multivariate mixture r is the number of selected linearly independent terms and p d is the cost of computing the inner product between two terms of a mixture in d variables for general gaussian mixtures p d d 3 since we need to diagonalize a d d matrix whereas for separated representations p d d there is no need for diagonalization due to conditioning issues the resulting accuracy is limited to about one half of the available significant digits for both algorithms we also describe an alternative algorithm that is capable of achieving higher accuracy but is only applicable in low dimensions or to multivariate mixtures in separated form we describe a number of initial applications of these algorithms to solve partial differential and integral equations and to address several problems in data science for data science applications in high dimensions we consider the kernel density estimation kde approach for constructing a probability density function pdf of a cloud of points a far field kernel summation method and the construction of equivalent sources for non oscillatory kernels used in both computational physics and data science and finally show how to use the new algorithm to produce seeds for subdividing a cloud of points into groups

p adic properties of division polynomials and elliptic divisibility sequences for a fixed rational point p e k on an elliptic curve we consider the sequence of values f n p n 1 of the division polynomials of e at p for a finite field open image in new window we prove that the sequence is periodic for a local field open image in new window we prove under certain hypotheses that there is a power q p e so that for all m 1 the limit of open image in new window exists in k and is algebraic over open image in new window we apply this result to prove an analogous p adic limit and algebraicity result for elliptic divisibility sequences

a fixed point iterative method for low n rank tensor pursuit the linearly constrained tensor n rank minimization problem is an extension of matrix rank minimization it is applicable in many fields which use the multi way data such as data mining machine learning and computer vision in this paper we adapt operator splitting technique and convex relaxation technique to transform the original problem into a convex unconstrained optimization problem and propose a fixed point iterative method to solve it we also prove the convergence of the method under some assumptions by using a continuation technique we propose a fast and robust algorithm for solving the tensor completion problem which is called fp lrtc fixed point for low n rank tensor completion our numerical results on randomly generated and real tensor completion problems demonstrate that this algorithm is effective especially for easy problems

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

sequential vs parallel complexity in simple gene assembly we investigate some differences between the general intramolecular model for gene assembly and its restricted simple model although both models satisfactorily sort all current experimental data we show that the general model offers assembly strategies for a given string that vary in both assembly length and the operations used while the simple model will always use the same number of each type of operation to sort a gene when simple operations are applied in parallel this is given a new twist we prove that for any n 1 there exists a string having maximally parallel assemblies of any length between n and 2n

sigma function solution of the initial value problem for somos 5 sequences the somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients for particular choices of coefficients and initial data integer sequences arise by making the connection with a second order nonlinear mapping with a first integral we prove that the two subsequences of odd even index terms each satisfy a somos 4 fourth order recurrence this leads directly to the explicit solution of the initial value problem for the somos 5 sequences in terms of the weierstrass sigma function for an associated elliptic curve

on two dimensional sparse matrix partitioning models methods and a recipe we consider two dimensional partitioning of general sparse matrices for parallel sparse matrix vector multiply operation we present three hypergraph partitioning based methods each having unique advantages the first one treats the nonzeros of the matrix individually and hence produces fine grain partitions the other two produce coarser partitions where one of them imposes a limit on the number of messages sent and received by a single processor and the other trades that limit for a lower communication volume we also present a thorough experimental evaluation of the proposed two dimensional partitioning methods together with the hypergraph based one dimensional partitioning methods using an extensive set of public domain matrices furthermore for the users of these partitioning methods we present a partitioning recipe that chooses one of the partitioning methods according to some matrix characteristics

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

parallel tensor methods for high dimensional linear pdes abstract high dimensional partial differential equations pdes arise in a number of fields of science and engineering where they are used to describe the evolution of joint probability functions their examples include the boltzmann and fokker planck equations we develop new parallel algorithms to solve such high dimensional pdes the algorithms are based on canonical and hierarchical numerical tensor methods combined with alternating least squares and hierarchical singular value decomposition both implicit and explicit integration schemes are presented and discussed we demonstrate the accuracy and efficiency of the proposed new algorithms in computing the numerical solution to both an advection equation in six variables plus time and a linearized version of the boltzmann equation

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

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

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

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

decomposing eeg data into space time frequency components using parallel factor analysis abstract finding the means to efficiently summarize electroencephalographic data has been a long standing problem in electrophysiology a popular approach is identification of component modes on the basis of the time varying spectrum of multichannel eeg recordings in other words a space frequency time atomic decomposition of the time varying eeg spectrum previous work has been limited to only two of these dimensions principal component analysis pca and independent component analysis ica have been used to create space time decompositions suffering an inherent lack of uniqueness that is overcome only by imposing constraints of orthogonality or independence of atoms conventional frequency time decompositions ignore the spatial aspects of the eeg framing of the data being as a three way array indexed by channel frequency and time allows the application of a unique decomposition that is known as parallel factor analysis parafac each atom is the tri linear decomposition into a spatial spectral and temporal signature we applied this decomposition to the eeg recordings of five subjects during the resting state and during mental arithmetic common to all subjects were two atoms with spectral signatures whose peaks were in the theta and alpha range these signatures were modulated by physiological state increasing during the resting stage for alpha and during mental arithmetic for theta furthermore we describe a new method source spectra imaging or ssi to estimate the location of electric current sources from the eeg spectrum the topography of the theta atom is frontal and the maximum of the corresponding ssi solution is in the anterior frontal cortex the topography of the alpha atom is occipital with maximum of the ssi solution in the visual cortex we show that the proposed decomposition can be used to search for activity with a given spectral and topographic profile in new recordings and that the method may be useful for artifact recognition and removal

formal systems for gene assembly in ciliates dna processing in ciliates a very ancient group of organisms is among the most sophisticated dna processing in living organisms it has a quite clear computational structure and even uses explicitly the linked list data structure particularly interesting from the computational point of view is the process of gene assembly from its micronuclear to its macronuclear form we investigate here the string rewriting and the graph rewriting models of this process involving three molecular operations which together form a universal set of operations in the sense that they can assembly any macronuclear gene from its micronuclear form in particular we prove that although the graph rewriting system is more abstract than the string rewriting system no essential information is lost in the sense that one can translate assembly strategies from one system into the other

independent component analysis and simultaneous third order tensor diagonalization comon s 1994 well known scheme for independent component analysis ica is based on the maximal diagonalization in a least squares sense of a higher order cumulant tensor in a previous paper we proved that for fourth order cumulants the computation of an elementary jacobi rotation is equivalent to the computation of the best rank 1 approximation of a fourth order tensor in this paper we show that for third order tensors the computation of an elementary jacobi rotation is again equivalent to a best rank 1 approximation however here it is a matrix that has to be approximated this favorable computational load makes it attractive to do something third order like for fourth order cumulant tensors as well we show that simultaneous optimal diagonalization of third order tensor slices of the fourth order cumulant is a suitable strategy this simultaneous third order tensor diagonalization approach stotd is similar in spirit to the efficient jade algorithm

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

a generalization of takane s algorithm for dedicom an algorithm is described for fitting the dedicom model for the analysis of asymmetric data matrices this algorithm generalizes an algorithm suggested by takane in that it uses a damping parameter in the iterative process takane s algorithm does not always converge monotonically based on the generalized algorithm a modification of takane s algorithm is suggested such that this modified algorithm converges monotonically it is suggested to choose as starting configurations for the algorithm those configurations that yield closed form solutions in some special cases finally a sufficient condition is described for monotonic convergence of takane s original algorithm

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

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

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

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

a least squares framework for component analysis over the last century component analysis ca methods such as principal component analysis pca linear discriminant analysis lda canonical correlation analysis cca locality preserving projections lpp and spectral clustering sc have been extensively used as a feature extraction step for modeling classification visualization and clustering ca techniques are appealing because many can be formulated as eigen problems offering great potential for learning linear and nonlinear representations of data in closed form however the eigen formulation often conceals important analytic and computational drawbacks of ca techniques such as solving generalized eigen problems with rank deficient matrices e g small sample size problem lacking intuitive interpretation of normalization factors and understanding commonalities and differences between ca methods this paper proposes a unified least squares framework to formulate many ca methods we show how pca lda cca lpp sc and its kernel and regularized extensions correspond to a particular instance of least squares weighted kernel reduced rank regression ls wkrrr the ls wkrrr formulation of ca methods has several benefits 1 provides a clean connection between many ca techniques and an intuitive framework to understand normalization factors 2 yields efficient numerical schemes to solve ca techniques 3 overcomes the small sample size problem 4 provides a framework to easily extend ca methods we derive weighted generalizations of pca lda sc and cca and several new ca techniques

fourth order cumulant based blind identification of underdetermined mixtures in this paper we study two fourth order cumulant based techniques for the estimation of the mixing matrix in underdetermined independent component analysis the first method is based on a simultaneous matrix diagonalization the second is based on a simultaneous off diagonalization the number of sources that can be allowed is roughly quadratic in the number of observations for both methods explicit expressions for the maximum number of sources are given simulations illustrate the performance of the techniques

pseudo spectra theory of tensors and tensor polynomial eigenvalue problems abstract this paper is devoted to the extension of the pseudo spectra theory from matrices to tensors based on the definition of an eigenpair of real symmetric tensors and results on the pseudo spectrum of square matrices we first introduce the pseudo spectrum of a complex tensor and investigate its fundamental properties such as its computational interpretations and the link with the stability of its related homogeneous dynamical system we then extend the pseudo spectrum to the setting of tensor polynomial eigenvalue problems we further derive basic structure of the pseudo spectrum for tensor polynomial eigenvalue problems including the symmetry boundedness and number of connected components under suitable mild assumptions finally we discuss the implication of the pseudo spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial

geometry and the complexity of matrix multiplication it has been known for some time that algebraic geometry and representation theory are useful for proving lower bounds on the complexity of the matrix multiplication operator in this talk i will explain how geometry can be used to prove both upper and lower bounds after a discussion of the problem and its history i will present very recent work with m michalek on border rank algorithms faculty host eric allender

the border rank of the multiplication of 2 2 matrices is seven one of the leading problems of algebraic complexity theory is matrix multiplication the naive multiplication of two n n matrices uses n multiplications in 1969 strassen 20 presented an explicit algorithm for multiplying 2 2 matrices using seven multiplications in the opposite direction hopcroft and kerr 12 and winograd 22 proved independently that there is no algorithm for multiplying 2 2 matrices using only six multiplications the precise number of multiplications needed to execute matrix multiplication or any given bilinear map is called the rank of the bilinear map a related problem is to determine the border rank of matrix multiplication or any given bilinear map first introduced in 6 5 roughly speaking some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small good perturbations of the map these perturbed maps can give rise to fast exact algorithms for matrix multiplication see 7 the border rank made appearances in the literature in the 1980s and early 1990s see e g 6 5 19 8 15 2 10 11 4 3 17 16 18 1 9 21 but to our knowledge there has not been much progress on the question since then more precisely for any complex projective variety x cp pv and point p pv define the x rank of p to be the smallest number r such that p is in the linear span of r points of x define r x the r th secant variety of x to be the zariski closure of the set of points of x rank r and define the x border rank of p to be the smallest r such that p r x the terminology is motivated by the case x seg pa 1 pb 1 p c c the segre variety of rank one matrices then the x rank of a matrix is just its usual rank let a b c be vector spaces and let f a b c be a bilinear map i e an element of a b c let x seg pa pb pc p a b c denote the segre variety of decomposable tensors in a b c the border rank of a bilinear map is its x border rank while for the segre product of two projective spaces border rank coincides with rank here they can be quite different in this paper we prove the theorem stated in the title let mmult c c c denote the matrix multiplication operator for 2 2 matrices strassen 18 showed that mmult 5 p p p our method of proof is to decompose

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

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

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

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

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

real rank two geometry abstract the real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points we seek a semi algebraic description of this set its algebraic boundary consists of the tangential variety and the edge variety our study of segre and veronese varieties yields a characterization of tensors of real rank two

partial and total matrix multiplication in 1979 considerable progress was made in estimating the complexity of matrix multiplication here the new techniques and recent results are presented based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication some of the entries of the matrices may be zero can efficiently be utilized to perform multiplication of large total matrices by combining pan s trilinear technique with a strong version of our compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of n times n matrices over arbitrary fields is possible in time o n beta where beta is a bit smaller than 3 ln 52 ln 110 approx 2 522

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

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

handwritten digit classification using higher order singular value decomposition in this paper we present two algorithms for handwritten digit classification based on the higher order singular value decomposition hosvd the first algorithm uses hosvd for construction of the class models and achieves classification results with error rate lower than 6 the second algorithm uses the hosvd for tensor approximation simultaneously in two modes classification results for the second algorithm are almost down at 5 even though the approximation reduces the original training data with more than 98 before the construction of the class models the actual classification in the test phase for both algorithms is conducted by solving a series least squares problems considering computational amount for the test presented the second algorithm is twice as efficient as the first one

reducibility of gene patterns in ciliates using the breakpoint graph gene assembly in ciliates is one of the most involved dna processings going on in any organism this process transforms one nucleus the micronucleus into another functionally different nucleus the macronucleus we continue the development of the theoretical models of gene assembly and in particular we demonstrate the use of the concept of the breakpoint graph known from another branch of dna transformation research more specifically 1 we characterize the intermediate gene patterns that can occur during the transformation of a given micronuclear gene pattern to its macronuclear form 2 we determine the number of applications of the loop recombination operation the most basic of the three molecular operations that accomplish gene assembly needed in this transformation 3 we generalize previous results and give elegant alternatives for some proofs concerning characterizations of the micronuclear gene patterns that can be assembled using a specific subset of the three molecular operations

parallel candecomp parafac decomposition of sparse tensors using dimension trees candecomp parafac cp decomposition of sparse tensors has been successfully applied to many problems in web search graph analytics recommender systems health care data analytics and many other domains in these applications efficiently computing the cp decomposition of sparse tensors is essential in order to be able to process and analyze data of massive scale for this purpose we investigate an efficient computation of the cp decomposition of sparse tensors and its parallelization we propose a novel computational scheme for reducing the cost of a core operation in computing the cp decomposition with the traditional alternating least squares cp als based algorithm we then effectively parallelize this computational scheme in the context of cp als in shared and distributed memory environments and propose data and task distribution models for better scalability we implement parallel cp als algorithms and compare our implementations with an efficient tensor factorization library using tensors form

a randomized algorithm for a tensor based generalization of the singular value decomposition an algorithm is presented and analyzed that when given as input a d mode tensor a computes an approximation a the approximation a is computed by performing the following for each of the d modes first form implicitly a matrix by unfolding the tensor along that mode then choose columns from the matrices thus generated and finally project the tensor along that mode onto the span of those columns an important issue affecting the quality of the approximation is the choice of the columns from the matrices formed by unfolding the tensor along each of its modes in order to address this issue two algorithms of independent interest are presented that given an input matrix a and a target rank k select columns that span a space close to the best rank k subspace of the matrix for example in one of the algorithms a number c that depends on k an error parameter and a failure probability of columns are chosen in c independent random trials according to a nonuniform probability distribution depending on the euclidean lengths of the columns when this algorithm for choosing columns is used in the tensor approximation then under appropriate assumptions bounds of the form

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

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

combining overlap and containment for gene assembly in ciliates abstract gene assembly in ciliates is an intricate biological process that has been studied formally and modeled through string and graph rewriting systems recently a restriction of the general intramolecular model called simple gene assembly has been introduced this restriction has subsequently been defined as a string rewriting system we show that by extending the notion of overlap graph to also represent containment it is possible to define an equivalent graph rewriting system for two of the three types of rules that make up simple gene assembly it turns out that this graph rewriting system is often less involved to study than its corresponding string rewriting system we illustrate this by giving characterizations of the power of both types of graph rewriting rules also a first step is made towards a characterization of the extended overlap graphs

some uniqueness results for parafac2 whereas the unique axes properties of parafac1 have been examined extensively little is known about uniqueness properties for the parafac2 model for covariance matrices this paper is concerned with uniqueness in the rank two case of parafac2 for this case harshman and lundy have recently shown subject to mild assumptions that parafac2 is unique when five covariance matrices are analyzed in the present paper this result is sharpened parafac2 is shown to be usually unique with four matrices with three matrices it is not unique unless a certain additional assumption is introduced if for instance the diagonal matrices of weights are constrained to be non negative three matrices are enough to have uniqueness in the rank two case of parafac2

exact recovery in the hypergraph stochastic block model a spectral algorithm abstract we consider the exact recovery problem in the hypergraph stochastic block model hsbm with k blocks of equal size more precisely we consider a random d uniform hypergraph h with n vertices partitioned into k clusters of size s n k hyperedges e are added independently with probability p if e is contained within a single cluster and q otherwise where 0 q p 1 we present a spectral algorithm which recovers the clusters exactly with high probability given mild conditions on n k p q and d our algorithm is based on the adjacency matrix of h which is a symmetric n n matrix whose u v th entry is the number of hyperedges containing both u and v to the best of our knowledge our algorithm is the first to guarantee exact recovery when the number of clusters k n

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

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

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

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