convexity of sample path performance and strong consistency of infinitesimal perturbation analysis estimates a strong consistency of infinitesimal perturbation analysis ipa estimates for a class of stochastic processes satisfying certain convex properties is established the approach is based on convex analysis of sample paths applications of the results to tandem and cyclic queueing systems are discussed possible extensions to more general systems are also illustrated

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

variance based extragradient methods with line search for stochastic variational inequalities in this paper we propose dynamic sampled stochastic approximated ds sa extragradient methods for stochastic variational inequalities svis that are robust with respect to an unknown lipschitz c

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

a simulation based approach to two stage stochastic programming with recourse in this paper we consider stochastic programming problems where the objective function is given as an expected value function we discuss monte carlo simulation based approaches to a numerical solution of such problems in particular we discuss in detail and present numerical results for two stage stochastic programming with recourse where the random data have a continuous multivariate normal distribution we think that the novelty of the numerical approach developed in this paper is twofold first various variance reduction techniques are applied in order to enhance the rate of convergence successful application of those techniques is what makes the whole approach numerically feasible second a statistical inference is developed and applied to estimation of the error validation of optimality of a calculated solution and statistically based stopping criteria for an iterative alogrithm 1998 the mathematical programming society inc published by elsevier science b v

stochastic optimization by simulation convergence proofs for the gi g 1 queue in steady state approaches like finite differences with common random numbers infinitesimal perturbation analysis and the likelihood ratio method have drawn a great deal of attention recently as ways of estimating the gradient of a performance measure with respect to continuous parameters in a dynamic stochastic system in this paper we study the use of such estimators in stochastic approximation algorithms to perform so called single run optimizations of steady state systems under mild conditions for an objective function that involves the mean system time in a gi g 1 queue we prove that many variant of these algorithms converge to the minimizer in most cases however the simulation length must be increased from iteration to iteration otherwise the algorithm may converge to the wrong value one exception is a particular implementation of infinitesimal perturbation analysis for which the single run optimization converges to the optimum even with a fixed and small number of ends of service per iteration as a by product of our convergence proofs we obtain some properties of the derivative estimators that could be of independent interest our analysis exploits the regenerative structure of the system but our derivative estimation and optimization algorithms do not always take advantage of that regenerative structure in a companion paper we report numerical experiments with an m m 1 queue which illustrate the basis convergence properties and possible pitfalls of the various techniques

approximation to optimization problems an elementary review during the last two decades the concept of epi convergence was introduced and then was used in various investigations in optimization and related areas the aim of this review is to show in an elementary way how closely the arguments in the epi convergence approach are related to those of the classical theory of convergence of functions

stochastic decomposition an algorithm for two stage linear programs with recourse we present a cutting plane algorithm for two stage stochastic linear programs with recourse motivated by benders decomposition our method uses randomly generated observations of random variables to construct statistical estimates of supports of the objective function in general the resulting piecewise linear approximations do not agree with the objective function in finite time however certain subsequences of the estimated supports are shown to accumulate at supports of the objective function with probability one from this we establish the convergence of the algorithm under relatively mild assumptions

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

sample path optimization of convex stochastic performance functions in this paper we propose a method for optimizing convex performance functions in stochastic systems these functions can include expected performance in static systems and steady state performance in discrete event dynamic systems they may be nonsmooth the method is closely related to retrospective simulation optimization it appears to overcome some limitations of stochastic approximation which is often applied to such problems we explain the method and give computational results for two classes of problems tandem production lines with up to 50 machines and stochastic pert program evaluation and review technique problems with up to 70 nodes and 110 arcs

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

analysis of sample path optimization sample path optimization is a method for optimizing limit functions occurring in stochastic modeling problems such as steady state functions in discrete event dynamic systems it is closely related to retrospective optimization techniques and to m estimation the method has been computationally tested elsewhere on problems arising in production and in project planning with apparent success in this paper we provide a mathematical justification for sample path optimization by showing that under certain assumptions which hold for the problems just mentioned the method will almost surely find a point that is in a specified sense sufficiently close to the set of optimizers of the limit function

the resolvent expansion for second order regular singular operators abstract the paper gives a scheme for computing the asymptotics of tr e tl as t 0 where l is an elliptic operator of the form l d 2 x 2 a x and a x is a family of operators satisfying appropriate ellipticity and smoothness conditions a principle example is the laplace operator for a manifold with an asymptotically conic singularity the expansion has the usual terms away from the singularity appropriately regularized at x 0 plus singular contributions determined by the function of a 0 1 4 1 2 applications to index theorems are given in a subsequent paper

asymptotic behavior of optimal solutions in stochastic programming asymptotic behavior of optimal solutions xi n of a sequence of stochastic programming problems is studied variational and generalized equations approaches are discussed an expansion of xi n in terms of a parametrized mathematical programming problem depending on a single random vector is given when optimal solutions of the parametrized program are directionally differentiable this expansion leads to a close form expression for the asymptotic distribution of xi n applicability of the involved regularity conditions to nondifferentiable cases and in particular to stochastic programming with recourse is discussed

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

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

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

variable number sample path optimization the sample path method is one of the most important tools in simulation based optimization the basic idea of the method is to approximate the expected simulation output by the average of sample observations with a common random number sequence in this paper we describe a new variant of powell s unconstrained optimization by quadratic approximation uobyqa method which integrates a bayesian variable number sample path vnsp scheme to choose appropriate number of samples at each iteration the statistically accurate scheme determines the number of simulation runs and guarantees the global convergence of the algorithm the vnsp scheme saves a significant amount of simulation operations compared to general purpose fixed number sample path methods we present numerical results based on the new 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

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

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

nondifferentiability of the steady state function in discrete event dynamic systems this paper suggests that expected value performance functions in discrete event dynamic systems can be nondifferentiable at dense sets of points in the parameter space when the sample performance functions are convex and the distributions of events times contain atoms a general result is first proved for regenerative processes and then applied to simple queueing examples where nondifferentiability at dense sets is established

stochastic optimization using a trust region method and random models in this paper we propose and analyze a trust region model based algorithm for solving unconstrained stochastic optimization problems our framework utilizes random models of an objective function f x obtained from stochastic observations of the function or its gradient our method also utilizes estimates of function values to gauge progress that is being made the convergence analysis relies on requirements that these models and these estimates are sufficiently accurate with high enough but fixed probability beyond these conditions no assumptions are made on how these models and estimates are generated under these general conditions we show an almost sure global convergence of the method to a first order stationary point in the second part of the paper we present examples of generating sufficiently accurate random models under biased or unbiased noise assumptions lastly we present some computational results showing the benefits of the proposed method compared to existing approaches that are based on sample averaging or stochastic gradients

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

stochastic nelder mead simplex method a new globally convergent direct search method for simulation optimization abstract nelder mead simplex method nm originally developed in deterministic optimization is an efficient direct search method that optimizes the response function merely by comparing function values while successful in deterministic settings the application of nm to simulation optimization suffers from two problems 1 it lacks an effective sample size scheme for controlling noise consequently the algorithm can be misled to the wrong direction because of noise and 2 it is a heuristic algorithm the quality of estimated optimal solution cannot be quantified we propose a new variant called stochastic nelder mead simplex method snm that employs an effective sample size scheme and a specially designed global and local search framework to address these two problems without the use of gradient information snm can handle problems where the response functions are nonsmooth or gradient does not exist this is complementary to the existing gradient based approaches we prove that snm can converge to the true global optima with probability one an extensive numerical study also shows that the performance snm is promising and is worthy of further investigation

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

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

uobyqa unconstrained optimization by quadratic approximation uobyqa is a new algorithm for general unconstrained optimization calculations that takes account of the curvature of the objective function f say by forming quadratic models by interpolation therefore because no first derivatives are required each model is defined by n 1 n 2 values of f where n is the number of variables and the interpolation points must have the property that no nonzero quadratic polynomial vanishes at all of them a typical iteration of the algorithm generates a new vector of variables widetilde underline x t say either by minimizing the quadratic model subject to a trust region bound or by a procedure that should improve the accuracy of the model then usually f widetilde underline x t is obtained and one of the interpolation points is replaced by widetilde underline x t therefore the paper addresses the initial positions of the interpolation points the adjustment of trust region radii the calculation of widetilde underline x t in the two cases that have been mentioned and the selection of the point to be replaced further uobyqa works with the lagrange functions of the interpolation equations explicitly so their coefficients are updated when an interpolation point is moved the lagrange functions assist the procedure that improves the model and also they provide an estimate of the error of the quadratic approximation to f which allows the algorithm to achieve a fast rate of convergence these features are discussed and a summary of the algorithm is given finally a fortran implementation of uobyqa is applied to several choices of f in order to investigate accuracy robustness in the presence of rounding errors the effects of first derivative discontinuities and the amount of work the numerical results are very promising for n 20 but larger values are problematical because the routine work of an iteration is of fourth order in the number of variables

some matrix variate distribution theory notational considerations and a bayesian application summary we introduce and justify a convenient notation for certain matrix variate distributions which by its emphasis on the important underlying parameters and the theory on which it is based eases greatly the task of manipulating such distributions important examples include the matrix variate normal t f and beta and the wishart and inverse wishart distributions the theory is applied to compound matrix distributions and to bayesian prediction in the multivariate linear model

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

line search methods with variable sample size for unconstrained optimization minimization of unconstrained objective functions in the form of mathematical expectation is considered the sample average approximation saa method transforms the expectation objective function into a real valued deterministic function using a large sample and thus deals with deterministic function minimization the main drawback of this approach is its cost a large sample of the random variable that defines the expectation must be taken in order to get a reasonably good approximation and thus the sample average approximation method requires a very large number of function evaluations we present a line search strategy that uses variable sample size and thus makes the process significantly cheaper two measures of progress lack of precision and a decrease of function value are calculated at each iteration based on these two measures a new sample size is determined the rule we present allows us to increase or decrease the sample size at each iteration until we reach some neighborhood of the solution an additional safeguard check is performed to avoid unproductive sample decrease eventually the maximal sample size is reached so that the variable sample size strategy generates a solution of the same quality as the saa method but with a significantly smaller number of function evaluations the algorithm is tested on a couple of examples including the discrete choice problem

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

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

perturbation analysis gives strongly consistent sensitivity estimates for the m g 1 queue the technique of perturbation analysis has recently been introduced as an efficient way to compute parameter sensitivities for discrete event systems thus far the statistical properties of perturbation analysis have been validated mainly through experiments this paper considers for an m g 1 queueing system the sensitivity of mean system time of a customer to a parameter of the arrival or service distribution it shows analytically that i the steady state value of the perturbation analysis estimate of this sensitivity is unbiased and ii a perturbation analysis algorithm implemented on a single sample path of the system gives asymptotically unbiased and strongly consistent estimates of this sensitivity no previous knowledge of perturbation analysis is assumed so the paper also serves to introduce this technique to the unfamiliar reader numerical extensions to gi g 1 queues and applications to optimization problems are also illustrated

sample size selection in optimization methods for machine learning this paper presents a methodology for using varying sample sizes in batch type optimization methods for large scale machine learning problems the first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the function and gradient we propose a criterion for increasing the sample size based on variance estimates obtained during the computation of a batch gradient we establish an o 1 epsilon complexity bound on the total cost of a gradient method the second part of the paper describes a practical newton method that uses a smaller sample to compute hessian vector products than to evaluate the function and the gradient and that also employs a dynamic sampling technique the focus of the paper shifts in the third part of the paper to l 1 regularized problems designed to produce sparse solutions we propose a newton like method that consists of two phases a minimalistic gradient projection phase that identifies zero variables and subspace phase that applies a subsampled hessian newton iteration in the free variables numerical tests on speech recognition problems illustrate the performance of the algorithms

proximity control in bundle methods for convex proximal bundle methods for minimizing a convex functionf generate a sequence x k by takingxk 1 to be the minimizer of hat f k x u k x x k 2 2 where hat f k is a sufficiently accurate polyhedral approximation tof andu k 0 the usual choice ofu k 1 may yield very slow convergence a technique is given for choosing u k adaptively that eliminates sensitivity to objective scaling some encouraging numerical experience is reported

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

stochastic optimization by simulation numerical experiments with the m m 1 queue in steady state this paper gives numerical illustrations of the behavior of stochastic approximation combined with different derivative estimation techniques to optimize a steady state system it is a companion paper to l ecuyer and glynn 1993 which gives convergence proofs for most of the variants experimented here the numerical experiments are made with a simple m m 1 queue which while simple serves to illustrate the basic convergence properties and possible pitfalls of the various techniques

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

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

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