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

two level multifactor designs for detecting the presence of interactions a design optimality criterion tr l optimality is applied to the problem of designing two level multifactor experiments to detect the presence of interactions among the controlled variables we give rules for constructing tr l optimal foldover designs and tr l optimal fractional factorial designs some results are given on the power of these designs for testing the hypothesis that there are no two factor interactions augmentation of the tr l optimal designs produces designs that achieve a compromise between the criteria of d optimality for parameter estimation in a first order model and tr l optimality for detecting lack of fit we give an example to demonstrate an application to the sensitivity analysis of a computer model

general tensor discriminant analysis and gabor features for gait recognition traditional image representations are not suited to conventional classification methods such as the linear discriminant analysis lda because of the undersample problem usp the dimensionality of the feature space is much higher than the number of training samples motivated by the successes of the two dimensional lda 2dlda for face recognition we develop a general tensor discriminant analysis gtda as a preprocessing step for lda the benefits of gtda compared with existing preprocessing methods such as the principal components analysis pca and 2dlda include the following 1 the usp is reduced in subsequent classification by for example lda 2 the discriminative information in the training tensors is preserved and 3 gtda provides stable recognition rates because the alternating projection optimization algorithm to obtain a solution of gtda converges whereas that of 2dlda does not we use human gait recognition to validate the proposed gtda the averaged gait images are utilized for gait representation given the popularity of gabor function based image decompositions for image understanding and object recognition we develop three different gabor function based image representations 1 gabord is the sum of gabor filter responses over directions 2 gabors is the sum of gabor filter responses over scales and 3 gaborsd is the sum of gabor filter responses over scales and directions the gabord gabors and gaborsd representations are applied to the problem of recognizing people from their averaged gait images a large number of experiments were carried out to evaluate the effectiveness recognition rate of gait recognition based on first obtaining a gabor gabord gabors or gaborsd image representation then using gdta to extract features and finally using lda for classification the proposed methods achieved good performance for gait recognition based on image sequences from the university of south florida usf humanid database experimental comparisons are made with nine state of the art classification methods in gait recognition

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

accelerated dynamic mri exploiting sparsity and low rank structure k t slr we introduce a novel algorithm to reconstruct dynamic magnetic resonance imaging mri data from under sampled k t space data in contrast to classical model based cine mri schemes that rely on the sparsity or banded structure in fourier space we use the compact representation of the data in the karhunen louve transform klt domain to exploit the correlations in the dataset the use of the data dependent kl transform makes our approach ideally suited to a range of dynamic imaging problems even when the motion is not periodic in comparison to current klt based methods that rely on a two step approach to first estimate the basis functions and then use it for reconstruction we pose the problem as a spectrally regularized matrix recovery problem by simultaneously determining the temporal basis functions and its spatial weights from the entire measured data the proposed scheme is capable of providing high quality reconstructions at a range of accelerations in addition to using the compact representation in the klt domain we also exploit the sparsity of the data to further improve the recovery rate validations using numerical phantoms and in vivo cardiac perfusion mri data demonstrate the significant improvement in performance offered by the proposed scheme over existing methods

dynamic mr image reconstruction separation from undersampled bf k t space via low rank plus sparse prior dynamic magnetic resonance imaging mri is used in multiple clinical applications but can stil lb enefi tf rom higher spatial or temporal resolution a dynamic mr image reconstruc tion method from partial space measurements is introduced that recovers and inherently separates the in formation in the dy namic scene the reconstructionmodel is based on a low rank plus sparse decomposition prior which is related to robust principal component analysis an algorithm is proposed to solve the convex optimization problem based on an alternating direction method of multipliers the method is validated with numerical phantom sim ulations and cardiac mri data against state of the art dynamic mri reconstruction methods results suggest that using the pro posed approach as a means of regularizing the inverse problem re mains competitive with state of the art reconstruction techniques additionally the decomposition induced by the reconstruction is shown to help in the context of motion estimation in dynamic con trast enhanced mri

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

multi dimensional low rank plus sparse decomposition for reconstruction of under sampled dynamic mri in this paper we introduce a multi dimensional approach to the problem of reconstruction of mr image sequences that are highly undersampled in k space by formulating the reconstruction as a high order low rank tensor plus sparse tensor decomposition problem we propose an efficient numerical algorithm based on the alternating direction method of multipliers admm to solve the optimization using tucker representation the sparse component is learnt efficiently with different sparsifying matrices along the modes of dynamic mr data to estimate the low rank tensor a convex cost function is defined to be the weighted sum of nuclear norms of its 3 unfolding matrices through extensive experimental results we show that our proposed method achieves superior reconstruction quality compared to the state of the art reconstruction methods graphical abstractdisplay omitted highlightsa novel multi dimensional analysis model is learnt to recover higher quality mri sequences the dynamic mri reconstruction is formulated as a higher dimensional low rank plus sparse tensor reconstruction problem an efficient numerical algorithm based on admm is proposed to solve the optimization problem

analysis and design of variable structure systems using a geometric approach multivariable variable structure systems in the sliding mode are studies using a geometric approach the properties of system order reduction and disturbance rejection are proved using projector theory new design methods for the choice of switching hyperplanes are derived for the closed loop eigenvalue eigenvector assignment problem

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

a new method of switching surface design for multivariable variable structure systems in this paper we discuss the problem of the switching surface for a multivariable variable structure system we first propose a new switching surface by which robustness of the controlled system to disturbances is provided based on the switching surface we develop control input to drive the system to track the desired trajectory this control input does not produce the chattering that normally occurs when a conventional control input is used we apply the proposed method to a multivariable linear system and discuss the simulation results to show the computational procedure and its effectiveness

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

tensor discriminant color space for face recognition recent research efforts reveal that color may provide useful information for face recognition for different visual tasks the choice of a color space is generally different how can a color space be sought for the specific face recognition problem to address this problem this paper represents a color image as a third order tensor and presents the tensor discriminant color space tdcs model the model can keep the underlying spatial structure of color images with the definition of n mode between class scatter matrices and within class scatter matrices tdcs constructs an iterative procedure to obtain one color space transformation matrix and two discriminant projection matrices by maximizing the ratio of these two scatter matrices the experiments are conducted on two color face databases ar and georgia tech face databases and the results show that both the performance and the efficiency of the proposed method are better than those of the state of the art color image discriminant model which involve one color space transformation matrix and one discriminant projection matrix specifically in a complicated face database with various pose variations

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

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

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

recovering low rank and sparse components of matrices from incomplete and noisy observations many problems can be characterized by the task of recovering the low rank and sparse components of a given matrix recently it was discovered that this nondeterministic polynomial time hard np hard task can be well accomplished both theoretically and numerically via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l 1 norm are utilized to induce low rank and sparsity this paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and gaussian noise we show that the resulting model falls into the applicable scope of the classical augmented lagrangian method moreover the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented lagrangian function hence some splitting numerical algorithms are developed for solving the new recovery model some preliminary numerical experiments verify that these augmented lagrangian based splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model

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

nonlinear integral type sliding surface for both matched and unmatched uncertain systems this note presents a new nonlinear integral type sliding surface which incorporates a virtual nonlinear nominal control to achieve prescribed specifications first the plant with matched uncertainties is considered the resultant closed loop system during ideal sliding mode behaves exactly like the nominal plant under the nonlinear nominal control which completely nullifies the matched uncertainties and consequently satisfies the prescribed specifications second the stability analysis of the proposed sliding mode for the systems with unmatched uncertainties is performed to exploit the stability conditions numerical results demonstrate the validity of the proposed concept

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

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

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

two stage sensitivity based group screening in computer experiments sophisticated computer codes that implement mathematical models of physical processes can involve large numbers of inputs and screening to determine the most active inputs is critical for understanding the input output relationship this article presents a new two stage group screening methodology for identifying active inputs in stage 1 groups of inputs showing low activity are screened out in stage 2 individual inputs from the active groups are identified inputs are evaluated through their estimated total effect sensitivity indices tsis which are compared with a benchmark null tsi distribution created from added low noise inputs examples show that compared with other procedures the proposed method provides more consistent and accurate results for high dimensional screening additional details and computer code are provided in supplementary materials available online

a graphical approach for evaluating response surface designs in terms of the mean squared error of prediction one of the fundamental purposes of response surface methodology is to develop a relatively simple linear model to explain a more complex phenomenon since the proposed model is only an approximation analysts almost always must address the potential of bias due to model misspecification this article proposes a graphical approach for evaluating response surface designs in terms of the mean squared error of prediction in particular this article develops measures of the maximum and minimum mean squared error of prediction over the surface of spheres through a region of interest these measures are then plotted against radius the resulting plots reveal at a glance how a response surface design performs in the presence of model misspecification these plots are used to investigate the performances of several common response surface designs

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

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

new methods to design an integral variable structure controller this paper presents two methods to design a single input single output integral variable structure system once an overall transfer function is selected two simple ways are presented to determine the required switching surface and the control function the proposed methods not only avoid transforming the original plant into a companion form but also enable the resulting control system to achieve the asymptotic tracking for a step reference signal and disturbance rejection for a bounded disturbance signal simultaneously two examples are given to illustrate the effectiveness of the proposed methods

parallel mr image reconstruction using augmented lagrangian methods magnetic resonance image mri reconstruction using sensitivity encoding sense requires regularization to suppress noise and aliasing effects edge preserving and sparsity based regularization criteria can improve image quality but they demand computation intensive nonlinear optimization in this paper we present novel methods for regularized mri reconstruction from undersampled sensitivity encoded data sense reconstruction using the augmented lagrangian al framework for solving large scale constrained optimization problems we first formulate regularized sense reconstruction as an unconstrained optimization task and then convert it to a set of equivalent constrained problems using variable splitting we then attack these constrained versions in an al framework using an alternating minimization method leading to algorithms that can be implemented easily the proposed methods are applicable to a general class of regularizers that includes popular edge preserving e g total variation and sparsity promoting e g norm of wavelet coefficients criteria and combinations thereof numerical experiments with synthetic and in vivo human data illustrate that the proposed al algorithms converge faster than both general purpose optimization algorithms such as nonlinear conjugate gradient ncg and state of the art mfista

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

choice of discontinuous control component for robust sliding mode performance the robustness properties of sliding mode behaviour are considered in terms of the choice of non linear control structure the performance of two classes of variable structure controllers in the presence of unmatched uncertainty is discussed criteria for suitable choice of a non linear control structure are presented and the concept of a performance sliding mode is introduced a numerical example is given to illustrate the phenomenon described

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

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

detection of interactions in experiments on large numbers of factors one of the main advantages of factorial experiments is the information that they can offer on interactions when there are many factors to be studied some or all of this information is often sacrificed to keep the size of an experiment economically feasible two strategies for group screening are presented for a large number of factors over two stages of experimentation with particular emphasis on the detection of interactions one approach estimates only main effects at the first stage classical group screening whereas the other new method interaction group screening estimates both main effects and key two factor interactions at the first stage three criteria are used to guide the choice of screening technique and also the size of the groups of factors for study in the first stage experiment the criteria seek to minimize the expected total number of observations in the experiment the probability that the size of the experiment exceeds a prespecified target and the proportion of active individual factorial effects which are not detected to implement these criteria results are derived on the relationship between the grouped and individual factorial effects and the probability distributions of the numbers of grouped factors whose main effects or interactions are declared active at the first stage examples are used to illustrate the methodology and some issues and open questions for the practical implementation of the results are discussed

hybrid cs dmri periodic time variant subsampling and omnidirectional total variation based reconstruction compressive sensing cs has been used to accelerate dynamic magnetic resonance imaging dmri currently the online cs dmri is faster whereas the offline cs dmri provides higher accuracy for image reconstruction to achieve good image reconstruction performance in terms of both speed and accuracy we propose a hybrid cs dmri method using periodic time variant subsampling for different frames in each period there is one reference frame that is sampled at a higher subsampling ratio the two nearby reference frames with good reconstruction quality can be used to provide rough predictions of the other frames between them to finely recover the current frame one structural regularization in the optimization model for reconstruction is a 2 d omnidirectional total variation otv for exploiting the sparsity of the difference between the predicted and estimated frames and the other is a 3 d otv as a regularization term for exploiting the bilateral spatio temporal coherence between the forward reference frame current frame and backward reference frame compared with classical total variation the proposed otv fully utilizes the correlations of all the possible directions of the data the formulated optimization model can be solved using iterative reweighted least squares with the pre conditioned conjugate gradient method numerical experiments demonstrate that the proposed method has better reconstruction accuracy than all the existing methods and low computational complexity that is comparable to the existing online methods

a new sliding mode control with fuzzy boundary layer this study develops a sliding mode controller based on fuzzy variable boundary layer with a control gain and boundary layer thickness as design parameters the control gain is an important factor affecting the control performance of variable structure system vss sliding mode controllers based on a variable boundary layer are superior to the fixed layer method for tracking in order to regulate the design parameters and increase operating efficiency the proposed methodologies make use of fuzzy inference which reduces the number of fuzzy inputs by using fuzzy algorithms in choosing a control gain and boundary layer we propose methods which have better tracking performance than the conventional method finally the results of simulation are given to demonstrate the validity of this algorithm

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

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

fast mr image reconstruction for partially parallel imaging with arbitrary k space trajectories both acquisition and reconstruction speed are crucial for magnetic resonance mr imaging in clinical applications in this paper we present a fast reconstruction algorithm for sense in partially parallel mr imaging with arbitrary k space trajectories the proposed method is a combination of variable splitting the classical penalty technique and the optimal gradient method variable splitting and the penalty technique reformulate the sense model with sparsity regularization as an unconstrained minimization problem which can be solved by alternating two simple minimizations one is the total variation and wavelet based denoising that can be quickly solved by several recent numerical methods whereas the other one involves a linear inversion which is solved by the optimal first order gradient method in our algorithm to significantly improve the performance comparisons with several recent parallel imaging algorithms indicate that the proposed method significantly improves the computation efficiency and achieves state of the art reconstruction quality

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