learning rate for convex support tensor machines tensors are increasingly encountered in prediction problems we extend previous results for high dimensional least squares convex tensor regression to classification problems with a hinge loss and establish its asymptotic statistical properties based on a general convex decomposable penalty the rate depends on both the intrinsic dimension and the rademacher complexity of the class of linear functions of tensor predictors

a bahadur representation of the linear support vector machine the support vector machine has been successful in a variety of applications also on the theoretical front statistical properties of the support vector machine have been studied quite extensively with a particular attention to its bayes risk consistency under some conditions in this paper we study somewhat basic statistical properties of the support vector machine yet to be investigated namely the asymptotic behavior of the coefficients of the linear support vector machine a bahadur type representation of the coefficients is established under appropriate conditions and their asymptotic normality and statistical variability are derived on the basis of the representation these asymptotic results do not only help further our understanding of the support vector machine but also they can be useful for related statistical inferences

on the interpretation and identification of dynamic takagi sugeno fuzzy models dynamic takagi sugeno fuzzy models are not always easy to interpret in particular when they are identified from experimental data it is shown that there exists a close relationship between dynamic takagi sugeno fuzzy models and dynamic linearization when using affine local model structures which suggests that a solution to the multiobjective identification problem exists however it is also shown that the affine local model structure is a highly sensitive parametrization when applied in transient operating regimes due to the multiobjective nature of the identification problem studied here special considerations must be made during model structure selection experiment design and identification in order to meet both objectives some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied these include constrained and regularized identification and locally weighted identification their usefulness in the present context is illustrated by examples

bias reduction for an explicit solution of source localization using tdoa this paper proposes two methods to reduce the bias of the well known algebraic explicit solution chan and ho a simple and efficient estimator for hyperbolic location ieee trans signal process vol 42 pp 1905 1915 aug 1994 for source localization using tdoa bias of a source location estimate is significant when the measurement noise is large and the geolocation geometry is poor bias also dominates performance when multiple times of independent measurements are available such as in uwb localization or in target tracking the paper starts by deriving the bias of the source location estimate from chan and ho the bias is found to be considerably larger than that of the maximum likelihood estimator two methods called biassub and biasred are developed to reduce the bias the biassub method subtracts the expected bias from the solution of chan and ho s work where the expected bias is approximated by the theoretical bias using the estimated source location and noisy data measurements the biasred method augments the equation error formulation and imposes a constraint to improve the source location estimate the biassub method requires the exact knowledge of the noise covariance matrix and biasred only needs the structure of it analysis shows that both methods reduce the bias considerably and achieve the crlb performance for distant source when the noise is gaussian and small the biassub method can nearly eliminate the bias and the biasred method is able to lower the bias to the same level as the maximum likelihood estimator the biasred method is extended for tdoa and fdoa positioning simulations corroborate the performance of the proposed methods

gaussian particle filtering sequential bayesian estimation for nonlinear dynamic state space models involves recursive estimation of filtering and predictive distributions of unobserved time varying signals based on noisy observations this paper introduces a new filter called the gaussian particle filter it is based on the particle filtering concept and it approximates the posterior distributions by single gaussians similar to gaussian filters like the extended kalman filter and its variants it is shown that under the gaussianity assumption the gaussian particle filter is asymptotically optimal in the number of particles and hence has much improved performance and versatility over other gaussian filters especially when nontrivial nonlinearities are present simulation results are presented to demonstrate the versatility and improved performance of the gaussian particle filter over conventional gaussian filters and the lower complexity than known particle filters

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

a stability property of nonlinear sampled data systems with slowly varying inputs a stability analysis is presented that deals with the response of a nonlinear sampled data system to a slowly varying exogenous input signal the main result similar to existing results for purely continuous time and discrete time systems establishes that if the system possesses a manifold of exponentially stable constant operating points equilibria corresponding to constant values of the input signal then an initial state close to this manifold and a slowly varying input signal yield a trajectory that remains close to the manifold the analysis involves casting the sampled data system as a continuous time system with discrete jumps at the sampling instants

image classification using correlation tensor analysis images as high dimensional data usually embody large variabilities to classify images for versatile applications an effective algorithm is necessarily designed by systematically considering the data structure similarity metric discriminant subspace and classifier in this paper we provide evidence that besides the fisher criterion graph embedding and tensorization used in many existing methods the correlation based similarity metric embodied in supervised multilinear discriminant subspace learning can additionally improve the classification performance in particular a novel discriminant subspace learning algorithm called correlation tensor analysis cta is designed to incorporate both graph embedded correlational mapping and discriminant analysis in a fisher type of learning manner the correlation metric can estimate intrinsic angles and distances for the locally isometric embedding which can deal with the case when euclidean metric is incapable of capturing the intrinsic similarities between data points cta learns multiple interrelated subspaces to obtain a low dimensional data representation reflecting both class label information and intrinsic geometric structure of the data distribution extensive comparisons with most popular subspace learning methods on face recognition evaluation demonstrate the effectiveness and superiority of cta parameter analysis also reveals its robustness

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

robust kalman filtering for discrete time varying uncertain systems with multiplicative noises in this paper a robust finite horizon kalman filter is designed for discrete time varying uncertain systems with both additive and multiplicative noises the system under consideration is subject to both deterministic and stochastic uncertainties sufficient conditions for the filter to guarantee an optimized upper bound on the state estimation error variance for admissible uncertainties are established in terms of two discrete riccati difference equations a numerical example is given to show the applicability of the presented method

a low rank approximation based transductive support tensor machine for semisupervised classification in the fields of machine learning pattern recognition image processing and computer vision the data are usually represented by the tensors for the semisupervised tensor classification the existing transductive support tensor machine tstm needs to resort to iterative technique which is very time consuming in order to overcome this shortcoming in this paper we extend the concave convex procedure based transductive support vector machine cccp tsvm to the tensor patterns and propose a low rank approximation based tstm in which the tensor rank one decomposition is used to compute the inner product of the tensors theoretically concave convex procedure based tstm cccp tstm is an extension of the linear cccp tsvm to tensor patterns when the input patterns are vectors cccp tstm degenerates into the linear cccp tsvm a set of experiments is conducted on 23 semisupervised classification tasks which are generated from seven second order face data sets three third order gait data sets and two third order image data sets to illustrate the performance of the cccp tstm the results show that compared with cccp tsvm and tstm cccp tstm provides significant performance gain in terms of test accuracy and training speed

gaussian filters for nonlinear filtering problems we develop and analyze real time and accurate filters for nonlinear filtering problems based on the gaussian distributions we present the systematic formulation of gaussian filters and develop efficient and accurate numerical integration of the optimal filter we also discuss the mixed gaussian filters in which the conditional probability density is approximated by the sum of gaussian distributions a new update rule of weights for gaussian sum filters is proposed our numerical tests demonstrate that new filters significantly improve the extended kalman filter with no additional cost and the new gaussian sum filter has a nearly optimal performance

brief analysis and design of gain scheduled sampled data control systems the existence of gain scheduled sampled data controllers that meet an important linearization requirement about a family of equilibria is investigated

improvements to platt s smo algorithm for svm classifier design this article points out an important source of inefficiency in platt s sequential minimal optimization smo algorithm that is caused by the use of a single threshold value using clues from the kkt conditions for the dual problem two threshold parameters are employed to derive modifications of smo these modified algorithms perform significantly faster than the original smo on all benchmark data sets tried

on a nonlinear multivariable servomechanism problem abstract multivariable nonlinear plants with both measured and unmeasured vector disturbance signals are considered output feedback dynamic control laws are designed for the problem of tracking a constant or slowly varying reference input while rejecting constant or slowly varying disturbances

regularized extended estimation with stabilized exponential forgetting this technical note concerns the problem of variable regularized estimation of time varying nonlinear systems from the bayesian viewpoint the questions of how to impose the posterior information being variably regularized and how to forget this information are carefully discussed the estimator design adopts the strategy of the iterated kalman filter but differs in that instead of the separated moments of the linearized system only the augmented covariance matrix is updated to suppress obsolete information a decision problem involving the kullback leibler divergence is solved the decision provides the best combination of a pair of time evolution model hypotheses in terms of the geometric mean as a result exponential forgetting with the adaptively tuned factor is inserted into the estimation process the regularization of the investigated statistics is induced through the processing of externally supplied information the presented estimator allows for absolute discarding or conversely retention of external information produced in terms of the normal wishart distribution

uncorrelated multilinear discriminant analysis with regularization and aggregation for tensor object recognition this paper proposes an uncorrelated multilinear discriminant analysis umlda framework for the recognition of multidimensional objects known as tensor objects uncorrelated features are desirable in recognition tasks since they contain minimum redundancy and ensure independence of features the umlda aims to extract uncorrelated discriminative features directly from tensorial data through solving a tensor to vector projection the solution consists of sequential iterative processes based on the alternating projection method and an adaptive regularization procedure is incorporated to enhance the performance in the small sample size sss scenario a simple nearest neighbor classifier is employed for classification furthermore exploiting the complementary information from differently initialized and regularized umlda recognizers an aggregation scheme is adopted to combine them at the matching score level resulting in enhanced generalization performance while alleviating the regularization parameter selection problem the umlda based recognition algorithm is then empirically shown on face and gait recognition tasks to outperform four multilinear subspace solutions mpca dater gtda tr1da and four linear subspace solutions bayesian lda ulda r jd lda

nonlinear estimation with state dependent gaussian observation noise we consider the problem of estimating the state of a system when measurement noise is a function of the system s state we propose generalizations of the extended kalman filter and the iterated extended kalman filter that can be utilized when the state estimate distribution is approximately gaussian the state estimate is computed by an iterative root searching method that maximizes a maximum likelihood function the new filter allows for the consistent treatment of a class of control problem involving nonlinear estimation from measurements with state dependent noise the effectiveness of the estimation algorithm is illustrated for a control problem with a mobile bearing only sensor

gain scheduling via linear fractional transformations abstract a linear finite dimensional plant with rational state space parameter dependence is controlled using a parameter dependent controller the parameters are known to take on values in a unit ball and are known in real time the goal of control is to stabilize the parameter dependent closed loop system and provide disturbance error attenuation as measured in induced l 2 norms the approach taken uses the optimally scaled small gain theorem and solves the control synthesis problem by reformulating the existence conditions into a finite dimensional convex optimization

design of controllers for linear parameter varying systems by the gain scheduling technique abstract design of controllers for linear multi input multi output systems whose dynamics depend on a time varying parameter is studied design algorithms for the state feedback control law based on the gain scheduling technique are proposed state feedback control laws are designed so that one of the following goals is achieved i the poles of the closed loop system for all fixed parameter values are placed in a small neighborhood of a set of desired closed loop poles ii the stability of the closed loop system for all fixed parameter values guarantees that the of the system with the time varying parameter independent of the rate of change of the parameter rigorously justified and computationally appealing design algorithms are developed

efficient classification for additive kernel svms we show that a class of nonlinear kernel svms admits approximate classifiers with runtime and memory complexity that is independent of the number of support vectors this class of kernels which we refer to as additive kernels includes widely used kernels for histogram based image comparison like intersection and chi squared kernels additive kernel svms can offer significant improvements in accuracy over linear svms on a wide variety of tasks while having the same runtime making them practical for large scale recognition or real time detection tasks we present experiments on a variety of datasets including the inria person daimler chrysler pedestrians uiuc cars caltech 101 mnist and usps digits to demonstrate the effectiveness of our method for efficient evaluation of svms with additive kernels since its introduction our method has become integral to various state of the art systems for pascal voc object detection image classification imagenet challenge trecvid etc the techniques we propose can also be applied to settings where evaluation of weighted additive kernels is required which include kernelized versions of pca lda regression k means as well as speeding up the inner loop of svm classifier training algorithms

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 novel technique for subpixel image classification based on support vector machine this paper presents a novel support vector machine classifier designed for subpixel image classification pixel spectral unmixing the proposed classifier generalizes the properties of svms to the identification and modeling of the abundances of classes in mixed pixels by using fuzzy logic this results in the definition of a fuzzy input fuzzy output support vector machine f2svm classifier that can 1 process fuzzy information given as input to the classification algorithm for modeling the subpixel information in the learning phase of the classifier and 2 provide a fuzzy modeling of the classification results allowing a relation many to one between classes and pixels the presented binary f2svm can address multicategory problems according to two strategies the fuzzy one against all foaa and the fuzzy one against one foao strategies these strategies generalize to the fuzzy case techniques based upon ensembles of binary classifiers used for addressing multicategory problems in crisp classification problems the effectiveness of the proposed f2svm classifier is tested on three problems related to image classification in presence of mixed pixels having different characteristics experimental results confirm the validity of the proposed subpixel classification method

approximate state feedback linearization using spline functions abstract this paper presents two numerical methods for synthesizing an approximate feedback linearizing control for single input nonlinear systems both techniques use part of the pseudo linearizing coordinate transformation as an output function t1 the first synthesis uses a well known recursive algorithm to compute the control given t1 splines represent functions that cannot be computed in closed form the second method yields a simplified controller that can be computed regardless of the plant s complexity experimental results from a rotating inverted pendulum show an improved performance of the approximate feedback linearization compared with pseudo linearization while highlighting the practical numerical difficulties that arise in the first method simulations of an acrobot illustrate the second method which is also improved over pseudo linearization

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

a stochastic analysis of a modified gain extended kalman filter with applications to estimation with bearings only measurements a new globally convergent nonlinear observer called the modified gain extended kalman observer mgeko is developed for a special class of systems this observer structure forms the basis of a new stochastic filter mechanization called the modified gain extended kalman filter mgekf a sufficient condition for the estimation errors of the mgekf to be exponentially bounded in the mean square is obtained finally the mgeko and the mgekf are applied to the three dimensional bearings only measurement problem where the extended kalman filter often shows erratic behavior

covariate adjusted tensor classification in high dimensions in contemporary scientific research it is often of great interest to predict a categorical response based on a high dimensional tensor i e multi dimensional array and additional covariates mo

the iterated kalman filter update as a gauss newton method it is shown that the iterated kalman filter ikf update is an application of the gauss newton method for approximating a maximum likelihood estimate an example is presented in which the iterated kalman filter update and maximum likelihood estimate show correct convergence behavior as the observation becomes more accurate whereas the extended kalman filter update does not

stabilization of nonlinear systems with moving equilibria this note provides a new method for the stabilization of nonlinear systems with parametric uncertainty unlike traditional techniques our approach does not assume that the equilibrium remains fixed for all parameter values the proposed method combines different optimization techniques to produce a robust control that accounts for uncertain parametric variations and the corresponding equilibrium shifts comparisons with analytical gain scheduling are provided

a computationally efficient lyapunov based scheduling procedure for control of nonlinear systems with stability guarantees we propose an alternative to gain scheduling for stabilization of nonlinear systems for a useful class of nonlinear systems the characterization of a region of stability based on a control lyapunov function is computationally tractable in the sense that computation times vary polynomially with the state dimension for a fixed number of scheduling variables using this fact we develop a procedure to expand the region of stability by constructing control lyapunov functions to various trim points of the system a lyapunov based control synthesis algorithm is used to construct a control law that guarantees closed loop stability for initial conditions in the expanded region of state space this control asymptotically recovers the optimal stability margin in the sense of a lyapunov derivative which in turn can be seen as a performance measure robustness to bounded disturbances and stabilization under bounded control are easily incorporated into this framework in the worst case the computational complexity of the analysis problem that develops in the new method is increased by an exponential in the disturbance dimension similarly we can handle control constraints with an increase in computational complexity of no more than an exponential in the control dimension we demonstrate the new control design procedure on an example

a survey of multilinear subspace learning for tensor data increasingly large amount of multidimensional data are being generated on a daily basis in many applications this leads to a strong demand for learning algorithms to extract useful information from these massive data this paper surveys the field of multilinear subspace learning msl for dimensionality reduction of multidimensional data directly from their tensorial representations it discusses the central issues of msl including establishing the foundations of the field via multilinear projections formulating a unifying msl framework for systematic treatment of the problem examining the algorithmic aspects of typical msl solutions and categorizing both unsupervised and supervised msl algorithms into taxonomies lastly the paper summarizes a wide range of msl applications and concludes with perspectives on future research directions

new support vector algorithms we propose a new class of support vector algorithms for regression and classification in these algorithms a parameter lets one effectively control the number of support vectors while this can be useful in its own right the parameterization has the additional benefit of enabling us to eliminate one of the other free parameters of the algorithm the accuracy parameter epsilon in the regression case and the regularization constant c in the classification case we describe the algorithms give some theoretical results concerning the meaning and the choice of and report experimental results

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

posterior cramer rao bounds for discrete time nonlinear filtering a mean square error lower bound for the discrete time nonlinear filtering problem is derived based on the van trees 1968 posterior version of the cramer rao inequality this lower bound is applicable to multidimensional nonlinear possibly non gaussian dynamical systems and is more general than the previous bounds in the literature the case of singular conditional distribution of the one step ahead state vector given the present state is considered the bound is evaluated for three important examples the recursive estimation of slowly varying parameters of an autoregressive process tracking a slowly varying frequency of a single cisoid in noise and tracking parameters of a sinusoidal frequency with sinusoidal phase modulation

robust h_ infty filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts in this paper the robust h filtering problem is studied for a class of uncertain nonlinear networked systems with both multiple stochastic time varying communication delays and multiple packet dropouts a sequence of random variables all of which are mutually independent but obey bernoulli distribution are introduced to account for the randomly occurred communication delays the packet dropout phenomenon occurs in a random way and the occurrence probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution in the interval the discrete time system under consideration is also subject to parameter uncertainties state dependent stochastic disturbances and sector bounded nonlinearities we aim to design a linear full order filter such that the estimation error converges to zero exponentially in the mean square while the disturbance rejection attenuation is constrained to a give level by means of the h performance index intensive stochastic analysis is carried out to obtain sufficient conditions for ensuring the exponential stability as well as prescribed h performance for the overall filtering error dynamics in the presence of random delays random dropouts nonlinearities and the parameter uncertainties these conditions are characterized in terms of the feasibility of a set of linear matrix inequalities lmis and then the explicit expression is given for the desired filter parameters simulation results are employed to demonstrate the effectiveness of the proposed filter design technique in this paper

technical communique h output feedback control of discrete time systems with state multiplicative noise linear discrete time systems with stochastic uncertainties in their state space model are considered the problem of dynamic output feedback control is solved via a new approach for both the finite horizon and the stationary cases in both cases a cost function is defined which is the expected value of the standard h performance index with respect to the uncertain parameters an example is given which demonstrates the applicability of the theory

factorization strategies for third order tensors abstract operations with tensors or multiway arrays have become increasingly prevalent in recent years traditionally tensors are represented or decomposed as a sum of rank 1 outer products using either the candecomp parafac cp or the tucker models or some variation thereof such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array the specifics of the approximate representation such as how many terms to use in the sum orthogonality constraints etc depend on the application in this paper we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem reminiscent of matrix factorizations we present a new factorization of a tensor as a product of tensors to derive the new factorization we define a closed multiplication operation between tensors a major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications specifically this new multiplication allows us to introduce concepts such as tensor transpose inverse and identity which lead to the notion of an orthogonal tensor the multiplication also gives rise to a linear operator and the null space of the resulting operator is identified we extend the concept of outer products of vectors to outer products of matrices all derivations are presented for third order tensors however they can be easily extended to the order p p 3 case we conclude with an application in image deblurring

brief stability preserving interpolation methods for the synthesis of gain scheduled controllers synthesis of gain scheduled controllers for nonlinear plants often requires that a parameter varying controller be generated from a finite set of linear time invariant controllers we propose interpolation methods for this task with the property that stability of the linearized closed loop system is preserved for all values of the scheduling parameter in addition slow variation arguments are presented that establish local stability of the nonlinear closed loop system with gain scheduled controller

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

low rank tensor train coefficient array estimation for tensor on tensor regression the tensor on tensor regression can predict a tensor from a tensor which generalizes most previous multilinear regression approaches including methods to predict a scalar from a tensor and a tensor from a scalar however the coefficient array could be much higher dimensional due to both high order predictors and responses in this generalized way compared with the current low candecomp parafac cp rank approximation based method the low tensor train tt approximation can further improve the stability and efficiency of the high or even ultrahigh dimensional coefficient array estimation in the proposed low tt rank coefficient array estimation for tensor on tensor regression we adopt a tt rounding procedure to obtain adaptive ranks instead of selecting ranks by experience besides an ell 2 constraint is imposed to avoid overfitting the hierarchical alternating least square is used to solve the optimization problem numerical experiments on a synthetic data set and two real life data sets demonstrate that the proposed method outperforms the state of the art methods in terms of prediction accuracy with comparable computational complexity and the proposed method is more computationally efficient when the data are high dimensional with small size in each mode

an enhanced velocity based algorithm for safe implementations of gain scheduled controllers abstractthis paper presents an enhanced velocity based algorithm to implement gain scheduled controllers for nonlinear and parameter dependent systems a new scheme including pre and post filtering is proposed with the assumption that the time derivative of the controller inputs is not available for feedback control it is shown that the proposed control structure can preserve the input output properties of the linearised closed loop system in the neighbourhood of each equilibrium point avoiding the emergence of the so called hidden coupling terms moreover it is guaranteed that this implementation will not introduce unobservable or uncontrollable unstable modes and hence the internal stability will not be affected a case study dealing with the design of a pitch axis missile autopilot is carried out and the numerical simulation results confirm the validity of the proposed approach

an lmi approach to control oriented identification and model in validation of lpv systems this note proposes a control oriented identification framework for a class of linear parameter varying systems that takes into account both the dependence of part of the model on time varying parameters as well as the possible existence of a nonparametric component the main results of the note show that the problems of obtaining and validating a model for these systems can be recast as linear matrix inequality feasibility problems moreover as the information is completed the algorithm is shown to converge in the l sub 2 induced topology to the actual plant additional results include deterministic bounds on the identification error these results are illustrated with a practical example arising in the context of active vision

extended kalman filtering with stochastic nonlinearities and multiple missing measurements in this paper the extended kalman filtering problem is investigated for a class of nonlinear systems with multiple missing measurements over a finite horizon both deterministic and stochastic nonlinearities are included in the system model where the stochastic nonlinearities are described by statistical means that could reflect the multiplicative stochastic disturbances the phenomenon of measurement missing occurs in a random way and the missing probability for each sensor is governed by an individual random variable satisfying a certain probability distribution over the interval 0 1 such a probability distribution is allowed to be any commonly used distribution over the interval 0 1 with known conditional probability the aim of the addressed filtering problem is to design a filter such that in the presence of both the stochastic nonlinearities and multiple missing measurements there exists an upper bound for the filtering error covariance subsequently such an upper bound is minimized by properly designing the filter gain at each sampling instant it is shown that the desired filter can be obtained in terms of the solutions to two riccati like difference equations that are of a form suitable for recursive computation in online applications an illustrative example is given to demonstrate the effectiveness of the proposed filter design scheme

the gain scheduling and the robust state feedback stabilization problems we consider dynamical uncertain systems of the form x spl dot a x w b x w u where w t spl isin w is an unknown but bounded time varying parameter we consider the robust state feedback stabilization problem in which we consider a control of the form u spl phi x and the gain scheduling stabilization problem in which a control of the form u spl phi x w full information control is admitted we show that for convex processes those systems in which for fixed x the set of all a x w b x w is convex the two problems are equivalent i e if there exists a locally lipschitz gain scheduling stabilizing control then there exists a robustly stabilizing control continuous everywhere except possibly at the origin thus for convex process stabilization knowledge of w t is not an advantage for the compensator then we consider the special class of polytopic lpv systems and show that there is no loss of regularity as in the general case if we pass from a gain scheduling controller to a state feedback controller in particular no discontinuity at the origin may occur we show that the existence of a dynamic controller always implies the existence of a static one finally we show that we can always find a linear gain scheduling controller for a stabilizable system this means that a possible advantage of the online measurement of the parameter w t is that this always allows for linear compensators whose implementation can be easier than that of nonlinear ones

geometric representation of high dimension low sample size data summary high dimension low sample size data are emerging in various areas of science we find a common structure underlying many such data sets by using a non standard type of asymptotics the dimension tends to while the sample size is fixed our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex essentially all the randomness in the data appears only as a random rotation of this simplex this geometric representation is used to obtain several new statistical insights

multilinear discriminant analysis for face recognition there is a growing interest in subspace learning techniques for face recognition however the excessive dimension of the data space often brings the algorithms into the curse of dimensionality dilemma in this paper we present a novel approach to solve the supervised dimensionality reduction problem by encoding an image object as a general tensor of second or even higher order first we propose a discriminant tensor criterion whereby multiple interrelated lower dimensional discriminative subspaces are derived for feature extraction then a novel approach called k mode optimization is presented to iteratively learn these subspaces by unfolding the tensor along different tensor directions we call this algorithm multilinear discriminant analysis mda which has the following characteristics 1 multiple interrelated subspaces can collaborate to discriminate different classes 2 for classification problems involving higher order tensors the mda algorithm can avoid the curse of dimensionality dilemma and alleviate the small sample size problem and 3 the computational cost in the learning stage is reduced to a large extent owing to the reduced data dimensions in k mode optimization we provide extensive experiments on orl cmu pie and feret databases by encoding face images as second or third order tensors to demonstrate that the proposed mda algorithm based on higher order tensors has the potential to outperform the traditional vector based subspace learning algorithms especially in the cases with small sample sizes

two variations on fisher s linear discriminant for pattern recognition discriminants are often used in pattern recognition to separate clusters of points in some multidimensional feature space the paper provides two fast and simple techniques for improving on the classification performance provided by fisher s linear discriminant for two classes both of these methods are also extended to nonlinear decision surfaces through the use of mercer kernels

linear estimation of continuous discrete linear state space models with multiplicative noise this paper deals with the estimation of the state variable of continuous discrete linear state space models with multiplicative noise specifically the optimal minimum variance linear filter for that class of models is constructed moreover the solutions of the differential equations that describe the evolution of the two first conditional moments between observations are obtained and an algorithm for their numerical computation is also given the performance of the linear filter is illustrated by means of numerical experiments

polynomial filtering of discrete time stochastic linear systems with multiplicative state noise the problem of finding an optimal polynomial state estimate for the class of stochastic linear models with a multiplicative state noise term is studied for such models a technique of state augmentation is used leading to the definition of a general polynomial filter the theory is developed for time varying systems with nonstationary and non gaussian noises moreover the steady state polynomial filter for stationary systems is also studied numerical simulations show the high performances of the proposed method with respect to the classical linear filtering techniques

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

nonlinear system stabilization via hierarchical switching control a nonlinear control system design framework predicated on a hierarchical switching controller architecture parameterized over a set of moving system equilibria is developed specifically using equilibria dependent lyapunov functions a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems the switching nonlinear controller architecture is designed based on a generalized lower semicontinuous lyapunov function obtained by minimizing a potential function over a given switching set induced by the parameterized system equilibria the proposed framework provides a rigorous alternative to designing gain scheduled feedback controllers and guarantees local and global closed loop system stability for general nonlinear systems

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

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

a linear support higher order tensor machine for classification there has been growing interest in developing more effective learning machines for tensor classification at present most of the existing learning machines such as support tensor machine stm involve nonconvex optimization problems and need to resort to iterative techniques obviously it is very time consuming and may suffer from local minima in order to overcome these two shortcomings in this paper we present a novel linear support higher order tensor machine shtm which integrates the merits of linear c support vector machine c svm and tensor rank one decomposition theoretically shtm is an extension of the linear c svm to tensor patterns when the input patterns are vectors shtm degenerates into the standard c svm a set of experiments is conducted on nine second order face recognition datasets and three third order gait recognition datasets to illustrate the performance of the proposed shtm the statistic test shows that compared with stm and c svm with the rbf kernel shtm provides significant performance gain in terms of test accuracy and training speed especially in the case of higher order tensors

gain scheduling dynamic linear controllers for a nonlinear plant abstract for a general type of nonlinear tracking problem we consider gain scheduling based on a family of linear dynamic controllers designed for a family of plant linearizations about constant operating points a necessary and sufficient condition for the existence of gain scheduled controllers satisfying appropriate requirements is presented using this condition the impact of linear controller configuration on the existence of a gain scheduled controller can be assessed and ad hoc approaches to scheduling can be analyzed

3 d discrete shearlet transform and video processing in this paper we introduce a digital implementation of the 3 d shearlet transform and illustrate its application to problems of video denoising and enhancement the shearlet representation is a multiscale pyramid of well localized waveforms defined at various locations and orientations which was introduced to overcome the limitations of traditional multiscale systems in dealing with multidimensional data while the shearlet approach shares the general philosophy of curvelets and surfacelets it is based on a very different mathematical framework which is derived from the theory of affine systems and uses shearing matrices rather than rotations this allows a natural transition from the continuous setting to the digital setting and a more flexible mathematical structure the 3 d digital shearlet transform algorithm presented in this paper consists in a cascade of a multiscale decomposition and a directional filtering stage the filters employed in this decomposition are implemented as finite length filters and this ensures that the transform is local and numerically efficient to illustrate its performance the 3 d discrete shearlet transform is applied to problems of video denoising and enhancement and compared against other state of the art multiscale techniques including curvelets and surfacelets

local rademacher complexities we propose new bounds on the error of learning algorithms in terms of a data dependent notion of complexity the estimates we establish give optimal rates and are based on a local and empirical version of rademacher averages in the sense that the rademacher averages are computed from the data on a subset of functions with small empirical error we present some applications to classification and prediction with convex function classes and with kernel classes in particular

alternating projection method for a class of tensor equations abstract this paper considers how to solve a class of tensor equations arising from the unified definition of tensor vector products of special interest is the order 3 tensor equation whose solutions are the intersection of a group of quadrics from a geometric point of view inspired by the method of alternating projections for set intersection problems we develop a hybrid alternating projection algorithm for solving order 3 tensor equations the local linear convergence of the alternating projection method is established under suitable conditions some numerical experiments are conducted to evaluate the effect of the proposed algorithm

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

robust estimation in non linear state space models with state dependent noise in this paper we present a robust estimation algorithm for non linear state space models driven by state dependent noise the algorithm is robust to outliers in the data we derive the algorithm step by step from first principles from theory to implementation the implementation is straightforward and consists mainly of two components 1 a slightly modified version of the rauch tung striebel recursions and 2 a backtracking line search strategy since it preserves the underlying chain structure of the problem its computational complexity grows linearly with the number of data the algorithm is iterative and is guaranteed to converge under mild assumptions to a local optimum from any starting point we validate our approach via experiments on synthetic data from a multi variate stochastic volatility model

guaranteed properties of gain scheduled control for linear parameter varying plants abstract gain scheduling has proven to be a successful design methodology in many engineering applications however in the absence of a sound theoretical analysis these designs come with no guarantees on the robustness performance or even nominal stability of the overall gain scheduled design this paper presents such an analysis for one type of gain scheduled system namely a linear parameter varying plant scheduling on its exogenous parameters conditions are given which guarantee that the stability robustness and performance properties of the fixed operating point designs carry over to the global gain scheduled design these conditions confirm and formalize popular notions regarding gain scheduled design such as the scheduling variable should vary slowly

generalized iterated kalman filter and its performance evaluation in this paper we present a generalized iterated kalman filter gikf algorithm for state estimation of a nonlinear stochastic discrete time system with state dependent multiplicative observation noise the gikf algorithm adopts the newton raphson iterative optimization steps to yield an approximate maximum a posteriori estimate of the states the mean square estimation error mse and the cramer rao lower bound crlb of the state estimates are also derived in particular the local convergence of mse of gikf is rigorously established it is also proved that the gikf yields a smaller mse than those of the generalized extended kalman filter and the traditional extended kalman filter the performance advantages and convergence of gikf are demonstrated using monte carlo simulations on a target tracking application in a range measuring sensor network

mpca multilinear principal component analysis of tensor objects this paper introduces a multilinear principal component analysis mpca framework for tensor object feature extraction objects of interest in many computer vision and pattern recognition applications such as 2d 3d images and video sequences are naturally described as tensors or multilinear arrays the proposed framework performs feature extraction by determining a multilinear projection that captures most of the original tensorial input variation the solution is iterative in nature and it proceeds by decomposing the original problem to a series of multiple projection subproblems as part of this work methods for subspace dimensionality determination are proposed and analyzed it is shown that the mpca framework discussed in this work supplants existing heterogeneous solutions such as the classical principal component analysis pca and its 2d variant 2d pca finally a tensor object recognition system is proposed with the introduction of a discriminative tensor feature selection mechanism and a novel classification strategy and applied to the problem of gait recognition results presented here indicate mpca s utility as a feature extraction tool it is shown that even without a fully optimized design an mpca based gait recognition module achieves highly competitive performance and compares favorably to the state of the art gait recognizers