on strictly positively invariant cones for a matrix a rn n it is shown that strict positive invariance of a proper cone c rn that is etac 0 int c t 0 implies the existence of a certain direct sum decomposition of rn into a invariant subspaces our results lead to a characterization of the set of initial points which give rise to solution curves that reach s under the differential equation ax also given is an application in stability theory

coding for the binary symmetric broadcast channel with two receivers block coding for the binary symmetric broadcast channel with two receivers is investigated a graph theoretic approach to the construction of a class of block codes with unequal error protection for two different sets of messages is presented a code in this class is a direct sum of two component codes each set of messages is encoded based on one component code the codes in this class are easy to implement decoding of these codes is presented and lower bounds on the achievable rates of these codes are derived the bounds are tighter than the katsman s bounds

semiglobal stabilization of linear discrete time systems subject to input saturation via linear feedback an are based approach we revisit the problem of semiglobal stabilization of linear discrete time systems subject to input saturation and give an algebraic riccati equation are based approach to the proof of a fact established earlier lin and saberi 1995 i e a linear discrete time system subject to input saturation is semiglobally stabilizable via linear feedback as long as the linear system in the absence of the saturation is stabilizable and detectable and all its open loop poles are located inside or on the unit circle moreover we drastically relax the requirements on the characteristic of the saturation elements as imposed in the earlier work

stability positive invariance and design of constrained regulators for networked control systems in this article the stability analysis the positive invariance of polyhedral sets and the design of state feedback regulators for networked control systems ncs with bounded transmission delays constant and unknown or time varying are investigated the dynamics of the ncs is described by autoregressive moving average arma models contrary to former approaches based on quadratic lyapunov functions in this article polyhedral lyapunov functions are used for both stability and positive invariance analysis and state feedback synthesis then based on the property that the exponential of a matrix can be expressed as a weighted sum of its constituent matrices it is proven that the problems of determination of stability margins or the design of stabilising controllers can be reduced to linear programming optimisation problems the use of arma models allows the development of methods for the design of state feedback controllers satisfying state constraints or convergence rate specifications defined on the

twin support vector machines for pattern classification we propose twin svm a binary svm classifier that determines two nonparallel planes by solving two related svm type problems each of which is smaller than in a conventional svm the twin svm formulation is in the spirit of proximal svms via generalized eigenvalues on several benchmark data sets twin svm is not only fast but shows good generalization twin svm is also useful for automatically discovering two dimensional projections of the data

positively invariant polyhedral sets of discrete time linear systems the problem of the existence of positively invariant polyhedral sets for linear discrete time dynamical systems is studied in the first part of the paper necessary and sufficient conditions for a

a design approach for constrained regulation in discrete singular systems abstract in the present study the problem of designing controllers for linear discrete time singular descriptor systems under initial state and control constraints is considered the aim is to eliminate any non causalities and make the closed loop system stable while the state feedback control vector must respect appropriate constraints the present approach constitutes an extension to the design of constrained linear state feedback controllers for linear regular discrete time systems

constrained stabilization of continuous time linear systems in this paper we consider the stabilization problem for linear continuous time systems under state and control constraints we show that the largest domain of attraction to the origin can be arbitrarily closely approximated by a polyhedral domain of attraction associated to a certain continuous feedback stabilizing control and we show how to use existing numeric procedures for discrete time systems to solve the continuous time problem we propose a new discontinuous stabilizing control law for scalar input systems which has the advantage of being successfully applicable to systems with quantized control

on lower and upper bounds in smooth and strongly convex optimization we develop a novel framework to study smooth and strongly convex optimization algorithms focusing on quadratic functions we are able to examine optimization algorithms as a recursive application of linear operators this in turn reveals a powerful connection between a class of optimization algorithms and the analytic theory of polynomials whereby new lower and upper bounds are derived whereas existing lower bounds for this setting are only valid when the dimensionality scales with the number of iterations our lower bound holds in the natural regime where the dimensionality is fixed lastly expressing it as an optimal solution for the corresponding optimization problem over polynomials as formulated by our framework we present a novel systematic derivation of nesterov s well known accelerated gradient descent method this rather natural interpretation of agd contrasts with earlier ones which lacked a simple yet solid motivation

subclass discriminant analysis over the years many discriminant analysis da algorithms have been proposed for the study of high dimensional data in a large variety of problems each of these algorithms is tuned to a specific type of data distribution that which best models the problem at hand unfortunately in most problems the form of each class pdf is a priori unknown and the selection of the da algorithm that best fits our data is done over trial and error ideally one would like to have a single formulation which can be used for most distribution types this can be achieved by approximating the underlying distribution of each class with a mixture of gaussians in this approach the major problem to be addressed is that of determining the optimal number of gaussians per class i e the number of subclasses in this paper two criteria able to find the most convenient division of each class into a set of subclasses are derived extensive experimental results are shown using five databases comparisons are given against linear discriminant analysis lda direct lda dlda heteroscedastic lda hlda nonparametric da nda and kernel based lda k lda we show that our method is always the best or comparable to the best

polyhedral regions of local stability for linear discrete time systems with saturating controls the study and the determination of polyhedral regions of local stability for linear systems subject to control saturation is addressed the analysis of the nonlinear behavior of the closed loop saturated system is made by dividing the state space in regions of saturation inside each of these regions the system evolution can be represented by a linear system with an additive disturbance from this representation a necessary and sufficient condition relative to the contractivity of a given convex compact polyhedral set is stated consequently the polyhedral set can be associated with a lyapunov function and the local asymptotic stability of the saturated closed loop system inside the set is guaranteed furthermore it is shown how in some particular cases the compactness condition can be relaxed in order to ensure the asymptotic stability in unbounded polyhedra finally an application of the contractivity conditions is presented in order to determine local asymptotic stability regions for the closed loop saturated system

multi class pairwise linear dimensionality reduction using heteroscedastic schemes linear dimensionality reduction ldr techniques have been increasingly important in pattern recognition pr due to the fact that they permit a relatively simple mapping of the problem onto a lower dimensional subspace leading to simple and computationally efficient classification strategies although the field has been well developed for the two class problem the corresponding issues encountered when dealing with multiple classes are far from trivial in this paper we argue that as opposed to the traditional ldr multi class schemes if we are dealing with multiple classes it is not expedient to treat it as a multi class problem per se rather we shall show that it is better to treat it as an ensemble of chernoff based two class reductions onto different subspaces whence the overall solution is achieved by resorting to either voting weighting or to a decision tree strategy the experimental results obtained on benchmark datasets demonstrate that the proposed methods are not only efficient but that they also yield accuracies comparable to that obtained by the optimal bayes classifier

on ip1 p2 antinorms of nonnegative matrices let 0 p p1 p2 1 x xi rn a aik rm n i e the vector x is elementwise positive and so is the matrix a the lp antinorm of x is defined by x p ixpi1p the lp1 p2 antinorm of a by a p1 p2 k iap1ikp2p11p2 and the lp1 p2 operator antinorm of a by a minp1 p2 infx rn ax p1 x p2 several properties of the antinorms p p1 p2 and minp1 p2 especially the supermultiplicativity of p1 p2 are studied and generalized to hold for nonnegative matrices the results are analogous with the well known submultiplicativity properties of lq1 q2 norms and lq1 q2 operator norms q1 q2 1 a generalization is outlined

improving bounds for eigenvalues of complex matrices using traces abstract let a be a complex matrix of order n with eigenvalues j j 1 2 n and m be any integer satisfying rank a m n the bound for j 2 by kress de vries and wegmann is strengthened furthermore new bounds are presented to estimate the spectral radius of a using m and traces of a a 2 a a and a a aa we also improve some wolkowicz styan bounds and previous localization of eigenvalues in rectangular or elliptic regions using traces several simple lower bounds for the spectral radius are given involving tr a tr a 2 tr a 3 and m

data dimensionality estimation methods a survey in this paper data dimensionality estimation methods are reviewed the estimation of the dimensionality of a data set is a classical problem of pattern recognition there are some good reviews algorithms for clustering data prentice hall englewood cliffs nj 1988 in literature but they do not include more recent developments based on fractal techniques and neural autoassociators the aim of this paper is to provide an up to date survey of the dimensionality estimation methods of a data set paying special attention to the fractal based methods

feedback control for linear time invariant systems with state and control bounds in the presence of disturbances the problem of the stabilizing linear control synthesis in the presence of state and input bounds for systems with additive unknown disturbances is considered the only information required about the disturbances is a finite convex polyhedral bound discrete and continuous time systems are considered the property of positive d invariance of a region is introduced and it is proved that a solution of the problem is achieved by the selection of a polyhedral set s and the computation of a feedback matrix k such that s is positively d invariant for the closed loop system it is shown that if polyhedral sets are considered the solution involves simple linear programming algorithms however the procedure suggested requires a great amount of computational work offline if the state space dimension is large because the feedback matrix k is obtained as a solution of a large set of linear inequalities all of the vertices of s are required

stochastic output feedback control convex lifting approach abstract this paper presents a method to design output feedback control for discrete time linear systems affected by bounded additive state output disturbances and subject to chance constraints on the state and hard constraints on the control input this method relies on a so called convex lifting which is a nonnegative convex piecewise affine function equal to 0 over a given stochastic positively invariant set and strictly positive outside this set accordingly it is shown that this function is strictly decreasing along the closed loop dynamics outside this invariant set and convergent to 0 as time tends to infinity consequently the state is convergent to the given invariant set while the method only requires solving a linear program at each sampling instant

reachability observability and realizability of continuous time positive systems this paper discusses reachability observability and realizability of single input single output linear time invariant systems in which state variables and or input output functions are restricted to be nonnegative to reflect physical constraints frequently encountered in real systems we define a set reachable from the origin with nonnegative inputs and also a set observable with nonnegative outputs we investigate geometrical structures of the sets through convex analysis and a duality relation between them is established next we consider positive realization of a given transfer function using the reachable set and the observable set we give a necessary and sufficient condition for positive realizability an example is given to demonstrate that a positive realizable transfer function does not in general have a jointly controllable and observable positive realization

model predictive control based on linear programming the explicit solution we study model predictive control mpc schemes for discrete time linear time invariant systems with constraints on inputs and states that can be formulated using a linear program lp in particular we focus our attention on performance criteria based on a mixed 1 norm namely 1 norm with respect to time and norm with respect to space first we provide a method to compute the terminal weight so that closed loop stability is achieved we then show that the optimal control profile is a piecewise affine and continuous function of the initial state and briefly describe the algorithm to compute it the piecewise affine form allows to eliminate online lp as the computation associated with mpc becomes a simple function evaluation besides practical advantages the availability of the explicit structure of the mpc controller provides an insight into the type of control action in different regions of the state space and highlights possible conditions of degeneracies of the lp such as multiple optima

more estimates for eigenvalues and singular values abstract there is an interesting relation between the angle that a matrix forms with the identity and its eigenvalues then we show for symmetric matrices that some information about the matrix can be known if the angle is computed a lower bound for the maximum eigenvalue is obtained for symmetric positive semidefinite matrices this inequality and the upper bound for eigenvalues given in 5 provide us an interval in which must lie the maximum eigenvalue finally some results from 5 and the first part of this paper are generalized for the non symmetric case

on infinity norms as lyapunov functions for linear systems the note presents a construction of the lyapunov function defined by l sub spl infin sup n vector norm for both continuous and discrete time linear systems in contrast to the previous results no restrictions on the positions or multiplicity of system matrix eigenvalues are required

feedback control of linear discrete time systems under state and control constraints in this paper the problem of stabilizing linear discrete time systems under state and control linear constraints is studied based on the concept of positive invariance existence conditions of linear state feedback control laws respecting both the constraints are established these conditions are then translated into an algorithm of linear programming

a reformative kernel fisher discriminant analysis abstract a reformative kernel fisher discriminant method is proposed which is directly derived from the naive kernel fisher discriminant analysis with superiority in classification efficiency in the novel method only a part of training patterns called significant nodes are necessary to be adopted in classifying one test pattern a recursive algorithm for selecting significant nodes which is the key of the novel method is presented in detail the experiment on benchmarks shows that the novel method is effective and much efficient in classifying

constrained regulation of linear systems abstract in this paper the linear constrained regulation problem without an assigned set of initial states is investigated this problem consists of the determination of a linear state feedback control law and of a set of admissible initial states d so that every trajectory of the resulting closed loop system which emanates from the set converges asymptotically to the origin without violation of the control constraints by applying well known results on the positive invariance of polyhedral sets an eigenstructure assignment approach to this problem is established if the number of unstable open loop eigenvalues does not exceed the number of control variables then this approach enables one to derive a linear state feedback stabilizing control law which makes the set of states where the control constraints are respected positively invariant this positively invariant set is the maximal admissible set of initial states for the control law under consideration for the case where the number of control variables is greater than the number of the unstable open loop eigenvalues a modified algorithm is proposed

a class of invariant regulators for the discrete time linear constrained regulation problem abstract stable dynamic systems admit positively invariant domains associated to their lyapunov functions conversely some domains can be made positively invariant for systems with state feedback controllers designed in such a way that some associated non negative definite functions are bound to decrease in particular this approach can be used to establish conditions on the gain matrix for linear constrained regulation problems lcrp we construct fixed and variable regulators easy to compute through linear programming for a class of constrained linear systems

a novel unified approach to invariance conditions for a linear dynamical system in this paper we propose a novel simple and unified approach to explore sufficient and necessary conditions i e invariance conditions under which four classic families of convex sets namely polyhedra polyhedral cones ellipsoids and lorenz cones are invariant sets for a linear discrete or continuous dynamical system for discrete dynamical systems we use the theorems of alternatives i e farkas lemma and s lemma to obtain simple and general proofs to derive invariance conditions this novel method establishes a solid connection between optimization theory and dynamical system also using the s lemma allows us to extend invariance conditions to any set represented by a quadratic inequality such sets include nonconvex and unbounded sets for continuous dynamical systems we use the forward or backward euler method to obtain the corresponding discrete dynamical systems while preserves invariance this enables us to develop a novel and elementary method to derive invariance conditions for continuous dynamical systems by using the ones for the corresponding discrete systems finally some numerical examples are presented to illustrate these invariance conditions

nonquadratic lyapunov functions for robust control abstract this paper deals with the problem of controlling a linear continuous time system with structured time varying parameter uncertainties and input disturbances with a lyapunov function approach in contrast with most of the previous results in the literature we do not confine our attention to the class of quadratic lyapunov functions conversely the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate lyapunov functions this question has a positive answer the proposed class is that of polyhedral norms or more generally of polyhedral minkowski functionals we show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a lyapunov function and a continuous state feedback compensator then it can be solved via a polyhedral lyapunov function and a possibly different continuous control moreover we show that the control can be piecewise linear a numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral

constrained stabilization via smooth lyapunov functions abstract we consider the problem of stabilizing a dynamic system by means of bounded controls we show that the largest domain of attraction can be arbitrarily closely approximated by a smooth domain of attraction for which we provide an analytic expression such an expression allows for the determination of a non linear control law in explicit form

enlargement of polytopic terminal region in nmpc by interpolation and partial invariance this paper uses the concept of partial invariance to derive a sequence of linear programs in order to maximize offline the volume of an invariant polytopic set with an arbitrary predefined number of vertices subject to a bound on closed loop performance interpolation techniques are used to determine a nonlinear control law which is optimal with respect to a closed loop cost bound through the on line solution of a linear program the invariant polytope is also used to define a receding horizon control law through an appropriate terminal constraint and cost

on unequal error protection of convolutional codes from an algebraic perspective in this paper convolutional codes are studied for unequal error protection uep from an algebraic theoretical viewpoint we first show that for every convolutional code there exists at least one optimal generator matrix with respect to uep the uep optimality of convolutional encoders is then combined with several algebraic properties e g systematic basic canonical and minimal to establish the fundamentals of convolutional codes for uep in addition a generic lower bound on the length of a uep convolutional code is proposed good uep codes with their lengths equal to the derived lower bound are obtained by computer search

robust mpc strategy with optimized polytopic dynamics for linear systems with additive and multiplicative uncertainty abstract polytopic dynamics have been introduced into the predictions in model predictive control mpc and have been optimized so as to maximize invariant ellipsoids for systems subject to multiplicative uncertainty this work was recently extended to the case of mixed additive and multiplicative uncertainty and sufficient conditions were provided to guarantee the invariance of an ellipsoid additionally under the assumption that the multiplicative uncertainty is known during a prediction horizon n this extension was used as the terminal control law of an overall robust mpc strategy that deployed an affine in the disturbance policy this assumption was needed to enable the handling of constraints in the overall mpc strategy the aims of this paper are to establish the necessity and sufficiency of the relevant invariance conditions of the polytopic dynamics and to propose a robust mpc strategy that uses polytopic sets to describe the evolution of predicted variables and therefore does not make an assumption that future multiplicative uncertainty is known the number of constraints and free variables of the resulting online optimization grows linearly with n the benefits of the strategy are illustrated by numerical simulations

some lower bounds for the spectral radius of matrices using traces abstract let a be an n n matrix with eigenvalues 1 2 n and let m be an integer satisfying rank a m n if a is real the best possible lower bound for its spectral radius in terms of m tr a and tr a 2 is obtained if a is any complex matrix two lower bounds for j 1 n j 2 are compared and furthermore a new lower bound for the spectral radius is given only in terms of tr a tr a 2 a a a aa n and m

design techniques for the control of discrete time systems subject to state and control constraints the regulation problem of linear discrete time systems under state and control constraints is investigated in the first part of the paper necessary and sufficient conditions for the existence of a solution to the constrained regulation problem are established the constructive form of the proof of this result also provides a method for the derivation of a control law transferring to the origin any state belonging to a given set of initial states while respecting the state and control constraints in the second part of the paper design techniques are proposed in which the determination of the control law is reduced to simple linear programming problems

lower bounds for the spectral radius of a matrix abstract we develop lower bounds for the spectral radius of symmetric skew symmetric and arbitrary real matrices our approach utilizes the well known leverrier faddeev algorithm for calculating the coefficients of the characteristic polynomial of a matrix in conjunction with a theorem by lucas which states that the critical points of a polynomial lie within the convex hull of its roots our results generalize and simplify a proof recently published by tarazaga for a lower bound on the spectral radius of a symmetric positive definite matrix in addition we provide new lower bounds for the spectral radius of skew symmetric matrices we apply these results to a problem involving the stability of fixed points in recurrent neural networks

dynamical statistical shape priors for level set based tracking in recent years researchers have proposed introducing statistical shape knowledge into level set based segmentation methods in order to cope with insufficient low level information while these priors were shown to drastically improve the segmentation of familiar objects so far the focus has been on statistical shape priors which are static in time yet in the context of tracking deformable objects it is clear that certain silhouettes such as those of a walking person may become more or less likely over time in this paper we tackle the challenge of learning dynamical statistical models for implicitly represented shapes we show how these can be integrated as dynamical shape priors in a bayesian framework for level set based image sequence segmentation we assess the effect of such shape priors with memory on the tracking of familiar deformable objects in the presence of noise and occlusion we show comparisons between dynamical and static shape priors between models of pure deformation and joint models of deformation and transformation and we quantitatively evaluate the segmentation accuracy as a function of the noise level and of the camera frame rate our experiments demonstrate that level set based segmentation and tracking can be strongly improved by exploiting the temporal correlations among consecutive silhouettes which characterize deforming shapes

why can lda be performed in pca transformed space abstract pca plus lda is a popular framework for linear discriminant analysis lda in high dimensional and singular case in this paper we focus on building a theoretical foundation for this framework moreover we point out the weakness of the previous lda based methods and suggest a complete pca plus lda algorithm experimental results on orl face image database indicate that the proposed method is more effective than the previous ones

a reduced order framework applied to linear systems with constrained controls a major issue in the control of dynamical systems is the integration of both technological constraints and some dynamic performance requirements in the design of the control system the authors show in this work that it is possible to solve a class of constrained control problems of linear systems by using a reduced order system obtained by the projection of the trajectories of the original system onto a subspace associated with the undesirable open loop eigenvalues the class of regulation schemes considered uses full state feedback to guarantee that any trajectory emanating from a given polyhedral set of admissible initial states remains in that set this set of admissible states is said to be positively invariant with respect to the closed loop system the authors also address the important issues of numerical stability and complexity of the computations

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

linear dimensionality reduction by maximizing the chernoff distance in the transformed space linear dimensionality reduction ldr techniques are quite important in pattern recognition due to their linear time complexity and simplicity in this paper we present a novel ldr technique which though linear aims to maximize the chernoff distance in the transformed space thus augmenting the class separability in such a space we present the corresponding criterion which is maximized via a gradient based algorithm and provide convergence and initialization proofs we have performed a comprehensive performance analysis of our method combined with two well known classifiers linear and quadratic on synthetic and real life data and compared it with other ldr techniques the results on synthetic and standard real life data sets show that the proposed criterion outperforms the latter when combined with both linear and quadratic classifiers

cyclic invariance for discrete time delay systems this technical communique introduces a new concept of set invariance with respect to linear discrete time dynamics affected by delay we are interested in the definition and characterization of sequences of cyclically invariant subsets in the state space the algebraic conditions established in the late 80s for linear dynamics are generalized to invariance analysis in the presence of delays for given sequences of polyhedral sets

stochastic viability and dynamic programming this paper deals with the stochastic control of nonlinear systems in the presence of state and control constraints for uncertain discrete time dynamics in finite dimensional spaces in the deterministic case the viability kernel is known to play a basic role for the analysis of such problems and the design of viable control feedbacks in the present paper we show how a stochastic viability kernel and viable feedbacks relying on probability or chance constraints can be defined and computed by a dynamic programming equation an example illustrates most of the assertions

kpca plus lda a complete kernel fisher discriminant framework for feature extraction and recognition this paper examines the theory of kernel fisher discriminant analysis kfd in a hilbert space and develops a two phase kfd framework i e kernel principal component analysis kpca plus fisher linear discriminant analysis lda this framework provides novel insights into the nature of kfd based on this framework the authors propose a complete kernel fisher discriminant analysis ckfd algorithm ckfd can be used to carry out discriminant analysis in double discriminant subspaces the fact that it can make full use of two kinds of discriminant information regular and irregular makes ckfd a more powerful discriminator the proposed algorithm was tested and evaluated using the feret face database and the cenparmi handwritten numeral database the experimental results show that ckfd outperforms other kfd algorithms

geometric mean for subspace selection subspace selection approaches are powerful tools in pattern classification and data visualization one of the most important subspace approaches is the linear dimensionality reduction step in the fisher s linear discriminant analysis flda which has been successfully employed in many fields such as biometrics bioinformatics and multimedia information management however the linear dimensionality reduction step in flda has a critical drawback for a classification task with c classes if the dimension of the projected subspace is strictly lower than c 1 the projection to a subspace tends to merge those classes which are close together in the original feature space if separate classes are sampled from gaussian distributions all with identical covariance matrices then the linear dimensionality reduction step in flda maximizes the mean value of the kullback leibler kl divergences between different classes based on this viewpoint the geometric mean for subspace selection is studied in this paper three criteria are analyzed 1 maximization of the geometric mean of the kl divergences 2 maximization of the geometric mean of the normalized kl divergences and 3 the combination of 1 and 2 preliminary experimental results based on synthetic data uci machine learning repository and handwriting digits show that the third criterion is a potential discriminative subspace selection method which significantly reduces the class separation problem in comparing with the linear dimensionality reduction step in flda and its several representative extensions

bayes optimality in linear discriminant analysis we present an algorithm that provides the one dimensional subspace where the bayes error is minimized for the c class problem with homoscedastic gaussian distributions our main result shows that the set of possible one dimensional spaces v for which the order of the projected class means is identical defines a convex region with associated convex bayes error function g v this allows for the minimization of the error function using standard convex optimization algorithms our algorithm is then extended to the minimization of the bayes error in the more general case of heteroscedastic distributions this is done by means of an appropriate kernel mapping function this result is further extended to obtain the d dimensional solution for any given d by iteratively applying our algorithm to the null space of the d l dimensional solution we also show how this result can be used to improve upon the outcomes provided by existing algorithms and derive a low computational cost linear approximation extensive experimental validations are provided to demonstrate the use of these algorithms in classification data analysis and visualization

an efficient approach to compute the threshold for multi dimensional linear classifiers abstract in this paper we theoretically analyze some properties that relate fisher s classifier and the optimal quadratic classifier when the latter is derived utilizing a particular covariance matrix for the classes we propose an efficient approach which is used to select the threshold after a linear transformation onto the one dimensional space is performed we achieve this by selecting the decision boundary that minimizes the classification error in the transformed space assuming that the univariate random variables are normally distributed our empirical results on synthetic and real life data sets show that our approach lead to smaller classification error than the traditional fisher s classifier the results also demonstrate that minimizing the classification error in the transformed space leads to smaller classification error in the original multi dimensional space

multiclass linear dimension reduction by weighted pairwise fisher criteria we derive a class of computationally inexpensive linear dimension reduction criteria by introducing a weighted variant of the well known k class fisher criterion associated with linear discriminant analysis lda it can be seen that lda weights contributions of individual class pairs according to the euclidean distance of the respective class means we generalize upon lda by introducing a different weighting function

on efficient computation of low complexity controlled invariant sets for uncertain linear systems a method is presented for determining invariant low complexity polytopic sets and associated linear feedback laws for linear systems with polytopic uncertainty conditions based on the relationship between 2 and norms are used to define an initial invariant low complexity polytope as the solution of a semi definite program the problem of computing a maximal controlled invariant low complexity polytopic set is then formulated as a bilinearly constrained problem and a relaxation of this problem is derived as an iterative sequence of convex programs the proposed method scales linearly with the state dimension which allows the possibility of determining low complexity robust controlled invariant sets for high order systems

on positively invariant sets for discrete time linear systems with disturbance an application of maximal disturbance sets this note concerns the investigation of maximal sets with respect to which a set is positively invariant and given a set a some properties of spl delta p i sets are analyzed the approach uses one step of the backward recursion introduced in d alessandro and de santis 1991 where feasibility and optimality conditions for linear dynamical input and or state constrained systems were analyzed in section ii the author gives the main definitions and some basic results in the subsequent section iii some properties of spl delta p i sets are investigated the particular case of polyhedral sets is analyzed in section iv in section v the author shows an application to linear systems with parametric uncertainties in the dynamic matrix a the author develops a technique to determine a set in the parameters space such that for any value of the parameter in the set a given set in the state space is positively invariant in the last section a numerical example is developed

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

simple lp type criteria for positively invariant polyhedral sets let s be a convex polyhedral set in the state space of a linear dynamical system a new proof of an algebraic condition for the set s to be positively invariant is given by the dual set of s the condition obtained in this correspondence can be examined by solving a set of standard lp type formulas it is also shown that the conventional algebraic conditions have redundancies in the lp type formulas discrete time systems as well as continuous time systems are considered

linear dimensionality reduction via a heteroscedastic extension of lda the chernoff criterion we propose an eigenvector based heteroscedastic linear dimension reduction ldr technique for multiclass data the technique is based on a heteroscedastic two class technique which utilizes the so called chernoff criterion and successfully extends the well known linear discriminant analysis lda the latter which is based on the fisher criterion is incapable of dealing with heteroscedastic data in a proper way for the two class case the between class scatter is generalized so to capture differences in co variances it is shown that the classical notion of between class scatter can be associated with euclidean distances between class means from this viewpoint the between class scatter is generalized by employing the chernoff distance measure leading to our proposed heteroscedastic measure finally using the results from the two class case a multiclass extension of the chernoff criterion is proposed this criterion combines separation information present in the class mean as well as the class covariance matrices extensive experiments and a comparison with similar dimension reduction techniques are presented

dependable control of uncertain linear systems based on set theoretic methods this article concerns the feedback control of discrete time systems subject to disturbances and uncertainties in both model parameters and signal measurements the uncertainties are assumed to be unknown but bounded and thus characterised by closed intervals or sets the main result is a new approach to design a feedback controller keeping the system state in a target set first a method is proposed that computes minimal enclosures of the set of reachable states which are consistent with the uncertain input and output measurements and the system dynamics then a control method to keep the current state set in the target set is developed which extends control techniques based on invariant polyhedra the method is illustrated by a mobile robot experiment

constrained stabilization with an assigned initial condition set the problem of stabilizing a linear discrete time system with control constraints is considered necessary and sufficient conditions are given for the existence of a state feedback controller which drives the state to the origin asymptotically from every initial state in an assigned compact polyhedral set these conditions can be checked via linear programming it is shown that when the problem has a solution a polyhedral function can be formed which turns out to be a lyapunov function if a proper nonlinear feedback controller is applied two procedures are presented for the construction of the lyapunov function the first is based on the property that the stabilizing feedback compensator exists if and only if for every initial condition chosen on a vertex of the set there exists an open loop control driving the state to its interior the second procedure is based on the construction of the controllability regions to the given polyhedral set this procedure can also be applied to systems with parameter u

two topics on linear unequal error protection codes bounds on their length and cyclic code classes it is possible for a linear block code to provide more protection for selected positions in the input message words than is guaranteed by the minimum distance of the code linear codes having this property are called linear unequal error protection luep codes bounds on the length of a luep code that ensures a given unequal error protection are derived a majority decoding method for certain classes of cyclic binary uep codes is treated a list of short i e of length less than 16 binary luep codes of optimal i e minimal length and a list of all cyclic binary uep codes of length less than 40 are included

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

convex liftings based robust control design this paper presents a new approach for control design of constrained linear systems affected by bounded additive disturbances and polytopic uncertainties this method hinges on so called convex liftings which emulate control lyapunov function by providing a constructive framework for optimization based control implementation it will be shown that this method can guarantee the recursive feasibility and robust stability finally a numerical example will be presented to illustrate this method

robust linear regulator design for discrete time systems under polyhedral constraints the robust linear constrained regulation problem for uncertain linear discrete time systems a b is investigated using the theory of positively invariant polyhedral sets closed and bounded polyhedral uncertain domains are considered involving parameters of both a and b matrices and polyhedral constraints on both state and control vectors new sufficient conditions for the solvability of the regulation problem are presented for computational solution of the regulation problem a linear programming approach is proposed where the constraints are given by the solvability conditions and the performance index represents a guaranteed trade off between the closed loop stability margin and the level of control effort

improved upper bounds for codes with unequal error protection asymptotic nonexistence bounds for unequal error protecting codes with two protection levels are considered we show that the improved estimates on the possible distance distributions for codes may sometimes yield sharper upper bounds than the previously known ones on the higher significance protection level of both nonlinear and linear codes having two protection levels

adaptive linear dimensionality reduction for classification abstract dimensionality reduction is the representation of high dimensional patterns in a low dimensional subspace based on a transformation which optimizes a specified criterion in the subspace for pattern classification the ideal criteria is the minimum achievable classification error the bayes error under strict assumptions of the pattern distribution the bayes error can be analytically expressed we use this as a starting point to develop an adaptive algorithm that computes a linear transformation based on the minimization of a cost function that approximates the bayes error in the subspace using kernel estimators we then relax the assumptions and extend the algorithm to more general pattern distributions our simulations with three synthetic and one real data set indicate that the proposed algorithm substantially outperforms fisher s linear discriminant

constrained predictive control of linear distributed parameter systems the model predictive control problem with guaranteed stability for a class of linear distributed parameter systems with simultaneous amplitude and rate control constraints is investigated the approach is developed using a finite dimensional input output modelling technique for distributed parameter systems based on a distributed output projection onto a finite basis set the paper employs a dual mode prediction strategy in conjunction with the projection based state space system description and proposes algorithms for the design of polyhedral positive invariant and feasible sets which serve as terminal target sets for predicted states an extension to predictive control with integral action is described the proposed algorithms are shown to work well by means of both simulations and experimental results

monotonicity properties for the viable control of discrete time systems this paper deals with the control of nonlinear systems in the presence of state and control constraints for discrete time dynamics in finite dimensional spaces the viability kernel is known to play a basic role for the analysis of such problems and the design of viable control feedbacks unfortunately this kernel may display very nonregular geometry and its computation is not an easy task in general in the present paper we show how monotonicity properties of both dynamics and constraints allow for relevant analytical upper and lower approximations of the viability kernel through weakly and strongly invariant sets an example on fish harvesting management illustrates some of the assertions

positive invariance monotonicity and comparison of nonlinear systems abstract in this paper the interrelation between positive invariance monotonicity and comparison of iterated nonlinear systems defined in partially ordered sets is studied first necessary and sufficient conditions guaranteeing the positive invariance of sets defined by relations of the form v x w with respect to nonlinear systems are established then various characterizations of monotone nonlinear systems are developed and a connection between positive invariance and monotonicity is established finally necessary and sufficient conditions for a not necessarily monotone nonlinear system to be a comparison system are established no specific algebraic structure is necessary for developing the main results of this work numerical examples illustrating the applicability of all these results to the case of discrete time dynamical systems are also given

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

a linear programming algorithm for invariant polyhedral sets of discrete time linear systems in this paper we formulate necessary and sufficient conditions for an arbitrary polyhedral set to be a positively invariant set of a linear discrete time system polyhedral cones and linear subspaces are included in the analysis a linear programming algorithm is presented that enables practical application of the results stated in this paper

the best possible lower bound for the perron root using traces abstract let a be an n n matrix with real eigenvalues wolkowicz and styan presented bounds for the eigenvalues using only n tr a and tr a 2 we show that their lower bound for the largest eigenvalue works also as a lower bound for the perron root of a if a is nonnegative and its eigenvalues are not necessarily real we also show that this bound is optimal under certain conditions finally we solve completely the problem to find the optimal lower bound for the perron root using only n tr a and tr a 2

about nonnegative realizations we will state sufficient conditions such that a discrete time time invariant finite dimensional system with a nonnegative impulse response has a nonnegative finite dimensional realization the conditions are in terms of the minimal polynomial

nonlinear unequal error protection codes are sometimes better than linear ones the parameters of some binary nonlinear two level unequal error protection uep codes are evaluated and compared with nonexistence bounds for linear uep codes the major points is that for uep codes linearity is an essential restriction the results demonstrate that at least in certain cases nonlinear codes perform better than linear ones this is in contrast to the situation for ordinary codes when in most cases the best possible result can be obtained even with the restriction of linearity

finite time stability and stabilisation with polyhedral domains for linear systems this paper deals with finite time stability boundedness and stabilisation of discrete time linear systems with regards to polyhedral domains the key observation made in this paper is that finite

on invariant polyhedra of continuous time linear systems this note presents some conditions of existence of positively invariant polyhedra for linear continuous time systems these conditions are first described algebraically then interpreted on the basis of the system eigenstructure then a simple state feedback placement method is described for solving some linear regulation problems under constraints

on the positive invariance of polyhedral sets for discrete time systems abstract in this note necessary and sufficient conditions for a polyhedral set to be a positively invariant set of a linear discrete time system are established

eigenstructure assignment for state constrained linear continuous time systems abstract linear state constraints define a polyhedron in the state space of a dynamical system if this polyhedron can be made positively invariant by state feedback the constraints are satisfied all along the trajectory for any initial point in this polyhedron in the context of continuous time linear systems some algebraic positive invariance conditions are established for unbounded symmetrical polyhedra the existence of a solution to the corresponding set of equations and inequalities depends on the structural properties of an associated input output system in particular the number of zeros of this system should be appropriate and they should be stable if these conditions are satisfied the problem can easily be solved by eigenvalue and eigenvector assignment

heteroscedastic linear feature extraction based on sufficiency conditions classification of high dimensional data typically requires extraction of discriminant features this paper proposes a linear feature extractor called whitened linear sufficient statistic wlss which is based on the sufficiency conditions for heteroscedastic gaussian distributions wlss approximates in the least squares sense an operator providing a sufficient statistic the proposed method retains covariance discriminance in heteroscedastic data while it reduces to the commonly used linear discriminant analysis lda in the homoscedastic case compared to similar heteroscedastic methods wlss imposes a low computational complexity and is highly generalizable as confirmed by its consistent competence over various data sets

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

on stochastic linear systems with zonotopic support sets abstract this paper analyzes stochastic linear discrete time processes whose process noise sequence consists of independent and uniformly distributed random variables on given zonotopes we propose a cumulant based approach for approximating both the transient and limit distributions of the associated state sequence the method relies on a novel class of k symmetric lyapunov equations which are used to construct explicit expressions for the cumulants the state distribution is recovered via a generalized gram charlier expansion with respect to products of a multivariate variant of wigner s semicircle distribution using chebyshev polynomials of the second kind this expansion converges uniformly under surprisingly mild conditions to the exact state distribution of the system a robust feedback control synthesis problem is used to illustrate the proposed approach

on optimal pairwise linear classifiers for normal distributions the two dimensional case optimal bayesian linear classifiers have been studied in the literature for many decades we demonstrate that all the known results consider only the scenario when the quadratic polynomial has coincident roots indeed we present a complete analysis of the case when the optimal classifier between two normally distributed classes is pairwise and linear we focus on some special cases of the normal distribution with nonequal covariance matrices we determine the conditions that the mean vectors and covariance matrices have to satisfy in order to obtain the optimal pairwise linear classifier as opposed to the state of the art in all the cases discussed here the linear classifier is given by a pair of straight lines which is a particular case of the general equation of second degree we also provide some empirical results using synthetic data for the minsky s paradox case and demonstrated that the linear classifier achieves very good performance finally we have tested our approach on real life data obtained from the uci machine learning repository the empirical results that we obtained show the superiority of our scheme over the traditional fisher s discriminant classifier

on matrices leaving invariant a nontrivial convex set abstract a real square matrix a leaves a nontrivial convex set invariant if there exists a convex set c which is not a linear subspace such that a c c it is shown that this is equivalent to the statement that a has an eigenvalue with 0 or 1

feedback stabilization of a class of distributed parameter systems with control constraints abstract the feedback stabilization problem for a class of infinite dimensional linear systems with control constraints is investigated the approach is developed using a state space system framework which is based on a semigroup formulation in contrast to the previous works in this direction it is not assumed that the c 0 semigroup we deal with is a semigroup of contractions the main result links the problem with the positive invariance of appropriate polyhedral sets verifiable necessary and sufficient conditions for this latter property are stated

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data