design of an adaptive observer and its application to an adaptive pole placement controller abstract in this paper an adaptive observer design method based upon the least square approach is presented for an unknown time invariant continuous linear plant this observer is then implemented in the design of an adaptive pole placement problem for arbitrarily positioning all the poles of the unknown plant the generated feedback control signal is shown to converge to a desired pole placing state feedback control signal

design and application of reduced order adaptive observer with exponential rate of convergence for single input single output linear systems abstract this paper is concerned with the design of an adaptive observer to estimate state variables and identify parameters of an unknown linear single input single output system with desired rate of exponential convergence the observer structure is based on the kreisselmeier types representation of parametrized form but the adaptive law is obtained by a new scheme the adaptive law is determined by equations forcing the errors into decreasing exponentially and it is designed to guarantee the asymptotic stability of the identification and estimation processes globally a method of reducing the order of the adaptive law is also proposed the reduction of the order is executed by using a priori information regarding relations between the unknown parameters an experimental application using a heated tank with water jacket is included to show the practical effectiveness of the reduced order adaptive scheme

simultaneous parameter tracking and state estimation in a linear system abstract the task of simultaneous tracking of time varying parameters and estimation of the state is treated for a linear system described by a time varying input output armax or delta model with known c noise parameters first a bayesian approach based conceptual solution is presented then it is shown that utilizing the properties of the observer canonical state model algebraic recursion operating on the joint parameter and state mean and covariance matrix can be obtained with no approximation involved several illustrative examples are included

rates of convergence for a class of global stochastic optimization algorithms inspired and motivated by the recent advances in simulated annealing algorithms this paper analyzes the convergence rates of a class of recursive algorithms for global optimization via monte carlo methods by using perturbed liapunov function methods stability results of the algorithms are established then the rates of convergence are ascertained by examining the asymptotic properties of suitably scaled estimation error sequences

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

information theoretic lower bounds on the oracle complexity of stochastic convex optimization relative to the large literature on upper bounds on complexity of convex optimization lesser attention has been paid to the fundamental hardn4516420ess of these problems given the extensive use of convex optimization in machine learning and statistics gaining an understanding of these complexity theoretic issues is important in this paper we study the complexity of stochastic convex optimization in an oracle model of computation we introduce a new notion of discrepancy between functions and use it to reduce problems of stochastic convex optimization to statistical parameter estimation which can be lower bounded using information theoretic methods using this approach we improve upon known results and obtain tight minimax complexity estimates for various function classes

applying experimental design and regression splines to high dimensional continuous state stochastic dynamic programming in stochastic dynamic programming sdp with continuous state and decision variables the future value function is computed at discrete points in the state spac e interpolation can be used to approximate the values of the future value function between these discrete points however for large dimensional problems the number of discrete points required to obtain a good approximation of the future value function can be prohibitively large statistical methods of experimental design and function estimation may be employed to overcome this curse of dimensionality in this paper we describe a method for estimating the future value function by multivariate adaptive regression splines mars fit over a discretization scheme based on orthogonal array oa experimental designs because orthogonal arrays only grow polynomially in the state space dimension our oa mars method is accurately able to solve higher dimensional sdp problems than previously possible to our knowledge the most efficient method published prior to this work employs tensor product cubic splines to approximate the future value function johnson et al 1993 the computational advantages of oa ma rs are demonstrated in comparisons with the method using tensor product cubic splines for applications of an inventory forecasting sdp with up to nine state variables computed on a small workstation in particular the storage of an adequate tensor product cubic spline for six dimensions exceeds the memory of our workstation and the run time for an accurate oa mars sdp solution would be at least an order of magnitude faster than using tensor product cubic splines for higher than six dimensions

unconditional stabilization of linear discrete systems via adaptive control this paper deals with the conditions of overall deterministic stability of the adaptive control schemes which may be obtained by associating any usual identification and control methods e g recursive minimum square stochastic approximation extended kalman filter for identification model reference pole placement quadratic optimization for control the deterministic and time invariance hypotheses are introduced here only for the analysis of the above schemes but not for their synthesis there are no assumptions of sufficiently exciting input or any requirement for parameter convergence nor an hypothesis of bounded input and output signals

truncated power method for sparse eigenvalue problems this paper considers the sparse eigenvalue problem which is to extract dominant largest sparse eigenvectors with at most k non zero components we propose a simple yet effective solution called truncated power method that can approximately solve the underlying nonconvex optimization problem a strong sparse recovery result is proved for the truncated power method and this theory is our key motivation for developing the new algorithm the proposed method is tested on applications such as sparse principal component analysis and the densest k subgraph problem extensive experiments on several synthetic and real world data sets demonstrate the competitive empirical performance of our method

regularization theory and neural networks architectures we had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units called regularization networks in particular s

a discrete adaptive observer for a system of unknown order a discrete adaptive observer for a single input single output linear system is described the order of the system is unknown except that it is not greater than some preassigned integer the observer finds the system order estimates the parameters in output canonical form and gives the estimated state vector at each sample step

a new class of incremental gradient methods for least squares problems the least mean squares lms method for linear least squares problems differs from the steepest descent method in that it processes data blocks one by one with intermediate adjustment of the parameter vector under optimization this mode of operation often leads to faster convergence when far from the eventual limit and to slower sublinear convergence when close to the optimal solution we embed both lms and steepest descent as well as other intermediate methods within a one parameter class of algorithms and we propose a hybrid class of methods that combine the faster early convergence rate of lms with the faster ultimate linear convergence rate of steepest descent these methods are well suited for neural network training problems with large data sets furthermore these methods allow the effective use of scaling based for example on diagonal or other approximations of the hessian matrix

stochastic first order methods with random constraint projection we consider convex optimization problems with structures that are suitable for sequential treatment or online sampling in particular we focus on problems where the objective function is an expect

a comparison of global and semi local approximation in t stage stochastic optimization the paper presents a comparison between two different flavors of nonlinear models to be used for the approximate solution of t stage stochastic optimization tso problems a typical paradigm of markovian decision processes specifically the well known class of neural networks is compared with a semi local approach based on kernel functions characterized by less demanding computational requirements to this purpose two alternative methods for the numerical solution of tso are considered one corresponding to the classic approximate dynamic programming adp and the other based on a direct optimization of the optimal control functions introduced here for the first time advantages and drawbacks in the tso context of the two classes of approximators are analyzed in terms of computational burden and approximation capabilities then their performances are evaluated through simulations in two important high dimensional tso test cases namely inventory forecasting and water reservoirs management

near optimal stochastic approximation for online principal component estimation principal component analysis pca has been a prominent tool for high dimensional data analysis online algorithms that estimate the principal component by processing streaming data are of tremendous practical and theoretical interests despite its rich applications theoretical convergence analysis remains largely open in this paper we cast online pca into a stochastic nonconvex optimization problem and we analyze the online pca algorithm as a stochastic approximation iteration the stochastic approximation iteration processes data points incrementally and maintains a running estimate of the principal component we prove for the first time a nearly optimal finite sample error bound for the online pca algorithm under the subgaussian assumption we show that the finite sample error bound closely matches the minimax information lower bound

continuous time weighted least squares algorithm and its application to adaptive feedback control a weighted least squares algorithm is examined for continuous time linear deterministic systems the convergence of the algorithm is discussed from several viewpoints the necessity of the persistently exciting condition is stressed for the exponential convergence of the algorithm the need and the effect of introducing the normalizing factor into the algorithm are also considered in relation to the global stability of the adaptive feedback control adaptive pole placement control by means of an adaptive observer is considered as a typical example of the application of the algorithm

error estimates for approximate optimization by the extended ritz method an alternative to the classical ritz method for approximate optimization is investigated in the extended ritz method sets of admissible solutions are approximated by their intersections with sets of linear combinations of all n tuples of functions from a given basis this alternative scheme called variable basis approximation includes functions computable by trigonometric polynomials with free frequencies free node splines neural networks and other nonlinear approximating families estimates of rates of approximate optimization by the extended ritz method are derived upper bounds on rates of convergence of suboptimal solutions to the optimal one are expressed in terms of the degree n of variable basis functions the modulus of continuity of the functional to be minimized the modulus of tikhonov well posedness of the problem and certain norms tailored to the type of basis the results are applied to convex best approximation and to kernel methods in machine learning

a simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training a general convergence criterion for certain iterative sequences in hilbert space is presented for an important subclass of these sequences estimates of the rate of convergence are given under very mild assumptions these results establish an 0 1 f4n nonsampling convergence rate for projection pursuit regression and neural network training where n represents the number of ridge functions neurons or coefficients in a greedy basis expansion

high dimensional analysis of semidefinite relaxations for sparse principal components principal component analysis pca is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix however pca is well known to behave poorly in the large p small n setting in which the problem dimension p is comparable to or larger than the sample size n this paper studies pca in this high dimensional regime but under the additional assumption that the maximal eigenvector is sparse say with at most k nonzero components we consider a spiked covariance model in which a base matrix is perturbed by adding a k sparse maximal eigenvector and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector as follows a a simple diagonal thresholding method which transitions from success to failure as a function of the rescaled sample size dia n p k n k 2 log p k and b a more sophisticated semidefinite programming sdp relaxation which succeeds once the rescaled sample size sdp n p k n klog p k is larger than a critical threshold in addition we prove that no method including the best method which has exponential time complexity can succeed in recovering the support if the order parameter sdp n p k is below a threshold our results thus highlight an interesting trade off between computational and statistical efficiency in high dimensional inference

convergent ritz approximations of the set of stabilizing controllers abstract a necessary and sufficient condition for linear exponential sums to be dense in l p is derived under this condition a ritz approximation converges which has been proposed to solve linear control problems state space representations of orthonormal basis functions for multivariable systems are given

incremental subgradient methods for nondifferentiable optimization we consider a class of subgradient methods for minimizing a convex function that consists of the sum of a large number of component functions this type of minimization arises in a dual context from lagrangian relaxation of the coupling constraints of large scale separable problems the idea is to perform the subgradient iteration incrementally by sequentially taking steps along the subgradients of the component functions with intermediate adjustment of the variables after processing each component function this incremental approach has been very successful in solving large differentiable least squares problems such as those arising in the training of neural networks and it has resulted in a much better practical rate of convergence than the steepest descent method in this paper we establish the convergence properties of a number of variants of incremental subgradient methods including some that are stochastic based on the analysis and computational experiments the methods appear very promising and effective for important classes of large problems a particularly interesting discovery is that by randomizing the order of selection of component functions for iteration the convergence rate is substantially improved

sparse pca optimal rates and adaptive estimation principal component analysis pca is one of the most commonly used statistical procedures with a wide range of applications this paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting under mild technical conditions we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate the lower bound is obtained by calculating the local metric entropy and an application of fano s lemma the rate optimal estimator is constructed using aggregation which however might not be computationally feasible we then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently it is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces a key idea in our construction is a reduction scheme which reduces the sparse pca problem to a high dimensional multivariate regression problem this method is potentially also useful for other related problems

variable structure model reference adaptive control systems model reference adaptive controllers when only input and output measurements are available are designed using the theory of variable structure systems state variable filters are used in order to obtain differentiator free controllers the proposed design allows the error transient to be prescribed in advance the drawbacks deriving from the typical high frequency chattering of the control signal during the sliding motion can be partially eliminated by means of a little modification of the adaptation law that does not cause the error signal to deteriorate considerably

minimization of error functionals over variable basis functions generalized tikhonov well posedness is investigated for the problem of minimization of error functionals over admissible sets formed by variable basis functions i e linear combinations of a fixed number of elements chosen from a given basis without a prespecified ordering for variable basis functions of increasing complexity rates of decrease of infima of error functionals are estimated upper bounds are derived on such rates which do not exhibit the curse of dimensionality with respect to the number of variables of admissible functions consequences are considered for boolean functions and decision trees

random algorithms for convex minimization problems this paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems where the constraint set is specified as the intersection of possibly infinitely many constraint sets each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function the proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance but it is rather learned in time through observations also the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite we analyze the proposed algorithm for the case when the objective function is differentiable with lipschitz gradients and the case when the objective function is not necessarily differentiable the behavior of the algorithm is investigated both for diminishing and non diminishing stepsize values the almost sure convergence to an optimal solution is established for diminishing stepsize for non diminishing stepsize the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set as well as for the expected sub optimality of the function values along the weighted averages

on adaptive state regulation an indirect adaptive state regulator consisting of an adaptive observer and an asymptotic feedback matrix synthesis is discussed in a rigorous analysis lyapunov stability of the resulting closed loop system without the need for external excitation possible global instabilities as well as global convergence subject to sufficient excitation are shown moreover the behavior of the closed loop system is analyzed for cases in which the true plant differs from the assumed linear time invariant model in that it is of higher order contains slight nonlinearities and time variation effects and is subject to plant and measurement disturbances

stable adaptive control of non minimum phase systems abstract an adaptive pole placement design for not necessarily minimum phase systems is analyzed with respect to stability conditions are given for boundedness of closed loop signals when the process is subject to bounded disturbances the most restrictive one can be avoided in a local stability result

recursive solutions to deadbeat observers for time varying multivariable systems this paper describes a new construction of the deadbeat observer for the state estimation of linear time varying multivariable systems the proposed design methods are computationally attractive because of their simplicity and recursive nature and are recommendable even for time invariant multivariable systems

the vss approach to the model reference control of nonminimum phase linear plants a nontrivial application of the discontinuous control system theory based on filippov s solution concept is presented the problem addressed is the model reference output feedback control of linear plants with non hurwitzian numerator in the transfer function a solution in the case of known systems is presented first an error state equation whose state is accessible is then derived for the case in which the plant parameters are unknown in the case in which some bounds on the plant characteristics are supposed known a pair of discontinuous signals acting on both the plant and the model is chosen causing the error state to vanish asymptotically

design of adaptive pole placement controllers for plants of unknown order by an indirect control approach an adaptive pole placement control problem is considered for a scalar discrete time linear plant assuming the knowledge of an upper bound of the plant order a class of models that can be regarded to be input output equivalent to the plant is first constructed based on the parameter estimate generated by a least squares type identification scheme a minimization problem is then derived that enables one to determine the model s coefficients and the pole placing feedback gains so that the desired closed loop poles are assigned in a limit

model reference adaptive control using only input and output measurements a new method of designing model reference adaptive controllers using only input and output measurements is presented the adjustable parameters are all contained in the control input signal to the process state variable filters are used to avoid taking pure derivatives of the process output while the hyperstability theorem ensures global stability of the parameter adaptations the control input can also contain a term to diminish the effect of external load disturbance the natural extensions to multivariable and certain non linear systems are then presented the performance of the adaptive controller in the presence of noise can be further improved by the use of the instrumental variable approach together with an adjustable series compensator the theoretical development is substantiated by extensive simulation results

deterministic learning for maximum likelihood estimation through neural networks in this paper a general method for the numerical solution of maximum likelihood estimation mle problems is presented it adopts the deterministic learning dl approach to find close approximations to ml estimator functions for the unknown parameters of any given density the method relies on the choice of a proper neural network and on the deterministic generation of samples of observations of the likelihood function thus avoiding the problem of generating samples with the unknown density under mild assumptions consistency and convergence with favorable rates to the true ml estimator function can be proved simulation results are provided to show the good behavior of the algorithm compared to the corresponding exact solutions

functional optimization through semilocal approximate minimization an approach based on semilocal approximation is introduced for the solution of a general class of operations research problems such as markovian decision problems multistage optimal control and maximum likelihood estimation because it is extremely hard to derive analytical solutions that minimize the cost in most instances of the problem we must look for approximate solutions here it is shown that good solutions can be obtained with a moderate computational effort by exploiting properties of semilocal approximation through kernel models and efficient sampling of the state space the convergence of the proposed method called semilocal approximate minimization slam is discussed and the consistency of the solution is derived simulation results show the efficiency of slam also through its application to a classic operations research problem i e inventory forecasting

discrete adaptive observer with exponential weighting properties an adaptive observer based on the exponentially weighted least squares method is studied and a method of selecting the state variable filters is proposed by this method all the past measurement data are weighted exponentially both with weighting coefficient and the state variable filters the proposed scheme guarantees fast convergence and is very stable

design of a deadbeat adaptive tracking system using an orthogonal decomposition technique a matrix pseudo in version lemma is presented to derive recursive algorithms for computing the best approximate solution to the sequential estimation problem a deadbeat convergence property of the algorithms is shown and is applied to the design of a deadbeat adaptive tracking system simulations illustrate perfect tracking of the system output to the periodic reference trajectory in finite steps

a parametrization for model reference adaptive pole placement a new parametrization for a linear system is presented this parametrization is the basis for two approaches to design an adaptive controller for pole placement one approach is based on parameter estimation and requires sufficient excitation while the other the model reference adaptive pole placement uses a reference model and an augmented error it is shown that the two distinct approaches result in identical error terms

exponential convergence of adaptive identification and control algorithms output and equation error adaptive identification algorithms are shown to be exponentially convergent under a deterministic or stochastic persistently exciting or spanning condition on the system inputs together with several other standard conditions an adaptive control algorithm is shown to be exponentially convergent under a deterministic or stochastic persistently exciting condition on the reference trajectory together with some standard conditions

stability result with application to adaptive control we prove a relatively simple stability result for a particular system of time varying difference equations we then display its usefulness in analysing the stability and convergence of discrete time adaptive control systems in particular we apply our result to demonstrate the stability and convergence of two disparate adaptive control schemes

estimation of continuous time linear system parameters from periodic data the parameters of a continuous time linear system are identified by use of integral functions satisfying periodicity conditions explicit equations are derived for the evaluation of integral functions in terms of measured periodic output data it is shown that a walsh function representation of the integral functions has modest computational requirements making implementation on a 16 bit microprocessor feasible simulation and experimental results are provided to illustrate truncation error estimation accuracies and output response errors

direct model reference adaptive pole placement control with exponential weighting properties a parameterization for a linear system is presented to design a direct model reference adaptive pole placement controller this parameterized model is one of the structured nonminimal models the exponentially weighted least squares algorithm is employed to estimate the control parameters the direct adaptive controller has the exponential weighting properties by the proposed method of selecting the characteristic polynomials of the sensitivity function filters in connection with the reference model

continuous time constrained least squares algorithms for recursive parameter estimation of stochastic linear systems by a stabilized output error method discrete time least squares algorithms for recursive parameter estimation have continuous time counterparts which minimize a quadratic functional the continuous time algorithms can also include in equality constraints asymptotic convergence is demonstrated by means of lyapunov methods the constrained algorithms are applied in a stabilized output error configuration for parameter estimation in stochastic linear systems

successive galerkin approximation algorithms for nonlinear optimal and robust control nonlinear optimal control and nonlinear h infinity control are two of the most significant paradigms in nonlinear systems theory unfortunately these problems require the solution of hamilton jacobi equations which are extremely difficult to solve in practice to make matters worse approximation techniques for these equations are inherently prone to the so called curse of dimensionality while there have been many attempts to approximate these equations solutions resulting in closed loop control with well defined stability and robustness have remained elusive this paper describes a recent breakthrough in approximating the hamilton jacobi bellman and hamilton jacobi isaacs equations successive approximation and galerkin approximation methods are combined to derive a novel algorithm that produces stabilizing closed loop control laws with well defined stability regions in addition we show how the structure of the algorithm can be exploited to reduce the amount of computation from exponential to po

an adaptive lq controller for non minimum phase systems abstract the paper studied the problem of the adaptive tracking of a reference signal by the output of any deterministic discrete siso linear system with unknown parameters an adaptive control scheme based on the lq control solution is proposed the stability of the resulting adaptively controlled system and conditions for a good tracking are studied several simulations illustrate the proposal

stochastic compositional gradient descent algorithms for minimizing compositions of expected value functions classical stochastic gradient methods are well suited for minimizing expected value objective functions however they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected value functions i e the problem min x mathbf e v left f v big mathbf e w g w x big right minxevfv ew gw x in order to solve this stochastic composition problem we propose a class of stochastic compositional gradient descent scgd algorithms that can be viewed as stochastic versions of quasi gradient method scgd update the solutions based on noisy sample gradients of f v g w fv gw and use an auxiliary variable to track the unknown quantity mathbf e w left g w x right ewgw x we prove that the scgd converge almost surely to an optimal solution for convex optimization problems as long as such a solution exists the convergence involves the interplay of two iterations with different time scales for nonsmooth convex problems the scgd achieves a convergence rate of mathcal o k 1 4 o k 1 4 in the general case and mathcal o k 2 3 o k 2 3 in the strongly convex case after taking k samples for smooth convex problems the scgd can be accelerated to converge at a rate of mathcal o k 2 7 o k 2 7 in the general case and mathcal o k 4 5 o k 4 5 in the strongly convex case for nonconvex problems we prove that any limit point generated by scgd is a stationary point for which we also provide the convergence rate analysis indeed the stochastic setting where one wants to optimize compositions of expected value functions is very common in practice the proposed scgd methods find wide applications in learning estimation dynamic programming etc

on stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix abstract in applications of signal processing and pattern recognition eigenvectors and eigenvalues of the statistical mean of a random matrix sequence are needed iterative methods are suggested and analyzed in which no sample moments are used convergence is shown by stochastic approximation theory

incremental constraint projection methods for variational inequalities we consider the solution of strongly monotone variational inequalities of the form f x x x ge 0 f x x x 0 for all x in x x x we focus on special structures that lend themselves to sampling such as when x x is the intersection of a large number of sets and or f f is an expected value or is the sum of a large number of component functions we propose new methods that combine elements of incremental constraint projection and stochastic gradient these methods are suitable for problems involving large scale data as well as problems with certain online or distributed structures we analyze the convergence and the rate of convergence of these methods with various types of sampling schemes and we establish a substantial rate of convergence advantage for random sampling over cyclic sampling

global stability proofs for continuous time indirect adaptive control schemes this note presents a general stability proof for continuous time adaptive control with very general assumptions on the identifier and controller we show that if the exogenous input signal is rich enough then both the identifier and the controller converge to their true values applications of the proof to pole placement design and design based on the factorization approach are discussed

an incremental gradient projection method with momentum term and adaptive stepsize rule we consider an incremental gradient method with momentum term for minimizing the sum of continuously differentiable functions this method uses a new adaptive stepsize rule that decreases the stepsize whenever sufficient progress is not made we show that if the gradients of the functions are bounded and lipschitz continuous over a certain level set then every cluster point of the iterates generated by the method is a stationary point in addition if the gradient of the functions have a certain growth property then the method is either linearly convergent in some sense or the stepsizes are bounded away from zero the new stepsize rule is much in the spirit of heuristic learning rules used in practice for training neural networks via backpropagation as such the new stepsize rule may suggest improvements on existing learning rules finally extension of the method and the convergence results to constrained minimization is discussed as are some implementation issues and numerical experience

sparse principal component analysis and iterative thresholding principal component analysis pca is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix however it behaves poorly when the number of features p is comparable to or even much larger than the sample size n in this paper we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse under a spiked covariance model we find that the new approach recovers the principal subspace and leading eigenvectors consistently and even optimally in a range of high dimensional sparse settings simulated examples also demonstrate its competitive performance

incremental proximal methods for large scale convex optimization we consider the minimization of a sum sum i 1 mf i x consisting of a large number of convex component functions f i for this problem incremental methods consisting of gradient or subgradient iterations applied to single components have proved very effective we propose new incremental methods consisting of proximal iterations applied to single components as well as combinations of gradient subgradient and proximal iterations we provide a convergence and rate of convergence analysis of a variety of such methods including some that involve randomization in the selection of components we also discuss applications in a few contexts including signal processing and inference machine learning

finite sample approximation results for principal component analysis a matrix perturbation approach principal component analysis pca is a standard tool for dimensional reduction of a set of n observations samples each with p variables in this paper using a matrix perturbation approach we study the nonasymptotic relation between the eigenvalues and eigenvectors of pca computed on a finite sample of size n and those of the limiting population pca as n oo as in machine learning we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample pca and population pca under a spiked covariance model in addition we also consider the relation between finite sample pca and the asymptotic results in the joint limit p n with p n c we present a matrix perturbation view of the phase transition phenomenon and a simple linear algebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit moreover our analysis also applies for finite p n where we show that although there is no sharp phase transition as in the infinite case either as a function of noise level or as a function of sample size n the eigenvector of sample pca may exhibit a sharp loss of tracking suddenly losing its relation to the true eigenvector of the population pca matrix this occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise whose eigenvector points in a random direction

minimax sparse principal subspace estimation in high dimensions we study sparse principal components analysis in high dimensions where p the number of variables can be much larger than n the number of observations and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix we prove optimal non asymptotic lower and upper bounds on the minimax subspace estimation error under two different but related notions of lq subspace sparsity for 0 q 1 our upper bounds apply to general classes of covariance matrices and they show that lq constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions 1 introduction principal components analysis pca was introduced in the early 20th century pearson 1901 hotelling 1933 and is arguably the most well known and widely used technique for dimension reduction it is part of the mainstream statistical repertoire and is routinely used in numerous and diverse areas of application however contemporary applications often involve much higher dimensional data than envisioned by the early developers of pca in such high dimensional situations where the number of variables p is of the same order or much larger than the number of observations n serious difficulties emerge standard pca can produce inconsistent estimates of the principal directions of variation and lead to unreliable conclusions johnstone and lu 2009 paul 2007 nadler 2008 the principal directions of variation correspond to the eigenvectors of the covariance matrix and in high dimensions consistent estimation of the eigenvectors is generally not possible without additional assumptions about the covariance matrix or its eigenstructure much of the recent development in pca has focused on methodology that applies the concept of sparsity to eigenvector estimation some examples include jolliffe trendafilov and uddin

adaptive control of nonminimum phase systems recent papers have established global convergence for a class of adaptive control algorithms for discrete time linear dynamic systems however in most cases studied to date it has been assumed that the system is stably invertible this assumption plays a major role in the proofs of convergence in this paper we consider the more general case in which it is not assumed that the system is either stable or stably invertible we establish local convergence for a class of adaptive control algorithms applied to general discrete deterministic linear time invariant systems by convergence in this context we mean that the system inputs and outputs remain bounded for all time and the closed loop poles are effectively assigned in the limit for a given desired trajectory

approximate stochastic dynamic programming for hydroelectric production planning abstract this paper presents a novel approach for approximate stochastic dynamic programming asdp over a continuous state space when the optimization phase has a near convex structure the approach entails a simplicial partitioning of the state space bounds on the true value function are used to refine the partition we also provide analytic formulae for the computation of the expectation of the value function in the uni basin case where natural inflows are strongly correlated the approach is experimented on several configurations of hydro energy systems it is also tested against actual industrial data

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea