random subspace based ensemble sparse representation a new random subspace based joint sparse representation model is proposed the sparse representations in multiple subspaces are integrated into an ensemble sparse representation the spectral clustering algorithm with random subspace based ensemble sparse representation is shown the semi supervised classification method with random subspace based ensemble sparse representation is described in this paper a new random subspace based ensemble sparse representation rs esr algorithm is proposed where the random subspace is introduced into sparse representation model for high dimensional data the random subspace method can not only reduce dimension of data but also make full use of effective information of data it is not like traditional dimensionality reduction methods that may lose some information of original data additionally a joint sparse representation model is emloyed to obtain the sparse representation of a sample set in the low dimensional random subspace then the sparse representations in multiple random subspaces are integrated as an ensemble sparse representation moreover the obtained rs esr is applied in classical clustering and semi supervised classification the experimental results on different real world data sets show the superiority of rs esr over traditional methods

robust nonnegative patch alignment for dimensionality reduction dimensionality reduction is an important method to analyze high dimensional data and has many applications in pattern recognition and computer vision in this paper we propose a robust nonnegative patch alignment for dimensionality reduction which includes a reconstruction error term and a whole alignment term we use correntropy induced metric to measure the reconstruction error in which the weight is learned adaptively for each entry for the whole alignment we propose locality preserving robust nonnegative patch alignment lp rna and sparsity preserviing robust nonnegative patch alignment sp rna which are unsupervised and supervised respectively in the lp rna we propose a locally sparse graph to encode the local geometric structure of the manifold embedded in high dimensional space in particular we select large p nearest neighbors for each sample then obtain the sparse representation with respect to these neighbors the sparse representation is used to build a graph which simultaneously enjoys locality sparseness and robustness in the sp rna we simultaneously use local geometric structure and discriminative information in which the sparse reconstruction coefficient is used to characterize the local geometric structure and weighted distance is used to measure the separability of different classes for the induced nonconvex objective function we formulate it into a weighted nonnegative matrix factorization based on half quadratic optimization we propose a multiplicative update rule to solve this function and show that the objective function converges to a local optimum several experimental results on synthetic and real data sets demonstrate that the learned representation is more discriminative and robust than most existing dimensionality reduction methods

stability of linear density flow hyperbolic systems under pi boundary control we consider a class of density flow systems described by linear hyperbolic conservation laws which can be monitored and controlled at the boundaries these control systems are open loop unstable and subject to unmeasured flow disturbances we address the issue of feedback stabilization and disturbance rejection under pi boundary control explicit necessary and sufficient stability conditions in the frequency domain are provided

the solution set of the algebraic riccati equation and the algebraic riccati inequality abstract we present a parametrization of the solution set of the algebraic riccati equation and the algebraic riccati inequality of optimal control where we assume only sign controllability of the underlying system furthermore we characterize the existence of maximal and minimal solutions of both the equation and the inequality by checkable conditions

pi regulation of a reaction diffusion equation with delayed boundary control the general context of this work is the feedback control of an infinite dimensional system so that the closed loop system satisfies a fading memory property and achieves the setpoint tracking of a given reference signal more specifically this paper is concerned with the proportional integral pi regulation control of the left neumann trace of a one dimensional reaction diffusion equation with a delayed right dirichlet boundary control in this setting the studied reaction diffusion equation might be either open loop stable or unstable the proposed control strategy goes as follows first a finite dimensional truncated model that captures the unstable dynamics of the original infinite dimensional system is obtained via spectral decomposition the truncated model is then augmented by an integral component on the tracking error of the left neumann trace after resorting to the artstein transformation to handle the control input delay the pi controller is designed by pole shifting stability of the resulting closed loop infinite dimensional system consisting of the original reaction diffusion equation with the pi controller is then established thanks to an adequate lyapunov function in the case of a time varying reference input and a time varying distributed disturbance our stability result takes the form of an exponential input to state stability iss estimate with fading memory finally another exponential iss estimate with fading memory is established for the tracking performance of the reference signal by the system output in particular these results assess the setpoint regulation of the left neumann trace in the presence of distributed perturbations that converge to a steady state value and with a time derivative that converges to zero numerical simulations are carried out to illustrate the efficiency of our control strategy

necessary and sufficient conditions for solvability of the riccati inequalities in the general case abstract this paper presents necessary and sufficient conditions for solvability of riccati inequalities in the general case in terms of purely imaginary eigenvalues of associated hamiltonian matrices

prime discriminant simplicial complex the structure representation of data distribution plays an important role in understanding the underlying mechanism of generating data in this paper we propose the prime discriminant simplicial complex pdsc by utilizing persistent homology to capture such structures assuming that each class is represented with a prime simplicial complex we classify unlabeled samples based on the nearest projection distances from the samples to the simplicial complexes we also extend the extrapolation ability of these complexes with a projection constraint term experiments in simulated and practical datasets indicate that compared with several published algorithms the proposed pdsc approaches achieve promising performance without losing structure representation

design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations the paper deals with the control and regulation by integral controllers for the nonlinear systems governed by scalar quasi linear hyperbolic partial differential equations both the control input and the measured output are located on the boundary the closed loop stabilization of the linearized model with the designed integral controller is proved first by using the method of spectral analysis and then by the lyapunov direct method based on the elaborated lyapunov function we prove local exponential stability of the nonlinear closed loop system with the same controller the output regulation to the set point with zero static error by the integral controller is shown upon the nonlinear system numerical simulations by the preissmann scheme are carried out to validate the robustness performance of the designed controller to face to unknown constant disturbances

dissipativity and the lur e problem for parabolic boundary control systems dissipativity is studied for a class of boundary control systems whose free evolution is described by a holomorphic semigroup it is proved that the system is dissipative if and only if there exists a bounded selfadjoint solution of the corresponding linear operator inequality in addition the lur e problem is solved under particular assumptions

robust recovery of subspace structures by low rank representation in this paper we address the subspace clustering problem given a set of data samples vectors approximately drawn from a union of multiple subspaces our goal is to cluster the samples into their respective subspaces and remove possible outliers as well to this end we propose a novel objective function named low rank representation lrr which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary it is shown that the convex program associated with lrr solves the subspace clustering problem in the following sense when the data is clean we prove that lrr exactly recovers the true subspace structures when the data are contaminated by outliers we prove that under certain conditions lrr can exactly recover the row space of the original data and detect the outlier as well for data corrupted by arbitrary sparse errors lrr can also approximately recover the row space with theoretical guarantees since the subspace membership is provably determined by the row space these further imply that lrr can perform robust subspace clustering and error correction in an efficient and effective way

efficient semidefinite spectral clustering via lagrange duality we propose an efficient approach to semidefinite spectral clustering ssc which addresses the frobenius normalization with the positive semidefinite p s d constraint for spectral clustering compared with the original frobenius norm approximation based algorithm the proposed algorithm can more accurately find the closest doubly stochastic approximation to the affinity matrix by considering the p s d constraint in this paper ssc is formulated as a semidefinite programming sdp problem in order to solve the high computational complexity of sdp we present a dual algorithm based on the lagrange dual formalization two versions of the proposed algorithm are proffered one with less memory usage and the other with faster convergence rate the proposed algorithm has much lower time complexity than that of the standard interior point based sdp solvers experimental results on both the uci data sets and real world image data sets demonstrate that 1 compared with the state of the art spectral clustering methods the proposed algorithm achieves better clustering performance and 2 our algorithm is much more efficient and can solve larger scale ssc problems than those standard interior point sdp solvers

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

matrix riccati inequality existence of solutions abstract we consider the various criteria including hamiltonian and frequency domain ones for solvability of the matrix riccati inequality arising from control problems some criteria are given in terms of symplectic algebra which is also used as an important technical tool

from local similarities to global coding a framework for coding applications feature coding has received great attention in recent years as a building block of many image processing algorithms in particular the importance of the locality assumption in coding approaches has been studied in many previous works we review this assumption and claim that using the similarity of data points to a more global set of anchor points does not necessarily weaken the coding method as long as the underlying structure of the anchor points is considered we propose to capture the underlying structure by assuming a random walker over the anchor points we also show that our method is a fast approximation to the diffusion map kernel experiments on various data sets show that with a knowledge of the underlying structure of anchor points different state of the art coding algorithms may boost their performance in different learning tasks by utilizing the proposed method

the algebraic riccati equation a polynomial approach abstract polynomial models are used to give a unified approach to the problem of classifying the set of all real symmetric solutions of the algebraic riccati equation

on a quadratic matrix inequality and the corresponding algebraic riccati equation abstract the purpose of this paper is to report some new results concerning the geometry of the set x1d4ab 0 of all solutions of a quadratic matrix inequality qmi arising in stochastic realization theory and the subset x1d4ab 0 of all solutions of the corresponding algebraic riccati equation are the boundary interior and extreme points of x1d4ab are studied some new results linking x1d4ab 0 to x1d4ab are presented a complete characterization of some subsets of x1d4ab 0 and x1d4ab are given additional properties of the feedback matrix corresponding to any element in x1d4ab 0 are obtained

algebraic riccati equation and symplectic algebra questions of existence uniqueness and the parametric dependence of solutions of the algebraic riccati equation are considered the different criteria for the solubility of this equation are obtained with the help of symplectic algebra

manifold constraint transfer for visual structure driven optimization abstract in this paper we leverage the manifold structure of visual data in order to improve performance in general optimization problems subject to linear constraints as the main theoretical result we show that manifold constraints can be transferred from the data to the optimized variables if these are linearly correlated we also show that the resulting optimization problem can be solved with an efficient alternating direction method of multipliers that can consistently integrate the manifold constraints during the optimization process this leads to a simplification of the approach which instead of directly optimizing on the manifold we can iteratively recast the problem as the projection over the manifold via an embedding method the proposed method is extremely versatile since it can be applied to different problems including kernel ridge regression krr and sparse coding which have numerous applications in machine learning and computer vision in particular we apply the methods to different problems such as tracking object recognition and categorization showing a consistent increase of performance with respect to the state of the art

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

robust controller for systems with exponentially stable strongly continuous semigroups abstract the theory of robust controllers is extended to the case where we have boundary and or distributed control and the system operator is an infinitesimal generator of a strongly continuous exponentially stable semigroup an example on hyperbolic systems is presented

low rank matrix factorization with adaptive graph regularizer in this paper we present a novel low rank matrix factorization algorithm with adaptive graph regularizer lmfagr we extend the recently proposed low rank matrix with manifold regularization mmf method with an adaptive regularizer different from mmf which constructs an affinity graph in advance lmfagr can simultaneously seek graph weight matrix and low dimensional representations of data that is graph construction and low rank matrix factorization are incorporated into a unified framework which results in an automatically updated graph rather than a predefined one the experimental results on some data sets demonstrate that the proposed algorithm outperforms the state of the art low rank matrix factorization methods

integral action in output feedback for multi input multi output nonlinear systems we address a particular problem of output regulation for multi input multi output nonlinear systems specifically we are interested in making the stability of an equilibrium point and the regulation to zero of an output robust to small unmodelled discrepancies between design model and actual system in particular those introducing an offset we propose a novel procedure which is intended to be relevant to realistic scenarios as illustrated by a non academic example

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

brief paper boundary control with integral action for hyperbolic systems of conservation laws stability and experiments a strict lyapunov function for boundary control with integral actions of hyperbolic systems of conservation laws that can be diagonalised with riemann invariants is presented the time derivative of this lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions and the integral control gains previous stability results are extended to guarantee the local convergence of the state towards a desired set point furthermore the control can be implemented as a feedback of the state measured only at the boundaries the control design method is illustrated with a hydraulic application namely the level and flow regulation in a reach of the sambre river and in the micro channel of valence respectively through simulations and experimentations

integrator forwarding a new recursive nonlinear robust design we consider the global stabilization of nonlinear systems in strict feedforward form we show that these systems while not feedback linearizable can be globally transformed by feedback and diffeomorphism into lower triangular form the transformation is explicit and recursive we employ this transformation to design a globally stabilizing controller with optimality properties and an input to state stability property with respect to matched uncertainties c 1997 elsevier science ltd

realistic action recognition via sparsely constructed gaussian processes realistic action recognition has been one of the most challenging research topics in computer vision the existing methods are commonly based on non probabilistic classification predicting category labels but not providing an estimation of uncertainty in this paper we propose a probabilistic framework using gaussian processes gps which can tackle regression problems with explicit uncertain models for action recognition a major challenge for gps when applied to large scale realistic data is that a large covariance matrix needs to be inverted during inference additionally from the manifold perspective the intrinsic structure of the data space is only constrained by a local neighborhood and data relationships with far distance usually can be ignored thus we design our gps covariance matrix via the proposed l1 construction and a local approximation la covariance weight updating method which are demonstrated to be robust to data noise automatically sparse and adaptive to the neighborhood extensive experiments on four realistic datasets i e ucf youtube ucf sports hollywood2 and hmdb51 show the competitive results of l1 gps compared with state of the art methods on action recognition tasks

label information guided graph construction for semi supervised learning in the literature most existing graph based semi supervised learning methods only use the label information of observed samples in the label propagation stage while ignoring such valuable information when learning the graph in this paper we argue that it is beneficial to consider the label information in the graph learning stage specifically by enforcing the weight of edges between labeled samples of different classes to be zero we explicitly incorporate the label information into the state of the art graph learning methods such as the low rank representation lrr and propose a novel semi supervised graph learning method called semi supervised low rank representation this results in a convex optimization problem with linear constraints which can be solved by the linearized alternating direction method though we take lrr as an example our proposed method is in fact very general and can be applied to any self representation graph learning methods experiment results on both synthetic and real data sets demonstrate that the proposed graph learning method can better capture the global geometric structure of the data and therefore is more effective for semi supervised learning tasks

noise robust semi supervised learning via fast sparse coding this paper presents a novel noise robust graph based semi supervised learning algorithm to deal with the challenging problem of semi supervised learning with noisy initial labels inspired by the successful use of sparse coding for noise reduction we choose to give new l1 norm formulation of laplacian regularization for graph based semi supervised learning since our l1 norm laplacian regularization is explicitly defined over the eigenvectors of the normalized laplacian matrix we formulate graph based semi supervised learning as an l1 norm linear reconstruction problem which can be efficiently solved by sparse coding furthermore by working with only a small subset of eigenvectors we develop a fast sparse coding algorithm for our l1 norm semi supervised learning finally we evaluate the proposed algorithm in noise robust image classification the experimental results on several benchmark datasets demonstrate the promising performance of the proposed algorithm highlightswe propose novel semi supervised learning based on fast sparse coding our algorithm achieves promising results in noise robust image classification our algorithm can readily be extended to many other challenging problems

h sub infinity control by state feedback and fast algorithms for the computation of optimal h sub infinity norms the suboptimality of some parameter for h sub infinity optimization by dynamic state feedback is characterized in terms of the solvability of riccati inequalities this is done without restricting the finite zero structure of the plant if there are no system zeros on the imaginary axis the h sub infinity problem can be treated in a complete and satisfactory way explicit characterizations optimum to be achieved are provided and a closed formula for the optimal value is derived in terms of the h sub infinity norm of some fixed transfer matrix if the optimum is not attained any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high gain a globally and quadratically convergent algorithm to compute the optimal value is proposed this algorithm is generalized to the h sub infinity optimization problem by measurement feedback

internal model theory for distributed parameter systems in this paper we consider robust output regulation of distributed parameter systems and the internal model principle the main purpose is to generalize the internal model principle by francis and wonham for infinite dimensional systems and clarify the relationships between different generalizations of the internal model we also construct a signal generator capable of generating infinite dimensional polynomially increasing signals

robust multivariable pi controller for infinite dimensional systems a robust multivariable controller is introduced for a class of distributed parameter systems the system to be controlled is given as dot x ax bu y cx in a banach space the purpose of the control which is based on the measurement y is to stabilize and regulate the system so that y t rightarrow y r as t rightarrow infty where y r is a constant reference vector under the assumptions that operator a generates a holomorphic stable semigroup b is linear and bounded c is linear and a bounded and the input and output spaces are of the same dimension a necessary and sufficient condition is found for the existence of a robust multivariable controller this controller appears to be a multivariable pi controller also a simple necessary criterion for the existence of a decentralized controller is derived the tuning of the controller is discussed and it is shown that the i part of the controller can be tuned on the basis of step responses without exact knowledge of the system s parameters the presented theory is then used as an example to control the temperature profile of a bar with the dirichlet boundary conditions

pi controllers for 1 d nonlinear transport equation in this paper we introduce a method to obtain necessary and sufficient stability conditions for systems governed by one dimensional nonlinear hyperbolic partial differential equations with closed loop integral controllers when the linear frequency analysis cannot be used anymore we study the stability of a general nonlinear transport equation where the control input and the measured output are both located on the boundaries the principle of the method is to extract the limiting part of the stability from the solution using a projector on a finite dimensional space and then use a lyapunov approach this paper improves a result of trinh andrieu and xu and gives an optimal condition for the design of the controller the results are illustrated with numerical simulations where the predicted stable and unstable regions can be clearly identified

inverse scattering experiments structured matrix inequalities and tensor algebra this paper deals with a general matrix extension problem with structural constraints and provides a recursive solution in terms of an inverse scattering experiment both the stationary and nonstationary cases are considered in addition to connections to tensor algebra

a robust pi controller for infinite dimensional systems in this paper we deal with single input single output systems of the form on a separable hilbert space h where the operator a is the generator of an exponentially stable c0 semigroup on h b h c is a a admissible linear operator and w is an arbitrary constant disturbance vector in h we propose a low gain pi controller which stabilizes and regulates the system such that for a given reference constant yr y t tends to yr independently of w as t our result generalizes the previous one of pohjolainen 1982 in that the semigroup is not necessarily holomorphic a numerical example will be given to illustrate the application of the theory

zeros of spectral factors the geometry of splitting subspaces and the algebraic riccati inequality in this paper it is shown how the zero dynamics of not necessarily square spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic riccati inequality the notion of output induced subspace of a minimal markovian splitting subspace which is the stochastic analogue of the supremal output nulling subspace in geometric control theory is introduced through this concept the analysis can be made coordinate free and straightforward geometric methods can be applied it is shown how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal markovian splitting subspaces in the sense that the relationship between the zeros of different spectral factors is reflected in the partial ordering of minimal splitting subspaces finally the well known characterization of the solutions of the algebraic riccati equation is generalized in terms of lagrangian subspaces invariant under the corresponding hamiltonian to the larger solution set of the algebraic riccati inequality

the general nonstrict algebraic riccati inequality abstract if a b is stabilizable it is well known how to verify the solvability or the existence of positive definite solutions of the nonstrict algebraic riccati inequality ari a x xa xbb x q 0 this paper serves to demonstrate how far it is presently possible to generalize these results if the uncontrollable modes of a b are not restricted we prove a new reduction principle which may be formulated as follows the ari has a solution iff a certain reduced order riccati equation for a stabilizable system a linear equation and a reduced order nonstrict lyapunov inequality corresponding to the uncontrollable modes on the imaginary axis have solutions basically the quadratic riccati inequality is hence reduced to a linear lyapunov inequality we discuss how this principle may be applied to actually check the solvability of the ari as the main motivation for our work we briefly explain the consequences of these results for the general state feedback h optimization problem

forwarding design with prescribed local behavior among the non linear control techniques some lyapunov design methods forwarding backstepping take advantage of the structure of the system feedforward form feedback form to formulate a continuous control law which stabilizes globally and asymptotically the equilibrium in addition to stabilization we focus on the local behaviour of the closed loop system providing conditions under which we can predetermine the behaviour around the origin for feedforward systems

pairwise sparsity preserving embedding for unsupervised subspace learning and classification two novel unsupervised dimensionality reduction techniques termed sparse distance preserving embedding sdpe and sparse proximity preserving embedding sppe are proposed for feature extraction and classification sdpe and sppe perform in the clean data space recovered by sparse representation and enhanced euclidean distances over noise removed data are employed to measure pairwise similarities of points in extracting informative features sdpe and sppe aim at preserving pairwise similarities between data points in addition to preserving the sparse characteristics this paper calculates the sparsest representation of all vectors jointly by a convex optimization the sparsest codes enable certain local information of data to be preserved and can endow sdpe and sppe a natural discriminating power adaptive neighborhood and robust characteristic against noise and errors in delivering low dimensional embeddings we also mathematically show sdpe and sppe can be effectively extended for discriminant learning in a supervised manner the validity of sdpe and sppe is examined by extensive simulations comparison with other related state of the art unsupervised algorithms show that promising results are delivered by our techniques

supervised neighborhood graph construction for semi supervised classification graph based methods are among the most active and applicable approaches studied in semi supervised learning the problem of neighborhood graph construction for these methods is addressed in this paper neighborhood graph construction plays a key role in the quality of the classification in graph based methods several unsupervised graph construction methods have been proposed that have addressed issues such as data noise geometrical properties of the underlying manifold and graph hyper parameters selection in contrast in order to adapt the graph construction to the given classification task many of the recent graph construction methods take advantage of the data labels however these methods are not efficient since the hypothesis space of their possible neighborhood graphs is limited in this paper we first prove that the optimal neighborhood graph is a subgraph of a k nn graph for a large enough k which is much smaller than the total number of data points therefore we propose to use all the subgraphs of k nns graph as the hypothesis space in addition we show that most of the previous supervised graph construction methods are implicitly optimizing the smoothness functional with respect to the neighborhood graph parameters finally we provide an algorithm to optimize the smoothness functional with respect to the neighborhood graph in the proposed hypothesis space experimental results on various data sets show that the proposed graph construction algorithm mostly outperforms the popular k nn based construction and other state of the art methods

linear matrix inequalities riccati equations and indefinite stochastic linear quadratic controls this paper deals with an optimal stochastic linear quadratic lq control problem in infinite time horizon where the diffusion term in dynamics depends on both the state and the control variables in contrast to the deterministic case we allow the control and state weighting matrices in the cost functional to be indefinite this leads to an indefinite lq problem which may still be well posed due to the deep nature of uncertainty involved the problem gives rise to a stochastic algebraic riccati equation sare which is however fundamentally different from the classical algebraic riccati equation as a result of the indefinite nature of the lq problem to analyze the sare we introduce linear matrix inequalities lmis whose feasibility is shown to be equivalent to the solvability of the sare moreover we develop a computational approach to the sare via a semi definite programming associated with the lmis finally numerical experiments are reported to illustrate the proposed approach

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

adding integral action for open loop exponentially stable semigroups and application to boundary control of pde systems the article deals with the output feedback regulation of exponentially stable systems by an integral controller we propose appropriate lyapunov functionals to prove exponential stability of the closed loop system an example of parabolic partial differential equation pde systems and an example of hyperbolic systems are worked out to show how exponentially stabilizing integral controllers are designed the proof is based on a novel lyapunov functional construction that employs the forwarding techniques

lagrangian relaxation graph matching abstract graph matching is a fundamental problem in computer vision area graph matching problem that incorporates pair wise constraints can be formulated as an integer quadratic programming iqp problem with affine mapping constraint since it is known to be np hard approximate relaxation methods are usually required to find approximate solutions in this paper we present a new effective graph matching relaxation method called lagrangian relaxation graph matching lrgm which aims to generate a relaxation model by incorporating the affine mapping constraint into the matching objective optimization there are three main benefits of the proposed lrgm method 1 the nonnegative affine mapping constraint encoding one to one mapping is naturally incorporated in lrgm relaxation via lagrangian regularization 2 by further adding a l 1 norm constraint lrgm can generate a sparse solution empirically and thus returns a desired discrete solution for original iqp matching problem 3 an effective update algorithm is derived to solve the proposed lrgm model theoretically the converged solution can be proven to be karush kuhn tucker kkt optimal experimental results on both synthetic data and real world image datasets show the effectiveness and benefits of the proposed lrgm method

subspace clustering using a low rank constrained autoencoder the performance of subspace clustering is affected by data representation data representation for subspace clustering maps data from the original space into another space with the property of better separability many data representation methods have been developed in recent years typical among them are low rank representation lrr and an autoencoder lrr is a linear representation method that captures the global structure of data with low rank constraint alternatively an autoencoder nonlinearly maps data into a latent space using a neural network by minimizing the difference between the reconstruction and input to combine the advantages of an lrr globality and autoencoder self supervision based locality we propose a novel data representation method for subspace clustering the proposed method called low rank constrained autoencoder lrae forces the latent representation of the neural network to be of low rank and the low rank constraint is computed as a prior from the input space one major advantage of the lrae is that the learned data representation not only maintains the local features of the data but also preserves the underlying low rank global structure extensive experiments on several datasets for subspace clustering were conducted they demonstrated that the proposed lrae substantially outperformed state of the art subspace clustering methods

homogeneous riccati equations versus quadratic inequalities it is proved that under certain hypotheses the existence of hermitian solutions of a homogenized algebraic riccati equation of continuous type is guaranteed if the corresponding quadratic inequality admits a hermitian solution both finite dimensional matrices and infinite dimensional hilbert space operators riccati equations are considered

on unifying multi view self representations for clustering by tensor multi rank minimization in this paper we address the multi view subspace clustering problem our method utilizes the circulant algebra for tensor which is constructed by stacking the subspace representation matrices of different views and then rotating to capture the low rank tensor subspace so that the refinement of the view specific subspaces can be achieved as well as the high order correlations underlying multi view data can be explored by introducing a recently proposed tensor factorization namely tensor singular value decomposition t svd kilmer et al in siam j matrix anal appl 34 1 148 172 2013 we can impose a new type of low rank tensor constraint on the rotated tensor to ensure the consensus among multiple views different from traditional unfolding based tensor norm this low rank tensor constraint has optimality properties similar to that of matrix rank derived from svd so the complementary information can be explored and propagated among all the views more thoroughly and effectively the established model called t svd based multi view subspace clustering t svd msc falls into the applicable scope of augmented lagrangian method and its minimization problem can be efficiently solved with theoretical convergence guarantee and relatively low computational complexity extensive experimental testing on eight challenging image datasets shows that the proposed method has achieved highly competent objective performance compared to several state of the art multi view clustering methods

mtc a fast and robust graph based transductive learning method despite the great success of graph based transductive learning methods most of them have serious problems in scalability and robustness in this paper we propose an efficient and robust graph based transductive classification method called minimum tree cut mtc which is suitable for large scale data motivated from the sparse representation of graph we approximate a graph by a spanning tree exploiting the simple structure we develop a linear time algorithm to label the tree such that the cut size of the tree is minimized this significantly improves graph based methods which typically have a polynomial time complexity moreover we theoretically and empirically show that the performance of mtc is robust to the graph construction overcoming another big problem of traditional graph based methods extensive experiments on public data sets and applications on web spam detection and interactive image segmentation demonstrate our method s advantages in aspect of accuracy speed and robustness

solution to the positive real control problem for linear time invariant systems in this paper we study the problem of synthesizing an internally stabilizing linear time invariant controller for a linear time invariant plant such that a given closed loop transfer function is extended strictly positive real necessary and sufficient conditions for the existence of a controller are obtained state space formulas for the controller design are given in terms of solutions to algebraic riccati equations or inequalities the order of the constructed controller does not exceed that of the plant

a collaborative representation based projections method for feature extraction in graph embedding based methods we usually need to manually choose the nearest neighbors and then compute the edge weights using the nearest neighbors via l2 norm e g lle it is difficult and unstable to manually choose the nearest neighbors in high dimensional space so how to automatically construct a graph is very important in this paper first we give a l2 graph like l1 graph l2 graph calculates the edge weights using the total samples avoiding manually choosing the nearest neighbors second a l2 graph based feature extraction method is presented called collaborative representation based projections crp like spp crp aims to preserve the collaborative representation based reconstruction relationship of data crp utilizes a l2 norm graph to characterize the local compactness information crp maximizes the ratio between the total separability information and the local compactness information to seek the optimal projection matrix crp is much faster than spp since crp calculates the objective function with l2 norm while spp calculate the objective function with l1 norm experimental results on feret ar yale face databases and the polyu finger knuckle print database demonstrate that crp works well in feature extraction and leads to a good recognition performance we give a l2 norm graph based on collaborative representation we propose a collaborative representation based projections crp for feature extraction crp is a rayleigh quotient form and can be calculated via generalized eigenvalue decomposition

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

approximate orthogonal sparse embedding for dimensionality reduction locally linear embedding lle is one of the most well known manifold learning methods as the representative linear extension of lle orthogonal neighborhood preserving projection onpp has attracted widespread attention in the field of dimensionality reduction in this paper a unified sparse learning framework is proposed by introducing the sparsity or l 1 norm learning which further extends the lle based methods to sparse cases theoretical connections between the onpp and the proposed sparse linear embedding are discovered the optimal sparse embeddings derived from the proposed framework can be computed by iterating the modified elastic net and singular value decomposition we also show that the proposed model can be viewed as a general model for sparse linear and nonlinear kernel subspace learning based on this general model sparse kernel embedding is also proposed for nonlinear sparse feature extraction extensive experiments on five databases demonstrate that the proposed sparse learning framework performs better than the existing subspace learning algorithm particularly in the cases of small sample sizes

principal graph and structure learning based on reversed graph embedding many scientific datasets are of high dimension and the analysis usually requires retaining the most important structures of data principal curve is a widely used approach for this purpose however many existing methods work only for data with structures that are mathematically formulated by curves which is quite restrictive for real applications a few methods can overcome the above problem but they either require complicated human made rules for a specific task with lack of adaption flexibility to different tasks or cannot obtain explicit structures of data to address these issues we develop a novel principal graph and structure learning framework that captures the local information of the underlying graph structure based on reversed graph embedding as showcases models that can learn a spanning tree or a weighted undirected ell 1 graph are proposed and a new learning algorithm is developed that learns a set of principal points and a graph structure from data simultaneously the new algorithm is simple with guaranteed convergence we then extend the proposed framework to deal with large scale data experimental results on various synthetic and six real world datasets show that the proposed method compares favorably with baselines and can uncover the underlying structure correctly

linear time subspace clustering via bipartite graph modeling we present a linear time subspace clustering approach that combines sparse representations and bipartite graph modeling the signals are modeled as drawn from a union of low dimensional subspaces and each signal is represented by a sparse combination of basis elements termed atoms which form the columns of a dictionary matrix the sparse representation coefficients are arranged in a sparse affinity matrix which defines a bipartite graph of two disjoint sets 1 atoms and 2 signals subspace clustering is obtained by applying low complexity spectral bipartite graph clustering that exploits the small number of atoms for complexity reduction the complexity of the proposed approach is linear in the number of signals thus it can rapidly cluster very large data collections performance evaluation of face clustering and temporal video segmentation demonstrates comparable clustering accuracies to state of the art at a significantly lower computational load

low rank embedding for robust image feature extraction robustness to noises outliers and corruptions is an important issue in linear dimensionality reduction since the sample specific corruptions and outliers exist the class special structure or the local geometric structure is destroyed and thus many existing methods including the popular manifold learning based linear dimensionality methods fail to achieve good performance in recognition tasks in this paper we focus on the unsupervised robust linear dimensionality reduction on corrupted data by introducing the robust low rank representation lrr thus a robust linear dimensionality reduction technique termed low rank embedding lre is proposed in this paper which provides a robust image representation to uncover the potential relationship among the images to reduce the negative influence from the occlusion and corruption so as to enhance the algorithm s robustness in image feature extraction lre searches the optimal lrr and optimal subspace simultaneously the model of lre can be solved by alternatively iterating the argument lagrangian multiplier method and the eigendecomposition the theoretical analysis including convergence analysis and computational complexity of the algorithms is presented experiments on some well known databases with different corruptions show that lre is superior to the previous methods of feature extraction and therefore it indicates the robustness of the proposed method the code of this paper can be downloaded from http www scholat com laizhihui

elastic net hypergraph learning for image clustering and semi supervised classification graph model is emerging as a very effective tool for learning the complex structures and relationships hidden in data in general the critical purpose of graph oriented learning algorithms is to construct an informative graph for image clustering and classification tasks in addition to the classical k nearest neighbor and r neighborhood methods for graph construction l 1 graph and its variants are emerging methods for finding the neighboring samples of a center datum where the corresponding ingoing edge weights are simultaneously derived by the sparse reconstruction coefficients of the remaining samples however the pairwise links of l 1 graph are not capable of capturing the high order relationships between the center datum and its prominent data in sparse reconstruction meanwhile from the perspective of variable selection the l 1 norm sparse constraint regarded as a lasso model tends to select only one datum from a group of data that are highly correlated and ignore the others to simultaneously cope with these drawbacks we propose a new elastic net hypergraph learning model which consists of two steps in the first step the robust matrix elastic net model is constructed to find the canonically related samples in a somewhat greedy way achieving the grouping effect by adding the l 2 penalty to the l 1 constraint in the second step hypergraph is used to represent the high order relationships between each datum and its prominent samples by regarding them as a hyperedge subsequently hypergraph laplacian matrix is constructed for further analysis new hypergraph learning algorithms including unsupervised clustering and multi class semi supervised classification are then derived extensive experiments on face and handwriting databases demonstrate the effectiveness of the proposed method

low rank preserving embedding abstract in this paper we consider the problem of linear dimensionality reduction with the novel technique of low rank representation which is a promising tool of discovering subspace structures of given data existing approaches based on graph embedding usually capture structure of data via stacking the local structure of each datum such as neighborhood graph l1 graph and l2 graph yet they lack explicit discrimination between those local structures and suffer from corrupted samples to this end we propose a new linear dimensionality reduction method by virtue of the lowest rank representation lrr of data which is dubbed low rank preserving embedding lrpe different from the traditional routes lrpe achieves all data self representations jointly and can thus extract the global structure of a data set as a whole the global low rank constraint explicitly enforces the lrr matrix to be block diagonal form so that the samples with a similar intrinsic structure which are more likely to be from the same class are described by a similar set of bases hence lrpe is discriminative even if no class labels are provided benefiting from the robust lrr lrpe is also robust to various noises and errors in data besides we rewritten all related methods into a unified formulation followed by a detailed solution and clear comparisons finally we conduct extensive experiments on publicly available data sets for data visualization and classification the inspiring experimental results show the effectiveness the cheap computation and the robustness of the proposed method

dissipativity of uncontrollable systems storage functions and lyapunov functions dissipative systems have played an important role in the analysis and synthesis of dynamical systems the commonly used definition of dissipativity often requires an assumption on the controllability of the system in this paper we use a definition of dissipativity that is slightly different and less often used in the literature to study a linear time invariant possibly uncontrollable dynamical system we provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not mixed i e no pair of uncontrollable poles is symmetric about the imaginary axis this condition is known to be related to the solvability of a lyapunov equation we link lyapunov functions for autonomous systems to storage functions of an uncontrollable system the set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature we show that for an uncontrollable system the set of storage functions is unbounded and that the unboundedness arises precisely due to the set of lyapunov functions for an autonomous linear system being unbounded further we show that stabilizability of a system results in this unbounded set becoming bounded from below positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive in this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions in most of the results in this paper we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity

an image segmentation method based on network clustering model abstract network clustering phenomena are ubiquitous in nature and human society in this paper a method involving a network clustering model is proposed for mass segmentation in mammograms first the watershed transform is used to divide an image into regions and features of the image are computed then a graph is constructed from the obtained regions and features the network clustering model is applied to realize clustering of nodes in the graph compared with two classic methods the algorithm based on the network clustering model performs more effectively in experiments

discriminative sparsity preserving projections for image recognition previous works have demonstrated that image classification performance can be significantly improved by manifold learning however performance of manifold learning heavily depends on the manual selection of parameters resulting in bad adaptability in real world applications in this paper we propose a new dimensionality reduction method called discriminative sparsity preserving projections dspp different from the existing sparse subspace algorithms which manually construct a penalty adjacency graph dspp employs sparse representation model to adaptively build both intrinsic adjacency graph and penalty graph with weight matrix and then integrates global within class structure into the discriminant manifold learning objective function for dimensionality reduction extensive experimental results on four image databases demonstrate the effectiveness of the proposed approach we analyzed the vertex points of penalty graph using sparse representation we adaptively constructed both penalty adjacency graph and intrinsic graph our method simultaneously considered the global and local geometric structure of data

regularized coplanar discriminant analysis for dimensionality reduction abstract the dimensionality reduction methods based on linear embedding such as neighborhood preserving embedding npe sparsity preserving projections spp and collaborative representation based projections crp try to preserve a certain kind of linear representation for each sample after projection however in the transformed low dimensional space the linear relationship between the samples may be changed which cannot make the linear representation based classifiers such as sparse representation based classifier src to achieve higher recognition accuracy in this paper we propose a new linear dimensionality reduction algorithm called regularized coplanar discriminant analysis rcda to address this problem it simultaneously seeks a linear projection matrix and some linear representation coefficients that make the samples from the same class coplanar and the samples from different classes not coplanar the proposed regularization term balances the bias from the optimal linear representation and that from the class mean to avoid overfitting the training data and overcomes the matrix singularity in solving the linear representation coefficients an alternative optimization approach is proposed to solve the rcda model experiments are done on several benchmark face databases and hyperspectral image databases and results show that rcda can obtain better performance than other dimensionality reduction methods

robust and low rank representation for fast face identification with occlusions in this paper we propose an iterative method to address the face identification problem with block occlusions our approach utilizes a robust representation based on two characteristics in order to model contiguous errors e g block occlusion effectively the first fits to the errors a distribution described by a tailored loss function the second describes the error image as having a specific structure resulting in low rank in comparison with image size we will show that this joint characterization is effective for describing errors with spatial continuity our approach is computationally efficient due to the utilization of the alternating direction method of multipliers a special case of our fast iterative algorithm leads to the robust representation method which is normally used to handle non contiguous errors e g pixel corruption extensive results on representative face databases in constrained and unconstrained environments document the effectiveness of our method over existing robust representation methods with respect to both identification rates and computational time

subspace learning by l0 induced sparsity subspace clustering methods partition the data that lie in or close to a union of subspaces in accordance with the subspace structure such methods with sparsity prior such as sparse subspace clustering ssc elhamifar and vidal in ieee trans pattern anal mach intell 35 11 2765 2781 2013 with the sparsity induced by the ell 1 norm are demonstrated to be effective in subspace clustering most of those methods require certain assumptions e g independence or disjointness on the subspaces however these assumptions are not guaranteed to hold in practice and they limit the application of existing sparse subspace clustering methods in this paper we propose ell 0 induced sparse subspace clustering ell 0 ssc in contrast to the required assumptions such as independence or disjointness on subspaces for most existing sparse subspace clustering methods we prove that ell 0 ssc guarantees the subspace sparse representation a key element in subspace clustering for arbitrary distinct underlying subspaces almost surely under the mild i i d assumption on the data generation we also present the no free lunch theorem which shows that obtaining the subspace representation under our general assumptions can not be much computationally cheaper than solving the corresponding ell 0 sparse representation problem of ell 0 ssc a novel approximate algorithm named approximate ell 0 ssc a ell 0 ssc is developed which employs proximal gradient descent to obtain a sub optimal solution to the optimization problem of ell 0 ssc with theoretical guarantee the sub optimal solution is used to build a sparse similarity matrix upon which spectral clustering is performed for the final clustering results extensive experimental results on various data sets demonstrate the superiority of a ell 0 ssc compared to other competing clustering methods furthermore we extend ell 0 ssc to semi supervised learning by performing label propagation on the sparse similarity matrix learnt by a ell 0 ssc and demonstrate the effectiveness of the resultant semi supervised learning method termed ell 0 sparse subspace label propagation ell 0 sslp

adaptive maximum margin analysis for image recognition abstract most existing discriminant manifold learning methods aim to maximize the margin among nearby data which is determined in the high dimensional original space as such they do not necessarily best maximize the margin between different classes in the low dimensional space which is a critically important property for image classification to handle this problem we propose an adaptive maximum margin analysis amma for feature extraction amma aims to seek a projection matrix that best maximize the margin which is calculated in the low dimensional space it uses sparse representation to adaptively construct the intrinsic and penalty graphs finally an iterative algorithm is developed to solve the projection matrix extensive experimental results on several image databases illustrate the effectiveness of the proposed approach

adaptive graph construction using data self representativeness for pattern classification graph construction from data constitutes a pre stage in many machine learning and computer vision tasks like semi supervised learning manifold learning and spectral clustering the influence of graph construction procedures on learning tasks and their related applications has only received limited study despite its critical impact on accuracy state of the art graphs are built via sparse coding adopting 1 regularization those graphs exhibit good performance in many computer vision applications however the locality and similarity among instances are not explicitly used in the coding scheme furthermore due to the use of 1 regularization these construction approaches can be computationally expensive in this paper we investigate graph construction using the data self representativeness property by incorporating a variant of locality constrained linear coding llc we introduce and derive four variants for graph construction these variants adopt a two phase llc tpllc compared with the recent 1 graphs our proposed objective function associated with three variants has an analytical solution and thus is more efficient a key element of the proposed methods is the second phase of coding that allows data closeness or locality to be naturally incorporated it performs a coding over some selected relevant samples and reinforces the individual regularization terms by exploiting the coefficients estimated in the first phase comprehensive experimental results using several benchmark datasets show that it can achieve or outperform existing state of the art results furthermore it is shown to be more efficient than the robust 1 graph construction schemes

multi view low rank sparse subspace clustering abstract most existing approaches address multi view subspace clustering problem by constructing the affinity matrix on each view separately and afterwards propose how to extend spectral clustering algorithm to handle multi view data this paper presents an approach to multi view subspace clustering that learns a joint subspace representation by constructing affinity matrix shared among all views relying on the importance of both low rank and sparsity constraints in the construction of the affinity matrix we introduce the objective that balances between the agreement across different views while at the same time encourages sparsity and low rankness of the solution related low rank and sparsity constrained optimization problem is for each view solved using the alternating direction method of multipliers furthermore we extend our approach to cluster data drawn from nonlinear subspaces by solving the corresponding problem in a reproducing kernel hilbert space the proposed algorithm outperforms state of the art multi view subspace clustering algorithms on one synthetic and four real world datasets

learning with ell 1 graph for image analysis the graph construction procedure essentially determines the potentials of those graph oriented learning algorithms for image analysis in this paper we propose a process to build the so called directed 1 graph in which the vertices involve all the samples and the ingoing edge weights to each vertex describe its 1 norm driven reconstruction from the remaining samples and the noise then a series of new algorithms for various machine learning tasks e g data clustering subspace learning and semi supervised learning are derived upon the 1 graphs compared with the conventional k nearest neighbor graph and ball graph the 1 graph possesses the advantages 1 greater robustness to data noise 2 automatic sparsity and 3 adaptive neighborhood for individual datum extensive experiments on three real world datasets show the consistent superiority of 1 graph over those classic graphs in data clustering subspace learning and semi supervised learning tasks

convex sparse spectral clustering single view to multi view spectral clustering sc is one of the most widely used methods for data clustering it first finds a low dimensional embedding u of data by computing the eigenvectors of the normalized laplacian matrix and then performs k means on text u top to get the final clustering result in this paper we observe that in the ideal case text u text u top should be block diagonal and thus sparse therefore we propose the sparse sc ssc method that extends the sc with sparse regularization on text u text u top to address the computational issue of the nonconvex ssc model we propose a novel convex relaxation of ssc based on the convex hull of the fixed rank projection matrices then the convex ssc model can be efficiently solved by the alternating direction method of multipliers furthermore we propose the pairwise ssc that extends ssc to boost the clustering performance by using the multi view information of data experimental comparisons with several baselines on real world datasets testify to the efficacy of our proposed methods

latent semantic learning with structured sparse representation for human action recognition this paper proposes a novel latent semantic learning method for extracting high level latent semantics from a large vocabulary of abundant mid level features i e visual keywords with structured sparse representation which can help to bridge the semantic gap in the challenging task of human action recognition to discover the manifold structure of mid level features we develop a graph based spectral embedding approach to latent semantic learning with the graph over mid level features being constructed using sparse representation moreover we define new l 1 norm hypergraph regularization to induce extra structured sparsity into sparse representation for graph construction due to the nice properties sparsity and noise robustness of such structured sparse representation our graph construction can capture dominant and robust relationships among mid level features which are crucial for the success of latent semantic learning in action recognition unlike the traditional latent semantic analysis based on topic models our latent semantic learning method can explore the manifold structure of mid level features in both graph construction and spectral embedding which results in compact but discriminative high level features the experimental results on the commonly used kth action dataset and unconstrained youtube action dataset show the promising performance of our method

laplacian regularized low rank representation and its applications low rank representation lrr has recently attracted a great deal of attention due to its pleasing efficacy in exploring low dimensional subspace structures embedded in data for a given set of observed data corrupted with sparse errors lrr aims at learning a lowest rank representation of all data jointly lrr has broad applications in pattern recognition computer vision and signal processing in the real world data often reside on low dimensional manifolds embedded in a high dimensional ambient space however the lrr method does not take into account the non linear geometric structures within data thus the locality and similarity information among data may be missing in the learning process to improve lrr in this regard we propose a general laplacian regularized low rank representation framework for data representation where a hypergraph laplacian regularizer can be readily introduced into i e a non negative sparse hyper laplacian regularized lrr model nshlrr by taking advantage of the graph regularizer our proposed method not only can represent the global low dimensional structures but also capture the intrinsic non linear geometric information in data the extensive experimental results on image clustering semi supervised image classification and dimensionality reduction tasks demonstrate the effectiveness of the proposed method

robust subspace segmentation via nonconvex low rank representation a nonconvex formulation to determine the low rank representation from contaminated data is proposed we provide a proximal iteratively reweighed algorithm for solving the nonconvex model the proposed nonconvex model can recover the underlying low rank structure of subspaces in spite of noisy corruptions recently low rank representation lrr has been successfully applied to explore subspace segmentation of data in this paper we propose a nonconvex formulation to determine the lrr from contaminated data unlike in traditional methods which directly utilize the nuclear norm to approximate the rank function and penalize noise using the 2 1 norm our method introduces the ky fan p k norm and the 2 q norm to better approximate the rank minimization problem and enhance the robustness against noise an efficient algorithm is derived for solving the novel objective function and this is followed by a rigorous theoretical proof of the convergence extensive experiments on face datasets clearly demonstrate that the proposed methods are more robust to illumination variations corruptions and occlusions

unsupervised multi class co segmentation via joint cut over l_ 1 manifold hyper graph of discriminative image regions this paper systematically advocates a robust and efficient unsupervised multi class co segmentation approach by leveraging underlying subspace manifold propagation to exploit the cross image coherency it can combat certain image co segmentation difficulties due to viewpoint change partial occlusion complex background transient illumination and cluttering texture patterns our key idea is to construct a powerful hyper graph joint cut framework which incorporates mid level image regions based intra image feature representation and l 1 manifold graph based inter image coherency exploration for local image region generation we propose a bi harmonic distance distribution difference metric to govern the super pixel clustering in a bottom up way it not only affords drastic data reduction but also gives rise to discriminative and structure meaningful feature representation as for the inter image coherency we leverage multi type features involved l 1 graph to detect the underlying local manifold from cross image regions as a result the implicit supervising information could be encoded into the unsupervised hyper graph joint cut framework we conduct extensive experiments and make comprehensive evaluations with other state of the art methods over various benchmarks including icoseg msrc and oxford flower all the results demonstrate the superiorities of our method in terms of accuracy robustness efficiency and versatility

geometry of the algebraic riccati equation part ii we prove that the set of real symmetric solutions of the algebraic riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related n times n matrix by characterizing the structure of this variety we obtain a detailed description of the geometric properties of the solution set of the algebraic riccati equation

semi supervised double sparse graphs based discriminant analysis for dimensionality reduction abstract discriminant analysis da is a well known dimensionality reduction tool in pattern classification with enough efficient labeled samples the optimal projections could be found by maximizing the between class scatter variance meanwhile minimizing the within class scatter variance however the acquisition of label information is difficult in practice so semi supervised discriminant analysis has attracted much attention in recent years where both few labeled samples and many unlabeled samples are utilized during learning process sparse graph learned by sparse representation contains local structure information about data and is widely employed in dimensionality reduction in this paper semi supervised double sparse graphs sdsg based dimensionality reduction is proposed which considers both the positive and negative structure relationship of data points by using double sparse graphs aiming to explore the discriminant information among unlabeled samples joint k nearest neighbor selection strategy is proposed to select pseudo labeled samples which contain some precise discriminant information in the following procedures the data subset consisting of labeled samples and pseudo labeled samples are used instead of the original data based on two different criterions two sdsg based discriminant analysis methods are designed and denoted by sdsg dda distance based da and sdsg rda reconstruction based da which also use different strategies to reduce the effect of pseudo labels inaccuracy finally the experimental results both on uci datasets and hyperspectral images validate the effectiveness and advantage of the proposed methods compared with some classical dimensionality reduction methods

acquiring linear subspaces for face recognition under variable lighting previous work has demonstrated that the image variation of many objects human faces in particular under variable lighting can be effectively modeled by low dimensional linear spaces even when there are multiple light sources and shadowing basis images spanning this space are usually obtained in one of three ways a large set of images of the object under different lighting conditions is acquired and principal component analysis pca is used to estimate a subspace alternatively synthetic images are rendered from a 3d model perhaps reconstructed from images under point sources and again pca is used to estimate a subspace finally images rendered from a 3d model under diffuse lighting based on spherical harmonics are directly used as basis images in this paper we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low dimensional linear space and that this subspace is close to those acquired by the other methods more specifically there exist configurations of k point light source directions with k typically ranging from 5 to 9 such that by taking k images of an object under these single sources the resulting subspace is an effective representation for recognition under a wide range of lighting conditions since the subspace is generated directly from real images potentially complex and or brittle intermediate steps such as 3d reconstruction can be completely avoided nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse harmonic light fields we validate the use of subspaces constructed in this fashion within the context of face recognition

structure constrained low rank representation benefiting from its effectiveness in subspace segmentation low rank representation lrr and its variations have many applications in computer vision and pattern recognition such as motion segmentation image segmentation saliency detection and semisupervised learning it is known that the standard lrr can only work well under the assumption that all the subspaces are independent however this assumption cannot be guaranteed in real world problems this paper addresses this problem and provides an extension of lrr named structure constrained lrr sc lrr to analyze the structure of multiple disjoint subspaces which is more general for real vision data we prove that the relationship of multiple linear disjoint subspaces can be exactly revealed by sc lrr with a predefined weight matrix as a nontrivial byproduct we also illustrate that sc lrr can be applied for semisupervised learning the experimental results on different types of vision problems demonstrate the effectiveness of our proposed method

sparse subspace clustering algorithm theory and applications many real world problems deal with collections of high dimensional data such as images videos text and web documents dna microarray data and more often such high dimensional data lie close to low dimensional structures corresponding to several classes or categories to which the data belong in this paper we propose and study an algorithm called sparse subspace clustering to cluster data points that lie in a union of low dimensional subspaces the key idea is that among the infinitely many possible representations of a data point in terms of other points a sparse representation corresponds to selecting a few points from the same subspace this motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of the data into subspaces since solving the sparse optimization program is in general np hard we consider a convex relaxation and show that under appropriate conditions on the arrangement of the subspaces and the distribution of the data the proposed minimization program succeeds in recovering the desired sparse representations the proposed algorithm is efficient and can handle data points near the intersections of subspaces another key advantage of the proposed algorithm with respect to the state of the art is that it can deal directly with data nuisances such as noise sparse outlying entries and missing entries by incorporating the model of the data into the sparse optimization program we demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real world problems of motion segmentation and face clustering

joint sparse representation and embedding propagation learning a framework for graph based semisupervised learning in this paper we propose a novel graph based semisupervised learning framework called joint sparse representation and embedding propagation learning jsrepl the idea of jsrepl is to join epl with sparse representation to perform label propagation like most of graph based semisupervised propagation learning algorithms jsrepl also constructs weights graph matrix from given data different from classical approaches which build weights graph matrix and estimate the labels of unlabeled data in sequence jsrepl simultaneously builds weights graph matrix and estimates the labels of unlabeled data we also propose an efficient algorithm to solve the proposed problem the proposed method is applied to the problem of semisupervised image clustering using the orl yale pie and yaleb data sets our experiments demonstrate the effectiveness of our proposed algorithm

learning locality constrained collaborative representation for robust face recognition abstract the models of low dimensional manifold and sparse representation are two well known concise models that suggest that each data can be described by a few characteristics manifold learning is usually investigated for dimension reduction by preserving some expected local geometric structures from the original space into a low dimensional one the structures are generally determined by using pairwise distance e g euclidean distance alternatively sparse representation denotes a data point as a linear combination of the points from the same subspace in practical applications however the nearby points in terms of pairwise distance may not belong to the same subspace and vice versa consequently it is interesting and important to explore how to get a better representation by integrating these two models together to this end this paper proposes a novel coding algorithm called locality constrained collaborative representation lccr which introduce a kind of local consistency into coding scheme to improve the discrimination of the representation the locality term derives from a biologic observation that the similar inputs have similar codes the objective function of lccr has an analytical solution and it does not involve local minima the empirical studies based on several popular facial databases show that lccr is promising in recognizing human faces with varying pose expression and illumination as well as various corruptions and occlusions