adaptive asymptotic control of multivariable systems based on a one parameter estimation approach multivariable adaptive control is an important and challenging research area where it usually requires designing adaptive laws to estimate a number of unknown parameters hence this process may excessively consume limitedly available computational resources and large amount of computational time which may degrade system performances particularly for the case of online updating of parameter estimates thus reducing the number of parameters to be estimated is a promising way to solve the problem the state of the art result in the area is to update just one adaptive parameter however the tracking errors are only ensured bounded and thus the resulting closed loop system is not asymptotically stable hence a question that arises is whether such a result of non zero tracking errors is the price paid for reducing the number of updating parameters up to now there is still no answer to this question in this paper we address such an issue and realize asymptotic stability with online estimation of just one parameter to achieve such a goal a novel dynamic loop gain function based approach is proposed to incorporate with the backstepping control design procedure which enables us to solve the algebraic loop problem caused by all the existing traditional nussbaum function based approaches and thus establish system stability a bound for the tracking error is explicitly established in terms of l2 norm which helps improve the transient performance by selecting the control parameters moreover guaranteed by the barbalats lemma the tracking error is further ensured to approach zero asymptotically finally simulation examples are conducted to testify the effectiveness of the proposed approach

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

nonnegative blind source separation by sparse component analysis based on determinant measure the problem of nonnegative blind source separation nbss is addressed in this paper where both the sources and the mixing matrix are nonnegative because many real world signals are sparse we deal with nbss by sparse component analysis first a determinant based sparseness measure named d measure is introduced to gauge the temporal and spatial sparseness of signals based on this measure a new nbss model is derived and an iterative sparseness maximization ism approach is proposed to solve this model in the ism approach the nbss problem can be cast into row to row optimizations with respect to the unmixing matrix and then the quadratic programming qp technique is used to optimize each row furthermore we analyze the source identifiability and the computational complexity of the proposed ism qp method the new method requires relatively weak conditions on the sources and the mixing matrix has high computational efficiency and is easy to implement simulation results demonstrate the effectiveness of our method

rate of convergence of the focuss algorithm focal underdetermined system solver focuss is a powerful method for basis selection and sparse representation where it employs the ell vphantom r j l p norm with p in 0 2 to measure the sparsity of solutions in this paper we give a systematical analysis on the rate of convergence of the focuss algorithm with respect to p in 0 2 we prove that the focuss algorithm converges superlinearly for 0 and linearly for 1 le p usually but may superlinearly in some very special scenarios in addition we verify its rates of convergence with respect to p by numerical experiments

sufficient subalgebras and the relative entropy of states of a von neumann algebra a subalgebram 0 of a von neumann algebram is called weakly sufficient with respect to a pair of states if the relative entropy of and coincides with the relative entropy of their restrictions tom 0 the main result says thatm 0 is weakly sufficient for if and only ifm 0 contains the radon nikodym cocycle d d t other conditions are formulated in terms of generalized conditional expectations and the relative hamiltonian

logarithmic sumt limits in convex programming the limits of a class of primal and dual solution trajectories associated with the sequential unconstrained minimization technique sumt are investigated for convex programming problems with non unique optima logarithmic barrier terms are assumed for linear programming problems such limits of both primal and dual trajectories are strongly optimal strictly complementary and can be characterized as analytic centers of loosely speaking optimality regions examples are given which show that those results do not hold in general for convex programming problems if the latter are weakly analytic bank et al 3 primal trajectory limits can be characterized in analogy to the linear programming case and without assuming differentiability that class of programming problems contains faithfully convex linear and convex quadratic programming problems as strict subsets in the differential case dual trajectory limits can be characterized similarly albeit under different conditions one of which suffices for strict complementarity

examples of ill behaved central paths in convex optimization this paper presents some examples of ill behaved central paths in convex optimization some contain infinitely many fixed length central segments others manifest oscillations with infinite variation these central paths can be encountered even for infinitely differentiable data

on the convergence of the central path in semidefinite optimization the central path in linear optimization always converges to the analytic center of the optimal set this result was extended to semidefinite optimization in d goldfarb and k scheinberg siam j optim 8 1998 pp 871 886 in this paper we show that this latter result is not correct in the absence of strict complementarity we provide a counterexample where the central path converges to a different optimal solution this unexpected result raises many questions we also give a short proof that the central path always converges in semidefinite optimization by using ideas from algebraic geometry

mixing matrix estimation from sparse mixtures with unknown number of sources in blind source separation many methods have been proposed to estimate the mixing matrix by exploiting sparsity however they often need to know the source number a priori which is very inconvenient in practice in this paper a new method namely nonlinear projection and column masking npcm is proposed to estimate the mixing matrix a major advantage of npcm is that it does not need any knowledge of the source number in npcm the objective function is based on a nonlinear projection and its maxima just correspond to the columns of the mixing matrix thus a column can be estimated first by locating a maximum and then deflated by a masking operation this procedure is repeated until the evaluation of the objective function decreases to zero dramatically thus the mixing matrix and the number of sources are estimated simultaneously because the masking procedure may result in some small and useless local maxima particle swarm optimization pso is introduced to optimize the objective function feasibility and efficiency of pso are also discussed comparative experimental results show the efficiency of npcm especially in the cases where the number of sources is unknown and the sources are relatively less sparse

on certain properties of the relative entropy of states of operator algebras the aim of the present paper is to establish certain properties of the relative entropy functional in the general context of c algebras most of these properties have been observed under more restrictive conditions in earlier papers the celebrated strong subadditivity of the entropy gives such an example the general approach is applied to prove the existence of the mean relative entropy for a noncommutative stationary markov chain with conditional expectations

a variational expression for the relative entropy we prove that for the relative entropy of faithful normal states and on the von neumann algebram the formula s varphi omega sup omega h log varphi h i h h in m holds

time frequency approach to underdetermined blind source separation this paper presents a new time frequency tf underdetermined blind source separation approach based on wigner ville distribution wvd and khatri rao product to separate n non stationary sources from m m n mixtures first an improved method is proposed for estimating the mixing matrix where the negative value of the auto wvd of the sources is fully considered then after extracting all the auto term tf points the auto wvd value of the sources at every auto term tf point can be found out exactly with the proposed approach no matter how many active sources there are as long as n 2m 1 further discussion about the extraction of auto term tf points is made and finally the numerical simulation results are presented to show the superiority of the proposed algorithm by comparing it with the existing ones

derivation of the euler equations from quantum dynamics we derive the euler equations from quantum dynamics for a class of fermionic many body systems we make two types of assumptions the first type are physical assumptions on the solution of the euler equations for the given initial data the second type are a number of reasonable conjectures on the statistical mechanics and dynamics of the fermion hamiltonian

subset selection in noise based on diversity measure minimization we develop robust methods for subset selection based on the minimization of diversity measures a bayesian framework is used to account for noise in the data and a maximum a posteriori map estimation procedure leads to an iterative procedure which is a regularized version of the focal underdetermined system solver focuss algorithm the convergence of the regularized focuss algorithm is established and it is shown that the stable fixed points of the algorithm are sparse we investigate three different criteria for choosing the regularization parameter quality of fit sparsity criterion l curve the l curve method as applied to the problem of subset selection is found not to be robust and we propose a novel modified l curve procedure that solves this problem each of the regularized focuss algorithms is evaluated through simulation of a detection problem and the results are compared with those obtained using a sequential forward selection algorithm termed orthogonal matching pursuit omp in each case the regularized focuss algorithm is shown to be superior to the omp in noisy environments

scalarization in vector optimization in this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem this will be done for weakly minimal and minimal solutions respectively finally the results will be applied to a certain class of approximation problems

two stage nonnegative sparse representation for large scale face recognition this paper proposes a novel nonnegative sparse representation approach called two stage sparse representation tsr for robust face recognition on a large scale database based on the divide and conquer strategy tsr decomposes the procedure of robust face recognition into outlier detection stage and recognition stage in the first stage we propose a general multisubspace framework to learn a robust metric in which noise and outliers in image pixels are detected potential loss functions including l1 l2 1 and correntropy are studied in the second stage based on the learned metric and collaborative representation we propose an efficient nonnegative sparse representation algorithm to find an approximation solution of sparse representation according to the l1 ball theory in sparse representation the approximated solution is unique and can be optimized efficiently then a filtering strategy is developed to avoid the computation of the sparse representation on the whole large scale dataset moreover theoretical analysis also gives the necessary condition for nonnegative least squares technique to find a sparse solution extensive experiments on several public databases have demonstrated that the proposed tsr approach in general achieves better classification accuracy than the state of the art sparse representation methods more importantly a significant reduction of computational costs is reached in comparison with sparse representation classifier this enables the tsr to be more suitable for robust face recognition on a large scale dataset

on convergence properties of the em algorithm for gaussian mixtures we build up the mathematical connection between the expectation maximization em algorithm and gradient based approaches for maximum likelihood learning of finite gaussian mixtures we show that the em step in parameter space is obtained from the gradient via a projection matrix p and we provide an explicit expression for the matrix we then analyze the convergence of em in terms of special properties of p and provide new results analyzing the effect that p has on the likelihood surface based on these mathematical results we present a comparative discussion of the advantages and disadvantages of em and other algorithms for the learning of gaussian mixture models

asymptotics of varadhan type and the gibbs variational principle for a large class of quantum models of mean field type the thermodynamic limit of the free energy density is proved to be given by the gibbs variational principle the latter is shown to be equivalent to a non commutative version of varadhan s asymptotic formula

strengthened monotonicity of relative entropy via pinched petz recovery map the quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy it is known that this inequality is only tight when there is a recovery map that exactly reverses the effects of the quantum channel on both states in this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version the recovery map is a convex combination of rotated petz recovery maps and perfectly reverses the quantum channel on the second state as a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved by fawzi and renner our proof only relies on the elementary properties of pinching maps and the operator logarithm

analysis of sparse representation and blind source separation in this letter we analyze a two stage cluster then l1 optimization approach for sparse representation of a data matrix which is also a promising approach for blind source separation bss in which fewer sensors than sources are present first sparse representation factorization of a data matrix is discussed for a given overcomplete basis matrix the corresponding sparse solution coefficient matrix with minimum l1 norm is unique with probability one which can be obtained using a standard linear programming algorithm the equivalence of the l1 norm solution and the l0 norm solution is also analyzed according to a probabilistic framework if the obtained l1 norm solution is sufficiently sparse then it is equal to the l0 norm solution with a high probability furthermore the l1 norm solution is robust to noise but the l0 norm solution is not showing that the l1 norm is a good sparsity measure these results can be used as a recoverability analysis of bss as discussed the basis matrix in this article is estimated using a clustering algorithm followed by normalization in which the matrix columns are the cluster centers of normalized data column vectors zibulevsky pearlmutter boll and kisilev 2000 used this kind of two stage approach in underdetermined bss our recoverability analysis shows that this approach can deal with the situation in which the sources are overlapped to some degree in the analyzed domain and with the case in which the source number is unknown it is also robust to additive noise and estimation error in the mixing matrix finally four simulation examples and an eeg data analysis example are presented to illustrate the algorithm s utility and demonstrate its performance

on the choice of parameters for the weighting method in vector optimization we present a geometrical interpretation of the weighting method for constrained finite dimensional vector optimization this approach is based on rigid movements which separate the image set from the negative of the ordering cone we study conditions on the existence of such translations in terms of the boundedness of the scalar problems produced by the weighting method finally using recession cones we obtain the main result of our work a sufficient condition under which weighting vectors yield solvable scalar problems

interpolation theory and the wigner yanase dyson lieb concavity the wigner yanase dyson lieb concavity is naturally captured in the frame of interpolation theory among other results a certain generalization involving operator monotone functions of this concavity in the context of general von neumann algebras is obtained also a close relationship between the above subjects and f hansen s inequality is clarified all results are proved by using simple variational expressions of involved quantities

a steepest descent method for vector optimization in this work we propose a cauchy like method for solving smooth unconstrained vector optimization problems when the partial order under consideration is the one induced by the nonnegative orthant we regain the steepest descent method for multicriteria optimization recently proposed by fliege and svaiter we prove that every accumulation point of the generated sequence satisfies a certain first order necessary condition for optimality which extends to the vector case the well known gradient equal zero condition for real valued minimization finally under some reasonable additional hypotheses we prove global convergence to a weak unconstrained minimizer as a by product we show that the problem of finding a weak constrained minimizer can be viewed as a particular case of the so called abstract equilibrium problem

on some properties and an application of the logarithmic barrier method we analyze the logarithmic barrier method for nonsmooth convex optimization in the setting of point to set theory this general framework allows us to both extend and include classical results we also propose an application for finding efficient points of nonsmooth constrained convex vector valued problems

penalty and barrier methods a unified framework it is established that many optimization problems may be formulated in terms of minimizing a function x rightarrow f 0 x h infty f 1 x f 2 x ldots f m x l infty ax b where the f i are closed functions defined on mathbb r n and where h infty and l infty are the recession functions of closed proper convex functions h and l a is a linear transformation from mathbb r n to a finite dimensional vector space y with b in y a generic algorithm based on the properties of recession functions is proposed this algorithm not only encompasses almost all penalty and barrier methods in nonlinear programming and in semidefinite programming but also generates new types of methods primal and dual convergence theorems are given

convergence analysis of the focuss algorithm focal underdetermined system solver focuss is a powerful and easy to implement tool for basis selection and inverse problems one of the fundamental problems regarding this method is its convergence which remains unsolved until now we investigate the convergence of the focuss algorithm in this paper we first give a rigorous derivation for the focuss algorithm by exploiting the auxiliary function following this we further prove its convergence by stability analysis

asymptotic analysis for penalty and barrier methods in convex and linear programming we consider a wide class of penalty and barrier methods for convex programming which includes a number of specific functions proposed in the literature we provide a systematic way to generate penalty and barrier functions in this class and we analyze the existence of primal and dual optimal paths generated by these penalty methods as well as their convergence to the primal and dual optimal sets for linear programming we prove that these optimal paths converge to single points

dictionary learning algorithms for sparse representation algorithms for data driven learning of domain specific overcomplete dictionaries are developed to obtain maximum likelihood and maximum a posteriori dictionary estimates based on the use of bayesian models with concave schur concave csc negative log priors such priors are appropriate for obtaining sparse representations of environmental signals within an appropriately chosen environmentally matched dictionary the elements of the dictionary can be interpreted as concepts features or words capable of succinct expression of events encountered in the environment the source of the measured signals this is a generalization of vector quantization in that one is interested in a description involving a few dictionary entries the proverbial 25 words or less but not necessarily as succinct as one entry to learn an environmentally adapted dictionary capable of concise expression of signals generated by the environment we develop algorithms that iterate between a representative set of sparse representations found by variants of focuss and an update of the dictionary using these sparse representations experiments were performed using synthetic data and natural images for complete dictionaries we demonstrate that our algorithms have improved performance over other independent component analysis ica methods measured in terms of signal to noise ratios of separated sources in the overcomplete case we show that the true underlying dictionary and sparse sources can be accurately recovered in tests with natural images learned overcomplete dictionaries are shown to have higher coding efficiency than complete dictionaries that is images encoded with an overcomplete dictionary have both higher compression fewer bits per pixel and higher accuracy lower mean square error

improved iteratively reweighted least squares for unconstrained smoothed ell_q minimization in this paper we first study ell q minimization and its associated iterative reweighted algorithm for recovering sparse vectors unlike most existing work we focus on unconstrained ell q minimization for which we show a few advantages on noisy measurements and or approximately sparse vectors inspired by the results in daubechies et al comm pure appl math 63 2010 pp 1 38 for constrained ell q minimization we start with a preliminary yet novel analysis for unconstrained ell q minimization which includes convergence error bound and local convergence behavior then the algorithm and analysis are extended to the recovery of low rank matrices the algorithms for both vector and matrix recovery have been compared to some state of the art algorithms and show superior performance on recovering sparse vectors and low rank matrices

sparse approximation to the eigensubspace for discrimination two dimensional 2 d image matrix based projection methods for feature extraction are widely used in many fields of computer vision and pattern recognition in this paper we propose a novel framework called sparse 2 d projections s2dp for image feature extraction different from the existing 2 d feature extraction methods s2dp iteratively learns the sparse projection matrix by using elastic net regression and singular value decomposition theoretical analysis shows that the optimal sparse subspace approximates the eigensubspace obtained by solving the corresponding generalized eigenequation with the s2dp framework many 2 d projection methods can be easily extended to sparse cases moreover when each row column of the image matrix is regarded as an independent high dimensional vector 1 d vector it is proven that the vector based eigensubspace is also approximated by the sparse subspace obtained by the same method used in this paper theoretical analysis shows that when compared with the vector based sparse projection learning methods s2dp greatly saves both computation and memory costs this property makes s2dp more tractable for real world applications experiments on well known face databases indicate the competitive performance of the proposed s2dp over some 2 d projection methods when facial expressions lighting conditions and time vary