a combinatorial approach to level of repair analysis abstract this paper presents an approach to optimise level of repair decisions taking into account submodular properties of standard life cycle cost functions which include fixed and variable costs it proposes an integer programming formulation to solve level of repair problems for multi echelon multi indenture level systems the method converges quickly to the optimum solution relying on heuristics to obtain tight bounds for a subsequent branch and bound procedure a software package called level of repair optimisation model lorom was developed to implement the branch and bound method that does not rely on linear programming relaxations this approach is rather generic and can be applied to a wide class of problems with convex total cost functions such as plant location problems or transportation problems with fixed costs

fast iterative kernel principal component analysis we develop gain adaptation methods that improve convergence of the kernel hebbian algorithm kha for iterative kernel pca kim et al 2005 kha has a scalar gain parameter which is either held constant or decreased according to a predetermined annealing schedule leading to slow convergence we accelerate it by incorporating the reciprocal of the current estimated eigenvalues as part of a gain vector an additional normalization term then allows us to eliminate a tuning parameter in the annealing schedule finally we derive and apply stochastic meta descent smd gain vector adaptation schraudolph 1999 2002 in reproducing kernel hilbert space to further speed up convergence experimental results on kernel pca and spectral clustering of usps digits motion capture and image denoising and image super resolution tasks confirm that our methods converge substantially faster than conventional kha to demonstrate scalability we perform kernel pca on the entire mnist data set

large solution to nonlinear elliptic equation with nonlinear gradient terms abstract the aim of this paper is to study the qualitative behavior of large solutions to the following problem u a x u q b x f u x u x x here a x c is a positive function with 0 1 b x c is a non negative function and may be singular near the boundary or vanish on the boundary and the nonlinear term f is a varying function whose variation at infinity is not regular we focus our investigation on the existence and asymptotic behavior close to the boundary of large solutions by karamata regular variation theory and the method of upper and lower solution the main results of this paper emphasize the central role played by the gradient term u q and the weight functions a x and b x

economic lot sizing an o n log n algorithm that runs in linear time in the wagner whitin case we consider the n period economic lot sizing problem where the cost coefficients are not restricted in sign in their seminal paper h m wagner and t m whitin proposed an o n2 algorithm for the special case of this problem where the marginal production costs are equal in all periods and the unit holding costs are nonnegative it is well known that their approach can also be used to solve the general problem without affecting the complexity of the algorithm in this paper we present an algorithm to solve the economic lot sizing problem in o n log n time and we show how the wagner whitin case can even be solved in linear time our algorithm can easily be explained by a geometrical interpretation and the time bounds are obtained without the use of any complicated data structure furthermore we show how wagner and whitin s and our algorithm are related to algorithms that solve the dual of the simple plant location formulation of the economic lot sizing problem

iterative kernel principal component analysis for image modeling in recent years kernel principal component analysis kpca has been suggested for various image processing tasks requiring an image model such as e g denoising or compression the original form of kpca however can be only applied to strongly restricted image classes due to the limited number of training examples that can be processed we therefore propose a new iterative method for performing kpca the kernel hebbian algorithm which iteratively estimates the kernel principal components with only linear order memory complexity in our experiments we compute models for complex image classes such as faces and natural images which require a large number of training examples the resulting image models are tested in single frame super resolution and denoising applications the kpca model is not specifically tailored to these tasks in fact the same model can be used in super resolution with variable input resolution or denoising with unknown noise characteristics in spite of this both super resolution and denoising performance are comparable to existing methods

specially structured uncapacitated facility location problems this paper considers a specially structured uncapacitated facility location problem we show that several problems including certain tool selection problems substitutable inventory problems supplier sourcing problems discrete lot sizing problems and capacity expansion problems can be formulated as instances of the problem we also show that the problem with m facilities and n customers can be solved in o mn as a shortest path problem on a directed graph

keller osserman type conditions for some elliptic problems with gradient terms in this paper we consider the elliptic boundary blow up problems u g u f u in u on where is a smooth bounded domain and the functions f and g are increasing and continuous our main concern will be to prove both existence and nonexistence of nonnegative solutions depending on new integral conditions of keller osserman type involving f and g we show in particular that the problem with a minus sign may have solutions inclusive for some functions g with slightly superquadratic growth at infinity that is somehow not expected we also obtain uniqueness of nonnegative solutions in some cases

the geometric median on riemannian manifolds with application to robust atlas estimation one of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images the study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space which map one anatomy to another it is now well established that representing the geometry of shapes or images in euclidian spaces undermines our ability to represent natural variability in populations in our previous work we have extended classical statistical analysis techniques such as averaging principal components analysis and regression to riemannian manifolds which are more appropriate representations for describing anatomical variability in this paper we extend the notion of robust estimation a well established and powerful tool in traditional statistical analysis of euclidian data to manifold valued representations of anatomical variability in particular we extend the geometric median a classic robust estimator of centrality for data in euclidean spaces we formulate the geometric median of data on a riemannian manifold as the minimizer of the sum of geodesic distances to the data points we prove existence and uniqueness of the geometric median on manifolds with non positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds generalizing the weiszfeld procedure for finding the geometric median of euclidean data we present an algorithm for computing the geometric median on an arbitrary manifold we show that this algorithm converges to the unique solution when it exists in this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images using this approach we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images

a dual based procedure for stochastic facility location in this paper we study how the uncapacitated facility location problem is transformed into a two stage stochastic program with recourse when uncertainty on demand selling prices production and transportation costs are introduced we then present a dual based procedure and indicate how the dual descent and primal dual adjustment procedures proposed by d erlenkotter 1978 in the static case can be made monotonically improving in the stochastic case results of computer experiments are reported

probability convexity and harmonic maps with small image i uniqueness and fine existence this paper uses probability theory to provide an approach to the dirichlet problem for harmonic maps the probabilistic tools used are those of manifold valued brownian motion and martingales probabilistic proofs are given of uniqueness continuous dependence on data and existence of generalized solutions finely harmonic maps for target domains which have convex geometry this class of domain includes all regular geodesic balls links are made with new results in other fields the martingale dirichlet problem riemannian centres of mass

equivalent instances of the simple plant location problem in this paper we deal with a pseudo boolean representation of the simple plant location problem we define instances of this problem that are equivalent in the sense that each feasible solution has the same goal function value in all such instances we further define a collection of polytopes whose union describes the set of instances equivalent to a given instance we use the concept of equivalence to develop a method by which we can extend the set of instances that we can solve using our knowledge of polynomially solvable special cases

boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms abstract in this paper we study the boundary behavior of solutions to boundary blow up elliptic problems u u q b x f u x u where is a bounded domain with smooth boundary in r n q 0 b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary and f is rapidly varying or normalized regularly varying at infinity

online kernel principal component analysis a reduced order model kernel principal component analysis kernel pca is an elegant nonlinear extension of one of the most used data analysis and dimensionality reduction techniques the principal component analysis in this paper we propose an online algorithm for kernel pca to this end we examine a kernel based version of oja s rule initially put forward to extract a linear principal axe as with most kernel based machines the model order equals the number of available observations to provide an online scheme we propose to control the model order we discuss theoretical results such as an upper bound on the error of approximating the principal functions with the reduced order model we derive a recursive algorithm to discover the first principal axis and extend it to multiple axes experimental results demonstrate the effectiveness of the proposed approach both on synthetic data set and on images of handwritten digits with comparison to classical kernel pca and iterative kernel pca

recursive robust pca or recursive sparse recovery in large but structured noise this paper studies the recursive robust principal components analysis problem if the outlier is the signal of interest this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors st in the presence of large but structured noise lt the structure that we assume on lt is that lt is dense and lies in a low dimensional subspace that is either fixed or changes slowly enough a key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background lt from moving foreground objects st on the fly to solve the above problem in recent work we introduced a novel solution called recursive projected cs reprocs in this paper we develop a simple modification of the original reprocs idea and analyze it this modification assumes knowledge of a subspace change model on the lt s under mild assumptions and a denseness assumption on the unestimated part of the subspace of lt at various times we show that with high probability the proposed approach can exactly recover the support set of st at all times and the reconstruction errors of both st and lt are upper bounded by a time invariant and small value in simulation experiments we observe that the last assumption holds as long as there is some support change of st every few frames

solving covering problems and the uncapacitated plant location problem on trees abstract given a tree network on n vertices a neighborhood subtree is defined as the set of all points on the tree within a certain radius of a given point called the center it is shown that for any two neighborhood subtrees containing the same endpoint of a longest path in the tree one is contained in the other this result is then used to obtain o n2 algorithms for the minimum cost covering problem and the minimum cost operating problem as well as an o n3 algorithm for the uncapacitated plant location problem on the tree

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

the simple plant location problem survey and synthesis abstract with emphasis on the simple plant location problem splp we consider an important family of discrete deterministic single criterion np hard and widely applicable optimization problems the introductory discussion on problem formulation aspects is followed by the establishment of relationships between splp and set packing set covering and set partitioning problems which all are among those structures in integer programming having the most wide spread applications an extensive discourse on solution properties and computational techniques spanning from early heuristics to the presumably most novel exact methods is then provided other subjects of concern include a subfamily of splp s solvable in polynomial time analyses of approximate algorithms transformability of p center and p median to splp and structural properties of the splp polytope along the way we attempt to synthesize these findings and relate them to other areas of integer programming

boundary blow up elliptic problems with nonlinear gradient terms and singular weights by karamata regular variation theory a perturbation method and construction of comparison functions we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x g u u 0 in u where is a bounded domain with smooth boundary in n q 0 g c 1 0 is increasing on 0 g 0 0 g is regularly varying at infinity with positive index and b is non negative in and is singular on the boundary

boundary blow up rates of large solutions for elliptic equations with convection terms abstract by karamata regular variation theory a perturbation method and constructing comparison functions we show the exact asymptotic behavior of large solutions to the semilinear elliptic equations with convection terms u u q b x f u x u x x where is a smooth bounded domain in r n the weight function b x is a non negative continuous function in the domain which may be vanishing on the boundary or be singular on the boundary f u c 2 0 is increasing on 0 satisfying the keller osserman condition and regularly varying at infinity with index 1

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed

intrinsic grassmann averages for online linear robust and nonlinear subspace learning principal component analysis pca and kernel principal component analysis kpca are fundamental methods in machine learning for dimensionality reduction the former is a technique for finding this approximation in finite dimensions and the latter is often in an infinite dimensional reproducing kernel hilbert space rkhs in this paper we present a geometric framework for computing the principal linear subspaces in both situations as well as for the robust pca case that amounts to computing the intrinsic average on the space of all subspaces the grassmann manifold points on this manifold are defined as the subspaces spanned by k tuples of observations the intrinsic grassmann average of these subspaces are shown to coincide with the principal components of the observations when they are drawn from a gaussian distribution we show similar results in the rkhs case and provide an efficient algorithm for computing the projection onto the this average subspace the result is a method akin to kpca which is substantially faster further we present a novel online version of the kpca using our geometric framework competitive performance of all our algorithms are demonstrated on a variety of real and synthetic data sets