structured multicategory support vector machines with analysis of variance decomposition the support vector machine has been a popular choice of classification method for many applications in machine learning while it often outperforms other methods in terms of classification accuracy the implicit nature of its solution renders the support vector machine less attractive in providing insights into the relationship between covariates and classes use of structured kernels can remedy the drawback borrowing the flexible model building idea of functional analysis of variance decomposition we consider multicategory support vector machines with analysis of variance kernels in this paper an additional penalty is imposed on the sum of weights of functional subspaces which encourages a sparse representation of the solution incorporation of the additional penalty enhances the interpretability of a resulting classifier with often improved accuracy the proposed method is demonstrated through simulation studies and an application to real data

algebraic relation of two meromorphic mappings on a kahler manifold having the same inverse images of hyperplanes abstract let m be a complete kahler manifold whose universal covering is biholomorphic to a ball b m r 0 0 r 0 in c m in this paper we will show that if two meromorphic mappings f and g from m into p n c have the same inverse images for 2 n 2 hyperplanes h i i 1 2 n 2 with multiplicities counted to level l 0 and satisfy the condition c then the map f g is algebraically degenerate over c where l 0 is a positive integer and is a non negative number explicitly estimated our result generalizes the previous result of h fujimoto for the case of mappings on c m to the case of mappings on a complete kahler manifold as above m and also improves his result by giving an explicit estimate for the number l 0

structured prediction dual extragradient and bregman projections we present a simple and scalable algorithm for maximum margin estimation of structured output models including an important class of markov networks and combinatorial models we formulate the estimation problem as a convex concave saddle point problem that allows us to use simple projection methods based on the dual extragradient algorithm nesterov 2003 the projection step can be solved using dynamic programming or combinatorial algorithms for min cost convex flow depending on the structure of the problem we show that this approach provides a memory efficient alternative to formulations based on reductions to a quadratic program qp we analyze the convergence of the method and present experiments on two very different structured prediction tasks 3d image segmentation and word alignment illustrating the favorable scaling properties of our algorithm

bi level path following for cross validated solution of kernel quantile regression we show how to follow the path of cross validated solutions to families of regularized optimization problems defined by a combination of a parameterized loss function and a regularization term a primary example is kernel quantile regression where the parameter of the loss function is the quantile being estimated even though the bi level optimization problem we encounter for every quantile is non convex the manner in which the optimal cross validated solution evolves with the parameter of the loss function allows tracking of this solution we prove this property construct the resulting algorithm and demonstrate it on real and artificial data this algorithm allows us to efficiently solve the whole family of bi level problems we show how it can be extended to cover other modeling problems like support vector regression and alternative in sample model selection approaches

the entire regularization path for the support vector machine the support vector machine svm is a widely used tool for classification many efficient implementations exist for fitting a two class svm model the user has to supply values for the tuning parameters the regularization cost parameter and the kernel parameters it seems a common practice is to use a default value for the cost parameter often leading to the least restrictive model in this paper we argue that the choice of the cost parameter can be critical we then derive an algorithm that can fit the entire path of svm solutions for every value of the cost parameter with essentially the same computational cost as fitting one svm model we illustrate our algorithm on some examples and use our representation to give further insight into the range of svm solutions

linear precoder optimization for mimo systems with joint power constraints this paper considers linear precoder optimization problems for multiple input multiple output mimo systems in addition to the conventionally used sum power constraint maximum eigenvalue constraint on the precoding matrix is also considered so as to account for power limitations imposed on each antenna by the linearity of its own power amplifier in practical implementations a framework employing directional derivative is developed to obtain optimal precoder designs for different criteria including maximizing the information rate and minimizing the sum of mean square error mse it turns out that power allocations in such situations are piecewise linear in sum power space the piecewise linear property allows us to generate the entire path of solution through finding out a finite number of breakpoints a homotopy type algorithm is then proposed to obtain the solution for an arbitrary sum power constraint the number of breakpoints to be determined in our exact piecewise linear solution is in fact only about two times of the number of transmit antennas so that our method is super fast and outperforms existing approximate solutions in the literature in both effectiveness and efficiency simulated experiments are performed to verify our theoretical analysis

an improved algorithm for the solution of the regularization path of support vector machine this paper describes an improved algorithm for the numerical solution to the support vector machine svm classification problem for all values of the regularization parameter c the algorithm is motivated by the work of hastie and follows the main idea of tracking the optimality conditions of the svm solution for ascending value of c it differs from hastie s approach in that the tracked path is not assumed to be 1 d instead a multidimensional feasible space for the optimality condition is used to solve the tracking problem such a treatment allows the algorithm to properly handle data sets which hastie s approach fails these data sets are characterized by the presence of linearly dependent points in the kernel space duplicate points or nearly duplicate points such data sets are quite common among many real world data especially those with nominal features other contributions of this paper include a unifying formulation of the tracking process in the form of a linear programming problem update formula for the linear programs considerations that guard against accumulation of errors resulting from the use of incremental updates and routines to speed up the algorithm the algorithm is implemented under the matlab environment and is available for download experiments with several data sets including data set having up to several thousand data points are reported

svm based feature extraction for face recognition the primary goal of linear discriminant analysis lda in face feature extraction is to find an effective subspace for identity discrimination the introduction of kernel trick has extended the lda to nonlinear decision hypersurface however there remained inherent limitations for the nonlinear lda to deal with physical applications under complex environmental factors these limitations include the use of a common covariance function among each class and the limited dimensionality inherent to the definition of the between class scatter since these problems are inherently caused by the definition of the fisher s criterion itself they may not be solvable under the conventional lda framework this paper proposes to adopt a margin based between class scatter and a regularization process to resolve the issue essentially we redesign the between class scatter matrix based on the svm margins to facilitate an effective and reliable feature extraction this is followed by a regularization of the within class scatter matrix extensive empirical experiments are performed to compare the proposed method with several other variants of the lda method using the feret ar and cmu pie databases

efficient computation and model selection for the support vector regression in this letter we derive an algorithm that computes the entire solution path of the support vector regression svr we also propose an unbiased estimate for the degrees of freedom of the svr model which allows convenient selection of the regularization parameter

an efficient implementation of an active set method for svms we propose an active set algorithm to solve the convex quadratic programming qp problem which is the core of the support vector machine svm training the underlying method is not new and is based on the extensive practice of the simplex method and its variants for convex quadratic problems however its application to large scale svm problems is new until recently the traditional active set methods were considered impractical for large svm problems by adapting the methods to the special structure of svm problems we were able to produce an efficient implementation we conduct an extensive study of the behavior of our method and its variations on svm problems we present computational results comparing our method with joachims svmlight see joachims 1999 the results show that our method has overall better performance on many svm problems it seems to have a particularly strong advantage on more difficult problems in addition this algorithm has better theoretical properties and it naturally extends to the incremental mode since the proposed method solves the standard svm formulation as does svmlight the generalization properties of these two approaches are identical and we do not discuss them in the paper

a simple method for solving the svm regularization path for semidefinite kernels the support vector machine svm remains a popular classifier for its excellent generalization performance and applicability of kernel methods however it still requires tuning of a regularization parameter c to achieve optimal performance regularization path following algorithms efficiently solve the solution at all possible values of the regularization parameter relying on the fact that the svm solution is piece wise linear in c the svmpath originally introduced by hastie et al while representing a significant theoretical contribution does not work with semidefinite kernels ong et al introduce a method improved svmpath isvmp algorithm which addresses the semidefinite kernel however singular value decomposition or qr factorizations are required and a linear programming solver is required to find the next c value at each iteration we introduce a simple implementation of the path following algorithm that automatically handles semidefinite kernels without requiring a method to detect singular matrices nor requiring specialized factorizations or an external solver we provide theoretical results showing how this method resolves issues associated with the semidefinite kernel as well as discuss in detail the potential sources of degeneracy and cycling and how cycling is resolved moreover we introduce an initialization method for unequal class sizes based upon artificial variables that work within the context of the existing path following algorithm and do not require an external solver experiments compare performance with the isvmp algorithm introduced by ong et al and show that the proposed method is competitive in terms of training time while also maintaining high accuracy

improvements to platt s smo algorithm for svm classifier design this article points out an important source of inefficiency in platt s sequential minimal optimization smo algorithm that is caused by the use of a single threshold value using clues from the kkt conditions for the dual problem two threshold parameters are employed to derive modifications of smo these modified algorithms perform significantly faster than the original smo on all benchmark data sets tried

sparse regression with output correlation for cardiac ejection fraction estimation traditional regression methods minimize the sum of errors of samples with various regularization terms such as the 1 norm and 2 norm for the diagnosis of cardiovascular diseases the cardiac ejection fraction ef represents an essential measure however existing regularization terms do not consider the output correlation the correlation between ground truth volumes and estimated volumes w r t each subject which is beneficial in estimating the cardiac ef in this paper we first propose a sparse regression with two regularization terms of the 1 norm and output correlation sroc then we propose a one dimensional solution path algorithm for quickly finding two good regulation parameters in the formulation of sroc the solution path algorithm can effectively handle singularities and infinities in the key matrix finally we conduct experiments on a clinical cardiac image dataset with 100 subjects the experimental results show that our method produces competitive and strong results for estimating the cardiac ef based on quantitative and qualitative analyses

model sparsity and brain pattern interpretation of classification models in neuroimaging interest is increasing in applying discriminative multivariate analysis techniques to the analysis of functional neuroimaging data model interpretation is of great importance in the neuroimaging context and is conventionally based on a brain map derived from the classification model in this study we focus on the relative influence of model regularization parameter choices on both the model generalization the reliability of the spatial patterns extracted from the classification model and the ability of the resulting model to identify relevant brain networks defining the underlying neural encoding of the experiment for a support vector machine logistic regression and fisher s discriminant analysis we demonstrate that selection of model regularization parameters has a strong but consistent impact on the generalizability and both the reproducibility and interpretable sparsity of the models for both 2 and 1 regularization importantly we illustrate a trade off between model spatial reproducibility and prediction accuracy we show that known parts of brain networks can be overlooked in pursuing maximization of classification accuracy alone with either 2 and or 1 regularization this supports the view that the quality of spatial patterns extracted from models cannot be assessed purely by focusing on prediction accuracy our results instead suggest that model regularization parameters must be carefully selected so that the model and its visualization enhance our ability to interpret the brain

optimization problems in statistical learning duality and optimality conditions abstract regularization methods are techniques for learning functions from given data we consider regularization problems the objective function of which consisting of a cost function and a regularization term with the aim of selecting a prediction function f with a finite representation f i 1 n c i k x i which minimizes the error of prediction here the role of the regularizer is to avoid overfitting in general these are convex optimization problems with not necessarily differentiable objective functions thus in order to provide optimality conditions for this class of problems one needs to appeal on some specific techniques from the convex analysis in this paper we provide a general approach for deriving necessary and sufficient optimality conditions for the regularized problem via the so called conjugate duality theory afterwards we employ the obtained results to the support vector machines problem and support vector regression problem formulated for different cost functions

better bounds for incremental medians in the incremental version of the well known k medianproblem the objective is to compute an incremental sequence of facility sets f email protected f email protected email protected f n where each f k contains at most k facilities we say that this incremental medians sequence is r competitive if the cost of each f k is at most r times the optimum cost of k facilities the smallest such r is called the competitive ratio of the sequence f k mettu and plaxton ramgopal r mettu c greg plaxton the online median problem in proc 41st symposium on foundations of computer science focs ieee 2000 pp 339 348 ramgopal r mettu c greg plaxton the online median problem siam journal on computing 32 3 2003 816 832 presented a polynomial time algorithm that computes an incremental sequence with competitive ratio 30 they also showed a lower bound of 2 the upper bound on the ratio was improved to 8 in guolong lin chandrashekha nagarajan rajmohan rajamaran david p williamson a general approach for incremental approximation and hierarchical clustering in proc 17th symposium on discrete algorithms soda 2006 pp 1147 1156 and marek chrobak claire kenyon john noga neal young online medians via online bidding in proc 7th latin american theoretical informatics symposium latin in lecture notes in computer science vol 3887 2006 pp 311 322 we improve both bounds in this paper we first show that no incremental sequence can have competitive ratio better than 2 01 and we give a probabilistic construction of a sequence whose competitive ratio is at most 2 42 7 656 we also propose a new approach to the problem that for instances that we refer to as equable achieves an optimal ratio of 2

general bounds for incremental maximization we propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems the defining characteristic of our framework is that the cardinality support of the solution is bounded by a value k in mathbb n that grows over time and we allow the solution to be extended one element at a time we investigate the best possible competitive ratio of such an incremental solution i e the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k we consider a large class of problems that contains many important cardinality constrained maximization problems like maximum matching knapsack and packing covering problems we provide a general 2 618 competitive incremental algorithm for this class of problems and we show that no algorithm can have competitive ratio below 2 18 in general in the second part of the paper we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step this algorithm is known to be 1 58 competitive for submodular objective functions but it has unbounded competitive ratio for the class of incremental problems mentioned above we define a relaxed submodularity condition for the objective function capturing problems like maximum weighted d dimensional matching maximum weighted b matching and a variant of the maximum flow problem we show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition our bound generalizes the tight bound of 1 58 slightly beyond sub modular functions and yields a tight bound of 2 313 for maximum weighted b matching our bound is also tight for a more general class of functions as the relevant parameter goes to infinity note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems and our bounds for the greedy algorithm carry over both

notes on submanifolds in warped products in this paper we study submanifolds in the warped product m fr where m is a riemannian manifold f is the warping function we investigate the graphs with positive gaussian curvature in m2 r the minimal graphs in m fr and the totally geodesic submanifolds in m fr

efficient revised simplex method for svm training existing active set methods reported in the literature for support vector machine svm training must contend with singularities when solving for the search direction when a singularity is encountered an infinite descent direction can be carefully chosen that avoids cycling and allows the algorithm to converge however the algorithm implementation is likely to be more complex and less computationally efficient than would otherwise be required for an algorithm that does not have to contend with the singularities we show that the revised simplex method introduced by rusin provides a guarantee of nonsingularity when solving for the search direction this method provides for a simpler and more computationally efficient implementation as it avoids the need to test for rank degeneracies and also the need to modify factorizations or solution methods based upon those rank degeneracies in our approach we take advantage of the guarantee of nonsingularity by implementing an efficient method for solving the search direction and show that our algorithm is competitive with svm qp and also that it is a particularly effective when the fraction of nonbound support vectors is large in addition we show competitive performance of the proposed algorithm against two popular svm training algorithms svmlight and libsvm

differential inequalities on complete riemannian manifolds and applications this paper treats various aspects of the asymptotic behavior of solutions of certain elliptic equations of geometric interest on complete riemannian manifolds sharp results relating the rate of volume growth of a complete riemannian manifold and the growth of its harmonic and subharmonic functions can be found in e22 together with references to related results in sect 2 of this paper we consider solutions of the more restrictive inequality au e o where div o grad and show that these must be unbounded if m n has even quadratically exponential volume growth and that u must have faster than linear growth if t

positive solutions of yamabe type equations on complete manifolds and applications we study the semilinear equation u a x u b x u 1 on a complete riemannian manifold we determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions a very general uniqueness result is also established our main results are valid without explicit curvature assumptions and appear to be new even in rm

prescribing curvature on open surfaces in this article we study the problem sometimes called the berger nirenberg problem of prescribing the curvature on a riemann surface that is on an oriented surface equipped with a conformal class of riemannian metrics

constant mean curvature spacelike hypersurfaces in lorentzian warped products and calabi bernstein type problems abstract in this paper we provide several uniqueness and non existence results for complete parabolic constant mean curvature spacelike hypersurfaces in lorentzian warped products under appropriate geometric assumptions as a consequence of this parametric study we obtain very general uniqueness and non existence results for a large family of uniformly elliptic edp s so solving the calabi bernstein problem in a wide family of spacetimes

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

on the svmpath singularity this paper proposes a novel ridge adding based approach for handling singularities that are frequently encountered in the powerful svmpath algorithm unlike the existing method that performs linear programming as an additional step to track the optimality condition path in a multidimensional feasible space our new approach provides a simpler and computationally more efficient implementation which needs no extra time consuming procedures other than introducing a random ridge term to each data point contrary to the existing ridge adding method which fails to avoid singularities as the ridge terms tend to zero our novel approach for any small random ridge terms guarantees the existence of the inverse matrix by ensuring that only one index is added into or removed from the active set the performance of the proposed algorithm in terms of both computational complexity and the ability of singularity avoidance is manifested by rigorous mathematical analyses as well as experimental results

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

piecewise linear regularized solution paths we consider the generic regularized optimization problem argmin l y x j efron hastie johnstone and tibshirani ann statist 32 2004 407 499 have shown that for the lasso that is if l is squared error loss and j 1 is the l 1 norm of the optimal coefficient path is piecewise linear that is is piecewise constant we derive a general characterization of the properties of loss l penalty j pairs which give piecewise linear coefficient paths such pairs allow for efficient generation of the full regularized coefficient paths we investigate the nature of efficient path following algorithms which arise we use our results to suggest robust versions of the lasso for regression and classification and to develop new efficient algorithms for existing problems in the literature including mammen and van de geer s locally adaptive regression splines

a new approach for uniqueness of complete maximal hypersurfaces in spatially parabolic grw spacetimes abstract complete maximal hypersurfaces in generalized robertson walker spacetimes whose fiber has a parabolic universal riemannian covering are studied under natural boundedness assumptions a spacelike hypersurface is shown to be parabolic when it is endowed with a conformal metric this fact allows us to get for complete maximal hypersurfaces several uniqueness results as an application all the entire solutions to the maximal hypersurface equation under certain constrains are obtained solving new calabi bernstein problems

determination of global minima of some common validation functions in support vector machine tuning of the regularization parameter c is a well known process in the implementation of a support vector machine svm classifier such a tuning process uses an appropriate validation function whose value evaluated over a validation set has to be optimized for the determination of the optimal c unfortunately most common validation functions are not smooth functions of c this brief presents a method for obtaining the global optimal solution of these non smooth validation functions the method is guaranteed to find the global optimum and relies on the regularization solution path of svm over a range of c values when the solution path is available the computation needed is minimal

a non integrated defect relation and the uniqueness theorem for meromorphic maps of a complete kahler manifold into pn c abstract in this paper a non integrated defect relation is established for meromorphic maps of an m dimensional complete kahler manifold m into p n c intersecting q hypersurfaces d 1 d q in p n c located in k subgeneral position where d 1 d q are said to be in k subgeneral position if at most k k n of these hypersurfaces meet at one point in p n c the related uniqueness theorem is also given

nonlinear regularization path for quadratic loss support vector machines regularization path algorithms have been proposed to deal with model selection problem in several machine learning approaches these algorithms allow computation of the entire path of solutions for every value of regularization parameter using the fact that their solution paths have piecewise linear form in this paper we extend the applicability of regularization path algorithm to a class of learning machines that have quadratic loss and quadratic penalty term this class contains several important learning machines such as squared hinge loss support vector machine svm and modified huber loss svm we first show that the solution paths of this class of learning machines have piecewise nonlinear form and piecewise segments between two breakpoints are characterized by a class of rational functions then we develop an algorithm that can efficiently follow the piecewise nonlinear path by solving these rational equations to solve these rational equations we use rational approximation technique with quadratic convergence rate and thus our algorithm can follow the nonlinear path much more precisely than existing approaches such as predictor corrector type nonlinear path approximation we show the algorithm performance on some artificial and real data sets

large margin semi supervised learning in classification semi supervised learning occurs when a large amount of unlabeled data is available with only a small number of labeled data in such a situation how to enhance predictability of classification through unlabeled data is the focus in this article we introduce a novel large margin semi supervised learning methodology using grouping information from unlabeled data together with the concept of margins in a form of regularization controlling the interplay between labeled and unlabeled data based on this methodology we develop two specific machines involving support vector machines and learning denoted as ssvm and spsi through difference convex programming in addition we estimate the generalization error using both labeled and unlabeled data for tuning regularizers finally our theoretical and numerical analyses indicate that the proposed methodology achieves the desired objective of delivering high performance in generalization particularly against some strong performers

efficient svm training using low rank kernel representations svm training is a convex optimization problem which scales with the training set size rather than the feature space dimension while this is usually considered to be a desired quality in large scale problems it may cause training to be impractical the common techniques to handle this difficulty basically build a solution by solving a sequence of small scale subproblems our current effort is concentrated on the rank of the kernel matrix as a source for further enhancement of the training procedure we first show that for a low rank kernel matrix it is possible to design a better interior point method ipm in terms of storage requirements as well as computational complexity we then suggest an efficient use of a known factorization technique to approximate a given kernel matrix by a low rank matrix which in turn will be used to feed the optimizer finally we derive an upper bound on the change in the objective function value based on the approximation error and the number of active constraints support vectors this bound is general in the sense that it holds regardless of the approximation method

a robust regularization path algorithm for nu support vector classification the nu support vector classification has the advantage of using a regularization parameter nu to control the number of support vectors and margin errors recently a regularization path algorithm for nu support vector classification nu svcpath suffers exceptions and singularities in some special cases in this brief we first present a new equivalent dual formulation for nu svc and then propose a robust nu svcpath based on lower upper decomposition with partial pivoting theoretical analysis and experimental results verify that our proposed robust regularization path algorithm can avoid the exceptions completely handle the singularities in the key matrix and fit the entire solution path in a finite number of steps experimental results also show that our proposed algorithm fits the entire solution path with fewer steps and less running time than original one does

learning the kernel function via regularization we study the problem of finding an optimal kernel from a prescribed convex set of kernels k for learning a real valued function by regularization we establish for a wide variety of regularization functionals that this leads to a convex optimization problem and for square loss regularization we characterize the solution of this problem we show that although k may be an uncountable set the optimal kernel is always obtained as a convex combination of at most m 2 basic kernels where m is the number of data examples in particular our results apply to learning the optimal radial kernel or the optimal dot product kernel

non integrated defect of meromorphic maps on kahler manifolds the purpose of this article is twofold the first is to establish a truncated non integrated defect relation for meromorphic mappings from a complete kahler manifold quotien of a ball into a projective variety intersecting hypersurfaces in subgeneral position we also apply it to the gauss mapping from a closed regular submanifold of mathbb c m the second aim is to establish an above type theorem with truncation level 1 for differentiably nondegenerate meromorphic mappings

properties of support vector machines support vector machines svms perform pattern recognition between two point classes by finding a decision surface determined by certain points of the training set termed support vectors sv this surface which in some feature space of possibly infinite dimension can be regarded as a hyperplane is obtained from the solution of a problem of quadratic programming that depends on a regularization parameter in this paper we study some mathematical properties of support vectors and show that the decision surface can be written as the sum of two orthogonal terms the first depending only on the em margin vectors which are svs lying on the margin the second proportional to the regularization parameter for almost all values of the parameter this enables us to predict how the decision surface varies for small parameter changes in the special but important case of feature space of finite dimension m we also show that there are at most m 1 margin vectors and observe that m 1 svs are usually sufficient to fully determine the decision surface for relatively small m this latter result leads to a consistent reduction of the sv number

a general approach for incremental approximation and hierarchical clustering we present a general framework and algorithmic approach for incremental approximation algorithms the framework handles cardinality constrained minimization problems such as the k median and k mst problems given some notion of ordering on solutions of different cardinalities k we give solutions for all values of k such that the solutions respect the ordering and such that for any k our solution is close in value to the value of an optimal solution of cardinality k for instance for the k median problem the notion of ordering is set inclusion and our incremental algorithm produces solutions such that for any k and k k k our solution of size k is a subset of our solution of size k we show that our framework applies to this incremental version of the k median problem introduced by mettu and plaxton r r mettu and c g plaxton siam j comput 32 2003 pp 816 832 and incremental versions of the k mst problem k vertex cover problem k set cover problem as well as the uncapacitated facility location problem which is not cardinality constrained for these problems we get either new incremental algorithms or improvements over what was previously known we also show that the framework applies to hierarchical clustering problems in particular we give an improved algorithm for a hierarchical version of the k median problem introduced by plaxton c g plaxton j comput system sci 72 2006 pp 425 443

a new solution path algorithm in support vector regression in this paper regularization path algorithms were proposed as a novel approach to the model selection problem by exploring the path of possibly all solutions with respect to some regularization hyperparameter in an efficient way this approach was later extended to a support vector regression svr model called epsiv svr however the method requires that the error parameter epsiv be set a priori this is only possible if the desired accuracy of the approximation can be specified in advance in this paper we analyze the solution space for epsiv svr and propose a new solution path algorithm called epsiv path algorithm which traces the solution path with respect to the hyperparameter epsiv rather than lambda although both two solution path algorithms possess the desirable piecewise linearity property our epsiv path algorithm overcomes some limitations of the original lambda path algorithm and has more advantages it is thus more appealing for practical use

non integrated defect relation for meromorphic maps from a kahler manifold intersecting hypersurfaces in subgeneral of pn c abstract in this article we establish a truncated non integrated defect relation for meromorphic mappings from an m dimensional complete kahler manifold into p n c intersecting q hypersurfaces q 1 q q in k subgeneral position of degree d i i e the intersection of any k 1 hypersurfaces is emptyset we will prove that i 1 q f u 1 q i k n 1 n 1 u u 1 d where u is explicitly estimated and d is the least common multiple of d i s our result generalizes and improves previous results in the last part of this paper we will apply this result to study the distribution of the gauss map of minimal surfaces

a consistent information criterion for support vector machines in diverging model spaces information criteria have been popularly used in model selection and proved to possess nice theoretical properties for classification claeskens et al 2008 proposed support vector machine information criterion for feature selection and provided encouraging numerical evidence yet no theoretical justification was given there this work aims to fill the gap and to provide some theoretical justifications for support vector machine information criterion in both fixed and diverging model spaces we first derive a uniform convergence rate for the support vector machine solution and then show that a modification of the support vector machine information criterion achieves model selection consistency even when the number of features diverges at an exponential rate of the sample size this consistency result can be further applied to selecting the optimal tuning parameter for various penalized support vector machine methods finite sample performance of the proposed information criterion is investigated using monte carlo studies and one real world gene selection problem

core vector machines fast svm training on very large data sets standard svm training has o m3 time and o m2 space complexities where m is the training set size it is thus computationally infeasible on very large data sets by observing that practical svm implementations only approximate the optimal solution by an iterative strategy we scale up kernel methods by exploiting such approximateness in this paper we first show that many kernel methods can be equivalently formulated as minimum enclosing ball meb problems in computational geometry then by adopting an efficient approximate meb algorithm we obtain provably approximately optimal solutions with the idea of core sets our proposed core vector machine cvm algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m experiments on large toy and real world data sets demonstrate that the cvm is as accurate as existing svm implementations but is much faster and can handle much larger data sets than existing scale up methods for example cvm with the gaussian kernel produces superior results on the kddcup 99 intrusion detection data which has about five million training patterns in only 1 4 seconds on a 3 2ghz pentium 4 pc