on adjunctions between lim sl top and sl lim abstract consider l 1 be a commutative strictly two sided quantale with the underlying lattice l being meet continuous two adjunctions one is between limit spaces and stratified l limit spaces and the other is between stratified l limit spaces and stratified l topological spaces are established the first adjunction can be viewed as an extension of lowen s adjunction between the category of topological spaces and stratified 0 1 topological spaces the second is an extension of an adjunction between limit spaces and stratified l topological spaces established in u hohle and t kubiak hohle kubiak semigroup forum 2007

lattice valued spaces fuzzy convergence abstract an alternative set of axioms is given for the study of lattice valued convergence spaces these axioms extend the notion of a probabilistic convergence space in the special case when the lattice is l 0 1 the category described here is shown to be topological cartesian closed and extensional

working set selection using functional gain for ls svm the efficiency of sequential minimal optimization smo depends strongly on the working set selection this letter shows how the improvement of smo in each iteration named the functional gain fg is used to select the working set for least squares support vector machine ls svm we prove the convergence of the proposed method and give some theoretical support for its performance empirical comparisons demonstrate that our method is superior to the maximum violating pair mvp working set selection

a fast iterative nearest point algorithm for support vector machine classifier design in this paper we give a new fast iterative algorithm for support vector machine svm classifier design the basic problem treated is one that does not allow classification violations the problem is converted to a problem of computing the nearest point between two convex polytopes the suitability of two classical nearest point algorithms due to gilbert and mitchell et al is studied ideas from both these algorithms are combined and modified to derive our fast algorithm for problems which require classification violations to be allowed the violations are quadratically penalized and an idea due to cortes and vapnik and friess is used to convert it to a problem in which there are no classification violations comparative computational evaluation of our algorithm against powerful svm methods such as platt s sequential minimal optimization shows that our algorithm is very competitive

rigorous proof of termination of smo algorithm for support vector machines sequential minimal optimization smo algorithm is one of the simplest decomposition methods for learning of support vector machines svms keerthi and gilbert have recently studied the convergence property of smo algorithm and given a proof that smo algorithm always stops within a finite number of iterations in this letter we point out the incompleteness of their proof and give a more rigorous proof

estimation of interclass correlations in familial data summary the maximum likelihood estimates of the parent child interclass correlation and other parameters from familial data when the families have unequal numbers of offspring is given also given are two alternative sets of estimates which are easier to compute than the maximum likelihood estimates using these estimates a test for the interclass correlation to be zero requires only the ordinary student s t distribution similarly other tests use only known distributions in the analysis of familial data the degree of mother sib resemblance is measured by the interclass correlation the correlation between two different types of family member several estimates of this correlation have been proposed in the literature see rosner donner hennekens 1977 for details and their monte carlo comparisons of mean squared errors in particular they gave maximum likelihood estimates when the sib sizes are equal however when the sib sizes are not equal rosner 1979 proposed an algorithm for finding the maximum likelihood estimates which involves iterative maximization of an implicit function of two parameters this algorithm is difficult to implement and may not even converge for some sets of data recently mak ng 1981 used the linear model approach of kempthorne tandan 1953 to obtain the maximum likelihood estimates this reduces the problem to iterative maximization of a function of one parameter however nothing is known about the convergence of the procedure another iterative method of finding the maximum likelihood estimates has been given by smith 1980 in this paper an alternative approach is given which requires solving only one equation also two alternative sets of estimates are given these estimates are easier to compute than the maximum likelihood estimates using these estimates a test for the interclass correlation to be zero is given it requires only the ordinary student s t distribution similarly other tests use only known results

on stratified l convergence spaces in the classical theory of convergence spaces fischer s diagonal axiom ensures that a generalized convergence space is topological in this note we present a lattice valued fischer diagonal axiom and show that this axiom leads to the conclusion that a stratified l convergence space is l topological

pretopological and topological lattice valued convergence spaces we show that the classical axiom which characterizes pretopological convergence spaces splits into two axioms in the general heyting algebra valued case furthermore we present a generalization of kowalski s diagonal condition to the lattice valued case

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

diagonal conditions for lattice valued uniform convergence spaces we define diagonal conditions for lattice valued uniform convergence spaces and show that these conditions are preserved under initial constructions further the forgetful functor from the category of stratified lattice valued uniform convergence spaces into the category of stratified lattice valued limit spaces maps these conditions to corresponding conditions we especially define uniform regularity and characterize this condition by certain closures of lattice valued filters as an application we generalize an extension theorem for uniformly continuous mappings

combined maximum likelihood estimates for the equicorrelation coefficient combined maximum likelihood estimates for equicorrelation covariance matrices are considered the case of a common equicorrelation rho and possibly different standard deviations sigma 1 sigma k among k experimental groups is examined first and the estimation of rho sigma 1 sigma k is discussed second under the assumption of a common standard deviation and possibly different equicorrelations the estimation of rho 1 rho k sigma is considered in each case maximum likelihood solutions and corresponding large sample variances are presented

the lower and upper p topological p regular modifications for lattice valued convergence spaces in this paper the notions of lower and upper topological regular modifications with respect to a stratified l generalized convergence structure and those with respect to a stratified l strong generalized convergence structure are defined and studied the basic properties and characterizations are presented in particular it is proved that the lower modifications behave reasonably well relative to final structures whereas the upper modifications exhibit comparable behavior relative to initial structures

diagonal conditions and continuous extension theorem abstract in this paper we obtain a generalization of kowalsky diagonal condition and that of fischer diagonal condition respectively namely kowalsky diagonal condition and fischer diagonal condition we show that our fischer diagonal condition assures a complete mv algebra valued convergence space proposed in this paper is strong l topological and kowalsky diagonal condition assures a principle or pretopological complete mv algebra valued convergence space is strong l topological also as applications we give a dual form of our fischer diagonal condition and obtain a concept of regular convergence space in addition we present an extension theorem for continuous maps from a dense subspace to a regular convergence space to show that our diagonal conditions works indeed

linear time algorithms for linear programming in r 3 and related problems linear time algorithms for linear programming in r 2 and r 3 are presented the methods used are applicable for other graphic and geometric problems as well as quadratic programming for exam

maximum likelihood model selection for 1 norm soft margin svms with multiple parameters adapting the hyperparameters of support vector machines svms is a challenging model selection problem especially when flexible kernels are to be adapted and data are scarce we present a coherent framework for regularized model selection of 1 norm soft margin svms for binary classification it is proposed to use gradient ascent on a likelihood function of the hyperparameters the likelihood function is based on logistic regression for robustly estimating the class conditional probabilities and can be computed efficiently overfitting is an important issue in svm model selection and can be addressed in our framework by incorporating suitable prior distributions over the hyperparameters we show empirically that gradient based optimization of the likelihood function is able to adapt multiple kernel parameters and leads to better models than four concurrent state of the art methods

asymptotic convergence of an smo algorithm without any assumptions the asymptotic convergence of c j lin 2001 can be applied to a modified smo sequential minimal optimization algorithm by s s keerthi et al 2001 with some assumptions the author shows that for this algorithm those assumptions are not necessary

p topologicalness and p regularity for lattice valued convergence spaces in this paper p topologicalness and p regularity for jager s stratified l generalized convergence spaces and those for boustique et al s stratified l convergence spaces are studied through two different methods one method generalizes the well known fischer s dual diagonal condition the other method uses the neighborhood closure of stratified l filter in addition the p topologicalness and p regularity w r t initial constructions and final constructions are also discussed

lagrangian support vector machines an implicit lagrangian for the dual of a simple reformulation of the standard quadratic program of a linear support vector machine is proposed this leads to the minimization of an unconstrained differentiable convex function in a space of dimensionality equal to the number of classified points this problem is solvable by an extremely simple linearly convergent lagrangian support vector machine lsvm algorithm lsvm requires the inversion at the outset of a single matrix of the order of the much smaller dimensionality of the original input space plus one the full algorithm is given in this paper in 11 lines of matlab code without any special optimization tools such as linear or quadratic programming solvers this lsvm code can be used as is to solve classification problems with millions of points for example 2 million points in 10 dimensional input space were classified by a linear surface in 82 minutes on a pentium iii 500 mhz notebook with 384 megabytes of memory and additional swap space and in 7 minutes on a 250 mhz ultrasparc ii processor with 2 gigabytes of memory other standard classification test problems were also solved nonlinear kernel classification can also be solved by lsvm although it does not scale up to very large problems it can handle any positive semidefinite kernel and is guaranteed to converge a short matlab code is also given for nonlinear kernels and tested on a number of problems

maximum gain working set selection for svms support vector machines are trained by solving constrained quadratic optimization problems this is usually done with an iterative decomposition algorithm operating on a small working set of variables in every iteration the training time strongly depends on the selection of these variables we propose the maximum gain working set selection algorithm for large scale quadratic programming it is based on the idea to greedily maximize the progress in each single iteration the algorithm takes second order information from cached kernel matrix entries into account we prove the convergence to an optimal solution of a variant termed hybrid maximum gain working set selection this method is empirically compared to the prominent most violating pair selection and the latest algorithm using second order information for large training sets our new selection scheme is significantly faster

optimization approaches to supervised classification abstract the supervised classification problem one of the oldest and most recurrent problems in applied data analysis has always been analyzed from many different perspectives when the emphasis is placed on its overall goal of developing classification rules with minimal classification cost supervised classification can be understood as an optimization problem on the other hand when the focus is in modeling the uncertainty involved in the classification of future unknown entities it can be formulated as a statistical problem other perspectives that pay particular attention to pattern recognition and machine learning aspects of supervised classification have also a long history that has lead to influential insights and different methodologies in this review two approaches to supervised classification strongly related to optimization theory will be discussed and compared in particular we will review methodologies based on mathematical programming models that optimize observable criteria linked to the true objective of misclassification error or cost minimization and approaches derived from the minimization of known bounds on the true misclassification error the former approach is known as the mathematical programming approach to supervised classification while the latter is in the origin of the well known classification support vector machines throughout the review two group as well as general multi group problems will be considered and the review will conclude with a discussion of the most promising research directions in this area

stratified lmn convergence tower groups and their stratified lmn uniform convergence tower structures abstract if l and m are frames and n is a quantale then using stratification mappings between frames we introduce a category of stratified lmn convergence tower groups a topological category we then prove that every stratified lmn limit tower group induces a stratified lmn uniform convergence tower space also we introduce a category of stratified lmn cauchy tower groups and show that the category of strongly normal stratified lmn limit tower groups is isomorphic to the category of stratified lmn cauchy tower groups we provide various examples in support of our theories so far developed herein the text

simple proof of convergence of the smo algorithm for different svm variants in this brief we give a new proof of the asymptotic convergence of the sequential minimum optimization smo algorithm for both the most violating pair and second order rules to select the pair of coefficients to be updated the proof is more self contained shorter and simpler than previous ones and has a different flavor partially building upon gilbert s original convergence proof of its algorithm to solve the minimum norm problem for convex hulls it is valid for both support vector classification svc and support vector regression which are formulated under a general problem that encompasses them moreover this general problem can be further extended to also cover other support vector machines svm related problems such as svc or one class svms while the convergence proof of the slight variant of smo needed for them remains basically unchanged

lattice valued convergence associated with cns spaces abstract in this paper in the context of meet continuous quantale we present a notion of conical topological lattice valued convergence space which provides the theory of lattice valued convergence associated with such conical neighborhood spaces

a note on the analysis of familial data because families have varying numbers of offspring the maximum likelihood estimators of the interclass or parent offspring and intraclass or offspring offspring correlations are difficult to compute srivastava 1984 for this reason alternative estimators have been proposed which are not necessarily asymptotically efficient however maximum likelihood estimators of the inter and intraclass correlations in families of constant size can be obtained in closed form

global convergence of decomposition learning methods for support vector machines decomposition methods are well known techniques for solving quadratic programming qp problems arising in support vector machines svms in each iteration of a decomposition method a small number of variables are selected and a qp problem with only the selected variables is solved since large matrix computations are not required decomposition methods are applicable to large qp problems in this paper we will make a rigorous analysis of the global convergence of general decomposition methods for svms we first introduce a relaxed version of the optimality condition for the qp problems and then prove that a decomposition method reaches a solution satisfying this relaxed optimality condition within a finite number of iterations under a very mild condition on how to select variables

estimation of the interclass correlation coefficient summary a noniterative method for estimating the interclass correlation coefficient is derived from the technique of weighted sums of squares the asymptotic variance of this estimator is derived under the assumption of normality through extensive monte carlo simulations the sample variance of this estimator is found not to differ greatly from that of the maximum likelihood estimator an important problem in genetics is the estimation of the degree of resemblance between a mother and her children as measured by the interclass correlation coefficient because parents produce differing numbers of offspring during their reproductive period the maximum likelihood estimation of the interclass correlation presents computational difficulties requiring an iterative solution therefore a number of noniterative estimators have been proposed in the literature these estimators will be shown to be associated with special cases of the generalized estimator derived from the weighted sums of squares of measurements on parents and offspring because our purpose is to provide a comprehensive approach to the noniterative point and interval estimation problems for the interclass correlation coefficient the asymptotic variance of the proposed estimator is derived under the assumption of normality moreover the sample variances of two special cases of the proposed estimator are compared to that of the maximum likelihood estimator through monte carlo simulations by a theoretical argument it is shown that the estimator proposed by srivastava 1984 has a uniformly smaller asymptotic variance than the ensemble estimator proposed by rosner donner hennekens 1977 smith 1957 used the weighted sums of squares approach for the estimation problem concerning the intraclass correlation coefficient which measures the degree to which siblings resemble one another our results together with those of smith 1957 provide a unified approach to estimation problems concerning familial correlations among a parent and offspring

on many valued stratified l fuzzy convergence spaces based on complete residuated lattices the relations between stratified l fuzzy convergence spaces and enriched l fuzzy topological spaces are studied it is shown that as a reflective subcategory the category of enriched l fuzzy topological spaces can embedded in the category of stratified l fuzzy convergence spaces also the specialization l preorder of stratified l fuzzy convergence spaces is investigated

relationships between l ordered convergence structures and strong l topologies by making use of the intrinsic fuzzy inclusion orders on the fuzzy power set the relationships between l ordered convergence spaces and strong l topological spaces are researched where l is a commutative unital quantale it is shown that the category of strong l topological spaces can be embedded in the category of l ordered convergence spaces as a reflective subcategory as a result we observe that there is a galois correspondence between the category of l ordered convergence spaces and the category of strong l topological spaces further it is proved that the class of all strong l topological l ordered convergence spaces precisely is the class of all strong l topological spaces and the class of spaces with non idempotent l ordered interior operators is characterized as a subclass of the class of l ordered convergence spaces

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

lattice valued convergence spaces extending the lattice context we define a category of stratified l generalized convergence spaces for the case where the lattice is an enriched cl premonoid we then investigate some of its categorical properties and those of its subcategories in particular the stratified l principal convergence spaces and the stratified l topological convergence spaces for some results we need to introduce a new condition on the lattice which is always true in the case where the lattice is a frame but not always true in the more general case as examples where we may apply the more general lattice context we examine the stratified l topological spaces and probabilistic limit spaces we show that the category of stratified l topological spaces is a reflective subcategory of our category and that the category of probabilistic limit spaces under a t norm is both a reflective and a coreflective subcategory of our category if we choose the lattice context appropriately

the relationship between various filter notions on agl monoid abstract the notion of a generalised filter is extended to the setting of a gl monoid it is shown that there exists a one to one correspondence between the collection of generalised filters on a set x and the collection of strongly stratified l filters on x specialising to the case where l is the closed unit interval 0 c viewed as a heyting algebra we show that any strongly stratified 0 c filter on x can be uniquely identified with a saturated filter on i x with characteristic value c in this way the notion of a generalised filter unifies various filter notions in particular necessity measures and finitely additive probability measures are specific examples of generalised filters

stratified l ordered convergence structures in this paper a new kind of lattice valued convergence structures on a universal set called stratified l ordered convergence structures are presented by modifying the axiom for stratified l generalized convergence structures in the fuzzy setting so as to make use of the intrinsic fuzzy inclusion order on the fuzzy power set the category of stratified l ordered convergence spaces described here is shown to be a reflective full subcategory in the category of stratified l generalized convergence spaces and hence it is topological and cartesian closed as preparation a further investigation of stratified l filters is presented from the viewpoint that latticed valued filters should be compatible with the intrinsic fuzzy inclusion order on the fuzzy power set

modifications lattice valued structures several important topological concepts such as regularity local compactness and local boundedness are investigated in the category of lattice valued convergence spaces these definitions coincide with the usual ones in familiar categories

on stratified l convergence spaces pretopological axioms and diagonal axioms this paper focuses on pretopological and diagonal axioms for stratified l convergence spaces recently defined by the authors in 2011 this notion is slightly stronger than jager s stratified l generalized convergence spaces in particular one pretopological axiom is stated without resorting to the notion of neighborhood l filters some generalizations of kowalsky s and fischer s diagonal conditions are presented the relationship between these axioms are investigated

generalized core vector machines kernel methods such as the support vector machine svm are often formulated as quadratic programming qp problems however given m training patterns a naive implementation of the qp solver takes o m 3 training time and at least o m2 space hence scaling up these qps is a major stumbling block in applying kernel methods on very large data sets and a replacement of the naive method for finding the qp solutions is highly desirable recently by using approximation algorithms for the minimum enclosing ball meb problem we proposed the core vector machine cvm algorithm that is much faster and can handle much larger data sets than existing svm implementations however the cvm can only be used with certain kernel functions and kernel methods for example the very popular support vector regression svr cannot be used with the cvm in this paper we introduce the center constrained meb problem and subsequently extend the cvm algorithm the generalized cvm algorithm can now be used with any linear nonlinear kernel and can also be applied to kernel methods such as svr and the ranking svm moreover like the original cvm its asymptotic time complexity is again linear in m and its space complexity is independent of m experiments show that the generalized cvm has comparable performance with state of the art svm and svr implementations but is faster and produces fewer support vectors on very large data sets

fast support vector data descriptions for novelty detection support vector data description svdd has become a very attractive kernel method due to its good results in many novelty detection problems however the decision function of svdd is expressed in terms of the kernel expansion which results in a run time complexity linear in the number of support vectors for applications where fast real time response is needed how to speed up the decision function is crucial this paper aims at dealing with the issue of reducing the testing time complexity of svdd a method called fast svdd f svdd is proposed unlike the traditional methods which all try to compress a kernel expansion into one with fewer terms the proposed f svdd directly finds the preimage of a feature vector and then uses a simple relationship between this feature vector and the svdd sphere center to re express the center with a single vector the decision function of f svdd contains only one kernel term and thus the decision boundary of f svdd is only spherical in the original space hence the run time complexity of the f svdd decision function is no longer linear in the support vectors but is a constant no matter how large the training set size is in this paper we also propose a novel direct preimage finding method which is noniterative and involves no free parameters the unique preimage can be obtained in real time by the proposed direct method without taking trial and error for demonstration several real world data sets and a large scale data set the extended mit face data set are used in experiments in addition a practical industry example regarding liquid crystal display micro defect inspection is also used to compare the applicability of svdd and our proposed f svdd when faced with mass data input the results are very encouraging

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

gahler s neighborhood condition for lattice valued convergence spaces we generalize a neighborhood condition from the category of convergence spaces to the category of lattice valued convergence spaces for a space in the category of convergence spaces this condition is equivalent to being a topological space it turns out that there are two meaningful generalizations of this condition to the category of lattice valued convergence spaces the first one guarantees that a space in a certain subcategory is a lattice valued topological space the other one is a levelwise condition which is equivalent to a generalization of fischer s diagonal condition the latter generalization is preserved under the embedding of the categories of convergence approach spaces and of probabilistic limit spaces into the category of lattice valued convergence spaces

a common framework for lattice valued uniform spaces and probabilistic uniform limit spaces we study a category of lattice valued uniform convergence spaces where the lattice is enriched by two algebraic operations this general setting allows us to view the category of lattice valued uniform spaces as a reflective subcategory of our category and the category of probabilistic uniform limit spaces as a coreflective subcategory

subcategories of lattice valued convergence spaces we study several subcategories of heyting algebra valued convergence spaces these categories generalize the well known categories of kent convergence spaces of limit spaces of pretopological spaces of pseudo topological spaces and of topological limit spaces to the many valued setting moreover a functorial mechanism is described for changing the basis lattice

correction to on many valued stratified l fuzzy convergence spaces fuzzy sets and systems 159 2008 2503 2519 the purpose of this note is to point out a mistake in w yao on many valued l fuzzy convergence spaces fuzzy sets and systems 159 2008 2503 2519 and to give a solution

lattice valued preuniform convergence spaces using the intrinsic lattice valued inclusion orders of l subsets and of stratified l filters the concept of a stratified l preuniform convergence structure is proposed this convergence structure is asymmetric and can be used to establish a framework of asymmetric lattice valued space structures the category of stratified l preuniform convergence spaces is introduced and some categorical properties are presented a reflective subcategory of stratified l preuniform convergence spaces is found that is categorically isomorphic to strong stratified l convergence spaces originally called stratified l ordered convergence spaces several subcategories of stratified l preuniform convergence spaces are established and relations between the different subcategories are presented we conclude that stratified l preuniform convergence structures introduced here could play a role in the framework of lattice valued asymmetric space structures in lattice valued topology

decomposition methods for linear support vector machines in this letter we show that decomposition methods with alpha seeding are extremely useful for solving a sequence of linear support vector machines svms with more data than attributes this strategy is motivated by keerthi and lin 2003 who proved that for an svm with data not linearly separable after c is large enough the dual solutions have the same free and bounded components we explain why a direct use of decomposition methods for linear svms is sometimes very slow and then analyze why alpha seeding is much more effective for linear than nonlinear svms we also conduct comparisons with other methods that are efficient for linear svms and demonstrate the effectiveness of alpha seeding techniques in model selection

estimating the support of a high dimensional distribution suppose you are given some data set drawn from an underlying probability distribution p and you want to estimate a simple subset s of input space such that the probability that a test point drawn from p lies outside of s equals some a priori specified value between 0 and 1 we propose a method to approach this problem by trying to estimate a function f that is positive on s and negative on the complement the functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data it is regularized by controlling the length of the weight vector in an associated feature space the expansion coefficients are found by solving a quadratic programming problem which we do by carrying out sequential optimization over pairs of input patterns we also provide a theoretical analysis of the statistical performance of our algorithm the algorithm is a natural extension of the support vector algorithm to the case of unlabeled data

a convergent hybrid decomposition algorithm model for svm training training of support vector machines svms requires to solve a linearly constrained convex quadratic problem in real applications the number of training data may be very huge and the hessian matrix cannot be stored in order to take into account this issue a common strategy consists in using decomposition algorithms which at each iteration operate only on a small subset of variables usually referred to as the working set training time can be significantly reduced by using a caching technique that allocates some memory space to store the columns of the hessian matrix corresponding to the variables recently updated the convergence properties of a decomposition method can be guaranteed by means of a suitable selection of the working set and this can limit the possibility of exploiting the information stored in the cache we propose a general hybrid algorithm model which combines the capability of producing a globally convergent sequence of points with a flexible use of the information in the cache as an example of a specific realization of the general hybrid model we describe an algorithm based on a particular strategy for exploiting the information deriving from a caching technique we report the results of computational experiments performed by simple implementations of this algorithm the numerical results point out the potentiality of the approach

a formal analysis of stopping criteria of decomposition methods for support vector machines in a previous paper the author 2001 proved the convergence of a commonly used decomposition method for support vector machines svms however there is no theoretical justification about its stopping criterion which is based on the gap of the violation of the optimality condition it is essential to have the gap asymptotically approach zero so we are sure that existing implementations stop in a finite number of iterations after reaching a specified tolerance here we prove this result and illustrate it by two extensions spl nu svm and a multiclass svm by crammer and singer 2001 a further result shows that in final iterations of the decomposition method only a particular set of variables are still being modified this supports the use of the shrinking and caching techniques in some existing implementations finally we prove the asymptotic convergence of a decomposition method for this multiclass svm discussions on the difference between this convergence proof and the one in another paper by lin are also included

enriched lattice valued convergence groups considering a category sl gconv of stratified enriched cl premonoid valued generalized convergence spaces we present the category sl gconvgrp of stratified l generalized convergence groups and some of its subcategories we present two natural examples on stratified enriched cl premonoid valued convergence groups among other results we show that every stratified strong l limit group is sl ucs uniformizable

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