very large scale linear programming a case study in combining interior point and simplex methods experience with solving a 12 753 313 variable linear program is described this problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application a scheme is described that requires successive solutions of small subproblems yielding a procedure that has little growth in solution time in terms of the number of variables experience using the simplex method as implemented in cplex an interior point method as implemented in obi and a hybrid interior point simplex approach is reported the resulting procedure illustrates the power of an interior point simplex combination for solving very large scale linear programs

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

considering cost asymmetry in learning classifiers receiver operating characteristic roc curves are a standard way to display the performance of a set of binary classifiers for all feasible ratios of the costs associated with false positives and false negatives for linear classifiers the set of classifiers is typically obtained by training once holding constant the estimated slope and then varying the intercept to obtain a parameterized set of classifiers whose performances can be plotted in the roc plane we consider the alternative of varying the asymmetry of the cost function used for training we show that the roc curve obtained by varying both the intercept and the asymmetry and hence the slope always outperforms the roc curve obtained by varying only the intercept in addition we present a path following algorithm for the support vector machine svm that can compute efficiently the entire roc curve and that has the same computational complexity as training a single classifier finally we provide a theoretical analysis of the relationship between the asymmetric cost model assumed when training a classifier and the cost model assumed in applying the classifier in particular we show that the mismatch between the step function used for testing and its convex upper bounds usually used for training leads to a provable and quantifiable difference around extreme asymmetries

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2

frobenius actions on the de rham cohomology of drinfeld modules we study the action of endomorphisms of a drinfeld a module phi on its de rham cohomology h dr phi l and related modules in the case where phi is defined over a field l of finite a characteristic mathfrak p among others we find that the nilspace h 0 of the total frobenius fr dr on h dr phi l mathfrak p has dimension h height of phi we define and study a pairing between the mathfrak p torsion mathfrak p phi of phi and h dr phi l which becomes perfect after dividing out h 0

zeros of eisenstein series quadratic class numbers and supersingularity for rational function fields the support of the divisor of certain eisenstein series on the modular curve for gl 2 fq t generate a big extension of fq t from this an upper bound for the number of supersingular primes of a xed degree for a given drinfeld module can be deduced in the course of the proof 2 power divisibility of the class number of certain hyperelliptic extensions of fq t is studied

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

the poisson kernel for drinfeld modular curves in an earlier work t 1 the author described a technique for constructing rigid analytic modular forms on the p adic upper half plane by means of an integral transform a poisson kernel in this paper we apply these methods to the study of rigid analytic modular forms on the upper half plane over a complete local field k of characteristic p arising out of the theory of drinfeld modules d these modular forms first studied by goss see gol go2 are in many ways analogous to the classical modular forms for sl2 z for a brief but effective introduction to the theory of these characteristic p modular forms see the introduction to gekeler s paper gekl also useful is gekeler s book gek2 we establish three main results first of all we show that if k is the function field of a complete geometrically irreducible curve c over the finite field lfq with q elements k is a completion of k f is an arithmetic subgroup of gl2 k and u is a measure on pi coming from a harmonic cocycle of weight n for i see definition 1 below then

self dual manifolds with positive ricci curvature we prove that the connected sums cp2 cp2 and cp2 cp2 cp2 admit self dual metrics with positive ricci curvature moreover every self dual metric of positive scalar curvature on cp2 cp2 is conformal to a metric with positive ricci curvature supported in part by nsf grant dms 9204093

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

sparse robust estimation and kalman smoothing with nonsmooth log concave densities modeling computation and theory we introduce a new class of quadratic support qs functions many of which already play a crucial role in a variety of applications including machine learning robust statistical inference sparsity promotion and inverse problems such as kalman smoothing well known examples of qs penalties include the l2 huber l1 and vapnik losses we build on a dual representation for qs functions using it to characterize conditions necessary to interpret these functions as negative logs of true probability densities this interpretation establishes the foundation for statistical modeling with both known and new qs loss functions and enables construction of non smooth multivariate distributions with specified means and variances from simple scalar building blocks the main contribution of this paper is a flexible statistical modeling framework for a variety of learning applications together with a toolbox of efficient numerical methods for estimation in particular a broad subclass of qs loss functions known as piecewise linear quadratic plq penalties has a dual representation that can be exploited to design interior point ip methods ip methods solve nonsmooth optimization problems by working directly with smooth systems of equations characterizing their optimality we provide several numerical examples along with a code that can be used to solve general plq problems the efficiency of the ip approach depends on the structure of particular applications we consider the class of dynamic inverse problems using kalman smoothing this class comprises a wide variety of applications where the aim is to reconstruct the state of a dynamical system with known process and measurement models starting from noisy output samples in the classical case gaussian errors are assumed both in the process and measurement models for such problems we show that the extended framework allows arbitrary plq densities to be used and that the proposed ip approach solves the generalized kalman smoothing problem while maintaining the linear complexity in the size of the time series just as in the gaussian case this extends the computational efficiency of the mayne fraser and rauch tung striebel algorithms to a much broader nonsmooth setting and includes many recently proposed robust and sparse smoothers as special cases

large scale linear programming techniques for the design of protein folding potentials we present large scale optimization techniques to model the energy function that underlies the folding process of proteins linear programming is used to identify parameters in the energy function model the objective being that the model predict the structure of known proteins correctly such trained functions can then be used either for ab initio prediction or for recognition of unknown structures in order to obtain good energy models we need to be able to solve dense linear programming problems with tens possibly hundreds of millions of constraints in a few hundred parameters which we achieve by tailoring and parallelizing the interior point code pcx

on identities for zeta values in tate algebras zeta values in tate algebras were introduced by pellarin in 2012 they are generalizations of carlitz s zeta values and play an increasingly important role in function field arithmetic in this paper we prove a conjecture of pellarin on identities for these zeta values the proof is based on arithmetic properties of carlitz s zeta values and an explicit formula for bernoulli type polynomials attached to pellarin s zeta values

compact self dual manifolds with positive scalar curvature on etudie l existence d une metrique auto duale a courbure scalaire positive sur une variete compacte simplement connexe qui n est pas conformement equivalente a ces metriques standard sur s 4 ou cp 2

successive overrelaxation for support vector machines successive overrelaxation sor for symmetric linear complementarity problems and quadratic programs is used to train a support vector machine svm for discriminating between the elements of two massive datasets each with millions of points because sor handles one point at a time similar to platt s sequential minimal optimization smo algorithm 1999 which handles two constraints at a time and joachims svm sup light 1998 which handles a small number of points at a time sor can process very large datasets that need not reside in memory the algorithm converges linearly to a solution encouraging numerical results are presented on datasets with up to 10 000 000 points such massive discrimination problems cannot be processed by conventional linear or quadratic programming methods and to our knowledge have not been solved by other methods on smaller problems sor was faster than svm sup light and comparable or faster than smo

a matrix free smoothing algorithm for large scale support vector machines this paper is concerned with the iterative solution of sequences of karush kuhn tucker kkt systems arising from smoothing newton type algorithms applied to support vector machines svms every perturbed newton step is crucial for the efficiency of the overall algorithm however the kkt systems based on smoothing reformulation become increasingly ill conditioned as the smoothing parameter approaches to zero by exploiting preconditioning techniques we propose a matrix free smoothing algorithm for solving large scale support vector machines numerical results and comparisons are given to demonstrate the effectiveness and speed of the algorithm

interior point methods for massive support vector machines we investigate the use of interior point methods for solving quadratic programming problems with a small number of linear constraints where the quadratic term consists of a low rank update to a positive semidefinite matrix several formulations of the support vector machine fit into this category an interesting feature of these particular problems is the volume of data which can lead to quadratic programs with between 10 and 100 million variables and if written explicitly a dense q matrix our code is based on ooqp an object oriented interior point code with the linear algebra specialized for the support vector machine application for the targeted massive problems all of the data is stored out of core and we overlap computation and input output to reduce overhead results are reported for several linear support vector machine formulations demonstrating that the method is reliable and scalable

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

values of certain l series in positive characteristic we introduce a class of deformations of the values of the goss zeta function we prove with the use of the theory of deformations of vectorial modular forms as well as with other techniques a formula for their value at 1 and some arithmetic properties of values at other positive integers our formulas involve anderson and thakur s function we discuss how our formulas may be used to investigate the existence of a kind of functional equation for the goss zeta function

robust linear and support vector regression the robust huber m estimator a differentiable cost function that is quadratic for small errors and linear otherwise is modeled exactly in the original primal space of the problem by an easily solvable simple convex quadratic program for both linear and nonlinear support vector estimators previous models were significantly more complex or formulated in the dual space and most involved specialized numerical algorithms for solving the robust huber linear estimator numerical test comparisons with these algorithms indicate the computational effectiveness of the new quadratic programming model for both linear and nonlinear support vector problems results are shown on problems with as many as 20000 data points with considerably faster running times on larger problems

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

special values of drinfeld modular forms and algebraic independence let a be a polynomial ring in one variable over a finite field and k be its fraction field let f be a drinfeld modular form of nonzero weight for a congruence subgroup of gl2 a so that the coefficients of the q expansion of f are algebraic over k we consider n cm points 1 n on the drinfeld upper half plane for which the quadratic fields k 1 k n are pairwise distinct suppose that f is non vanishing at these n points then we prove that f 1 f n are algebraically independent over k

transcendence and drinfeld modules it has been ten years since v g drinfeld introduced in 3 the concept of elliptic a module which we now call drinfeld a module these modules are analogues of elliptic curves for global function field k and we have witnessed a beautiful algebraic theory gradually taking shape in 13 we have asked a few natural t ranscendence questions conjectures concerning these modules having in mind the analogue of the classical work of siegel schneider about elliptic integrals and based on evidences provided by the old papers of carlitz and wade cf 1 11 here in this article we will prove those conjectures we will show that indeed there is an interesting transcendence theory for each algebraic curve over a finite field fol lowing drinfeld we let m be a lattice in i e a finitely generated discrete a module in k and form

active set support vector regression this paper presents active set support vector regression asvr a new active set strategy to solve a straightforward reformulation of the standard support vector regression problem this new algorithm is based on the successful asvm algorithm for classification problems and consists of solving a finite number of linear equations with a typically large dimensionality equal to the number of points to be approximated however by making use of the sherman morrison woodbury formula a much smaller matrix of the order of the original input space is inverted at each step the algorithm requires no specialized quadratic or linear programming code but merely a linear equation solver which is publicly available asvr is extremely fast produces comparable generalization error to other popular algorithms and is available on the web for download

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

exterior point method for support vector machines we present an exterior point method epm for training dual soft margin support vector machines svms the epm stems from nonlinear rescaling and augmented lagrangian methods and allows iterates to approach the solution of a constrained nonlinear optimization problem from the exterior of the feasible set furthermore the epm produces and solves a well conditioned system of linear equations at each iteration thus avoiding numerical inaccuracies that can occur when solving ill conditioned systems therefore the epm may be an attractive alternative to existing quadratic programming solvers for training svms we report numerical results for training the svm with the epm on data up to several thousand data points from the uc irvine machine learning repository

conformal deformations of integral pinched 3 manifolds abstract in this paper we prove that under an explicit integral pinching assumption between the l 2 norm of the ricci curvature and the l 2 norm of the scalar curvature a closed 3 manifold with positive scalar curvature admits a conformal metric of positive ricci curvature in particular using a result of hamilton this implies that the manifold is diffeomorphic to a quotient of s 3 the proof of the main result of the paper is based on ideas developed in an article by m gursky and j viaclovsky

feasible descent algorithms for mixed complementarity problems in this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems the main features of this framework are a it is well defined for an arbitrary mixed complementarity problem b it generates only feasible iterates c it has a strong global convergence theory and d it is locally fast convergent under standard regularity assumptions this framework is applied to the path solver in order to show viability of the approach numerical results for an appropriate modification of the path solver indicate that this framework leads to substantial computational improvements

gap phenomena and curvature estimates for conformally compact einstein manifolds author s li gang qing jie shi yuguang abstract in this paper we first use the result in 12 to remove the assumption of the l 2 boundedness of weyl curvature in the gap theorem in 9 and then obtain a gap theorem for a class of conformally compact einstein manifolds with very large renormalized volume we also uses the blow up method to derive curvature estimates for conformally compact einstein manifolds with large renormalized volume the second part of this paper is on conformally compact einstein manifolds with conformal infinities of large yamabe constants based on the idea in 15 we manage to give the complete proof of the relative volume inequality 1 9 on conformally compact einstein manifolds therefore we obtain the complete proof of the rigidity theorem for conformally compact einstein manifolds in general dimensions with no spin structure assumption cf 29 15 as well as the new curvature pinch estimates for conformally compact einstein manifolds with conformal infinities of very large yamabe constant we also derive the curvature estimates for conformally compact einstein manifolds with conformal infinities of large yamabe constant

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 conformally invariant gap theorem characterizing mathbb cp 2 via the ricci flow we extend the sphere theorem of chang et al publ math inst ht etudes sci 98 105 434 2003 to give a conformally invariant characterization of mathbb cp 2 g fs in particular we introduce a conformal invariant beta m 4 g ge 0 defined on conformal four manifolds satisfying a positivity condition it follows from chang et al 2003 that if 0 le beta m 4 g 0 and 4 le beta m 4 g 0 small enough then m 4 is diffeomorphic to mathbb cp 2 the ricci flow is used in a crucial way to pass from the bounds on beta to pointwise curvature information

an equation of monge ampere type in conformal geometry and four manifolds of positive ricci curvature we formulate natural conformally invariant conditions on a 4 manifold for the existence of a metric whose schouten tensor satisfies a quadratic inequality this inequality implies that the eigenvalues of the ricci tensor are positively pinched

the linear l 1 estimator and the huber m estimator relationships between a linear l1 estimation problem and the huber m estimator problem can be easily established by their dual formulations the least norm solution of a linear programming problem studied by mangasarian and meyer siam j control optim 17 1979 pp 745 752 provides a key link between the dual problems based on the dual formulations we establish a local linearity property of the huber m estimators with respect to the tuning parameter gamma and prove that the solution set of the huber m estimator problem is lipschitz continuous with respect to perturbations of the tuning parameter gamma as a consequence the set of the linear l1 estimators is the limit of the set of the huber m estimators as gamma to 0 thus the huber m estimator problem has many solutions for small tuning parameter gamma if the linear l1 estimation problem has multiple solutions a recursive version of madsen and nielsen s algorithm siam j optim 3 1993 pp 223 235 based on computation of the huber m estimator is proposed for finding a linear l1 estimator

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

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

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

the principal eigenvalue of a conformally invariant differential operator with an application to semilinear elliptic pde in this paper we give conditions under which the paneitz operator a conformally invariant fourth order differential operator is positive we also prove a sharp estimate for a related conformal invariant these results are combined to study the existence of a semilinear pde with critical sobolev growth

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

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 least squares approach to direct importance estimation we address the problem of estimating the ratio of two probability density functions which is often referred to as the importance the importance values can be used for various succeeding tasks such as covariate shift adaptation or outlier detection in this paper we propose a new importance estimation method that has a closed form solution the leave one out cross validation score can also be computed analytically therefore the proposed method is computationally highly efficient and simple to implement we also elucidate theoretical properties of the proposed method such as the convergence rate and approximation error bounds numerical experiments show that the proposed method is comparable to the best existing method in accuracy while it is computationally more efficient than competing approaches

estimating the order of vanishing at infinity of drinfeld quasi modular forms we introduce and study certain deformations of drinfeld quasi modular forms by using rigid analytic trivialisations of corresponding anderson s t motives we show that a sub algebra of these deformations has a natural graduation by the group z 2 x z q 1 z and an homogeneous automorphism and we deduce from this and other properties multiplicity estimates

transcendance de l invariant modulaire en caracteristique finie abstract let j be the fourier expansion at infinity of the modular invariant j associated to a drinfeld module of rank 2 defined on the algebraic closure bar k of mathbb f q t and t an element of the completion of bar k then at least one of the two elements t j t is transcendental over mathbb f q t

smoothing multivariate performance measures optimizing multivariate performance measure is an important task in machine learning joachims 2005 introduced a support vector method whose underlying optimization problem is commonly solved by cutting plane methods cpms such as svm perf and bmrm it can be shown that cpms converge to an e accurate solution in o 1 e iterations where is the trade off parameter between the regularizer and the loss function motivated by the impressive convergence rate of cpm on a number of practical problems it was conjectured that these rates can be further improved we disprove this conjecture in this paper by constructing counter examples however surprisingly we further discover that these problems are not inherently hard and we develop a novel smoothing strategy which in conjunction with nesterov s accelerated gradient method can find an e accurate solution in o min 1 e 1 e iterations computationally our smoothing technique is also particularly advantageous for optimizing multivariate performance scores such as precision recall break even point and rocarea the cost per iteration remains the same as that of cpms empirical evaluation on some of the largest publicly available data sets shows that our method converges significantly faster than cpms without sacrificing generalization ability

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

frobenius distributions of drinfeld modules over finite fields we express the weighted class number of drinfeld a modules of rank two with given characteristic polynomial over the finite field f p a p p spec a where a fq t as an infinite product of local terms some auxiliary results of independent interest about characteristic polynomials of drinfeld modules are given

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm

minimal kernel classifiers a finite concave minimization algorithm is proposed for constructing kernel classifiers that use a minimal number of data points both in generating and characterizing a classifier the algorithm is theoretically justified on the basis of linear programming perturbation theory and a leave one out error bound as well as effective computational results on seven real world datasets a nonlinear rectangular kernel is generated by systematically utilizing as few of the data as possible both in training and in characterizing a nonlinear separating surface this can result in substantial reduction in kernel data dependence over 94 in six of the seven public datasets tested on and with test set correctness equal to that obtained by using a conventional support vector machine classifier that depends on many more data points this reduction in data dependence results in a much faster classifier that requires less storage to eliminate data points the proposed approach makes use of a novel loss function the pound function which is a linear combination of the 1 norm and the step function that measures both the magnitude and the presence of any error