a note on reductions of the dispersionless toda hierarchy we investigate the algebraic structure of the dispersionless hirota equations for the m 1 variable reductions of the dispersionless toda hierarchy which contains the special case for m 1 considered by kodama and pierce we demonstrate the m 2 case to illustrate the obtained result

an iterative algorithm learning the maximal margin classifier abstract a simple learning algorithm for maximal margin classifiers also support vector machines with quadratic cost function is proposed we build our iterative algorithm on top of the schlesinger kozinec algorithm s k algorithm from 1981 which finds a maximal margin hyperplane with a given precision for separable data we suggest a generalization of the s k algorithm i to the non linear case using kernel functions and ii for non separable data the requirement in memory storage is linear to the data this property allows the proposed algorithm to be used for large training problems the resulting algorithm is simple to implement and as the experiments showed competitive to the state of the art algorithms the implementation of the algorithm in matlab is available we tested the algorithm on the problem aiming at recognition poor quality numerals

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

towards lax formulation of integrable hierarchies of topological type to each partition function of cohomological field theory one can associate an hamiltonian integrable hierarchy of topological type the givental group acts on such partition functions and consequently on the associated integrable hierarchies we consider the hirota and lax formulations of the deformation of the hierarchy of n copies of kdv obtained by an infinitesimal action of the givental group by first deforming the hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo differential operators we show that such deformed hierarchy admits an explicit lax formulation we then compare the deformed hamiltonians obtained from the lax equations with the analogous formulas obtained in buryak et al j differ geom 92 153 185 2012 buryak et al j geom phys 62 1639 1651 2012 to find that they agree we finally comment on the possibility of extending the hirota and lax formulation on the whole orbit of the givental group action

the extended bigraded toda hierarchy we generalize the toda lattice hierarchy by considering n m dependent variables we construct roots and logarithms of the lax operator which are uniquely defined operators with coefficients that are series of differential polynomials in the dependent variables and we use them to provide a lax pair definition of the extended bigraded toda hierarchy generalizing 4 using r matrix theory we give the bi hamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions finally we study the dispersionless limit and its connection with a class of frobenius manifolds on the orbit space of the extended affine weyl groups of the a series defined by dubrovin and zhang 1998 compos math 111 167

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

steady state kalman filtering with an h error bound abstract an estimator design problem is considered which involves both l 2 least squares and h worst case frequency domain aspects specifically the goal of the problem is to minimize an l 2 state estimation error criterion subject to a prespecified h constraint on the state estimation error the h estimation error constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on the l 2 state estimation error the principal result is a sufficient condition for characterizing fixed order i e full and reduced order estimators with bounded l 2 and h estimation error the sufficient condition involves a system of modified riccati equations coupled by an oblique projection i e idempotent matrix when the h constraint is absent the sufficient condition specializes to the l 2 state estimation result given in 2

a novel geometric approach to binary classification based on scaled convex hulls geometric methods are very intuitive and provide a theoretical foundation to many optimization problems in the fields of pattern recognition and machine learning in this brief the notion of scaled convex hull sch is defined and a set of theoretical results are exploited to support it these results allow the existing nearest point algorithms to be directly applied to solve both the separable and nonseparable classification problems successfully and efficiently then the popular s k algorithm has been presented to solve the nonseparable problems in the context of the sch framework the theoretical analysis and some experiments show that the proposed method may achieve better performance than the state of the art methods in terms of the number of kernel evaluations and the execution time

closed formulae for a parametric mixed sensitivity problem abstract mixed sensitivity optimization is a two block problem which normally requires an iterative solution the purpose of this paper is to give an explicit closed formula for the optimal cost in the case of a parametric mixed sensitivity problem we also show that all the controllers which satisfy an arbitrary h norm bound may be parameterized in terms of the solutions to the standards h 2 riccati equations we hope that these results will help progress towards closed formulae for the optimal cost in a wider class of problems

a dual result to kharitonov s theorem a dual problem to kharitonov s problem involving a diamond instead of a rectangle is considered the results show that a family of polynomials with coefficients varying in the diamond is strictly hurwitz if and only if eight one dimensional exposed edges of the diamond are strictly hurwitz

h sub infinity minimum error state estimation of linear stationary processes a state estimator is derived which minimizes the h sub infinity norm of the estimation error power spectrum matrix two approaches are presented the first achieves the optimal estimator in the frequency domain by finding the filter transfer function matrix that leads to an equalizing solution the second approach establishes a duality between the problem of h sub infinity filtering and the problem of unconstrained input h sub infinity optimal regulation using this duality previously published results for the latter regulation problem are applied which lead to an optimal filter that possess the structure of the corresponding kalman filter the two approaches usually lead to different results they are compared by a simple example which also demonstrates a clear advantage of the h sub infinity estimate over the conventional l sub 2 estimate

a geometric approach to support vector machine svm classification the geometric framework for the support vector machine svm classification problem provides an intuitive ground for the understanding and the application of geometric optimization algorithms leading to practical solutions of real world classification problems in this work the notion of reduced convex hull is employed and supported by a set of new theoretical results these results allow existing geometric algorithms to be directly and practically applied to solve not only separable but also nonseparable classification problems both accurately and efficiently as a practical application of the new theoretical results a known geometric algorithm has been employed and transformed accordingly to solve nonseparable problems successfully

convergence of the dre solution to the are strong solution in this paper we use the boundary solutions to a linear matrix inequality lmi associated with kalman filtering to investigate the convergence of the solution of a differential riccati equation dre to the so called strong solution of an algebraic riccati equation are we furthermore extend our results to kalman filtering with indefinite input noise covariances

a general soft method for learning svm classifiers with l 1 norm penalty based on the geometric interpretation of support vector machines svms this paper presents a general technique that allows almost all the existing l 2 norm penalty based geometric algorithms including gilbert s algorithm schlesinger kozinec s sk algorithm and mitchell dem yanov malozemov s mdm algorithm to be softened to achieve the corresponding learning l 1 svm classifiers intrinsically the resulting soft algorithms are to find e optimal nearest points between two soft convex hulls theoretical analysis has indicated that our proposed soft algorithms are essentially generalizations of the corresponding existing hard algorithms and consequently they have the same properties of convergence and almost the identical cost of computation as a specific example the problem of solving n svms by the proposed soft mdm algorithm is investigated and the corresponding solution procedure is specified and analyzed to validate the general soft technique several real classification experiments are conducted with the proposed l 1 norm based mdm algorithms and numerical results have demonstrated that their performance is competitive to that of the corresponding l 2 norm based algorithms such as sk and mdm algorithms

the spaces of laurent polynomials gromov witten theory of ℙ1 orbifolds and integrable hierarchies let mk m be the space of laurent polynomials in one variable x t1x tk mx where k m 1 are fixed integers and tk m 6 0 according to b dubrovin 11 mk m can be equipped with a semi simple frobenius structure in this paper we prove that the corresponding descendent and ancestor potentials of mk m defined as in 16 satisfy hirota quadratic equations hqe for short let ck m be the orbifold obtained from p by cutting small discs d1 z and d2 z 1 around z 0 and z and gluing back the orbifolds d1 zk and d2 zm in the obvious way we show that the orbifold quantum cohomology of ck m coincides with mk m as frobenius manifolds modulo some yet to be clarified details this implies that the descendent respectively the ancestor potential of mk m is a generating function for the descendent respectively ancestor orbifold gromov witten invariants of ck m there is a certain similarity between our hqe and the lax operators of the extended bi graded toda hierarchy introduced by g carlet in 7 therefore it is plausible that our hqe characterize the tau functions of this hierarchy and we expect that the extended bi graded toda hierarchy governs the gromov witten theory of ck m

two extensions of 1d toda hierarchy the extended toda hierarchy of carlet dubrovin and zhang is reconsidered in light of a 2 1d extension of the 1d toda hierarchy constructed by ogawa these two extensions of the 1d toda hierarchy turn out to have a very similar structure and the former may be thought of as a kind of dimensional reduction of the latter in particular this explains an origin of the mysterious structure of the bilinear formalism proposed by milanov

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

an effective method to determine whether a point is within a convex hull and its generalized convex polyhedron classifier abstract a convex polyhedron classifier that encloses the minority class using a combination of hyperplanes is potentially effective in imbalanced classification to construct an easy to use convex polyhedron classifier this paper first presents a theoretical foundation for determining whether a point is within the convex hull of a finite point set this foundation corresponds to a geometric method in which the result is expressed as a separating hyperplane if the given point and the given convex hull are located on either side of the learned hyperplane this indicates that the point is outside of the convex hull otherwise the conclusion that the point is within the convex hull can be obtained as a generalization of the geometric method a convex polyhedron classifier is further proposed for binary classification if two finite point sets are polyhedrally separable a series of hyperplanes can be learned as a combined piecewise linear classifier which surrounds a point set that is inside using a convex polyhedron and excludes the other point set that is outside experimental results on twelve real world datasets show that the proposed classifier is generally better than the other two piecewise linear classifiers moreover a comparison with several types of support vector machines confirms its competitiveness

additional symmetries of the extended bigraded toda hierarchy the extended bigraded toda hierarchy ebth is an integrable system satisfied by the total descendant potential of with two orbifold points we construct additional symmetries of the ebth and describe explicitly their action on the lax operator wave operators and tau function of the hierarchy in particular we obtain infinitesimal symmetries of the ebth that act on the tau function as a subalgebra of the virasoro algebra generalizing those of dubrovin and zhang

a geometric nearest point algorithm for the efficient solution of the svm classification task geometric methods are very intuitive and provide a theoretically solid approach to many optimization problems one such optimization task is the support vector machine svm classification which has been the focus of intense theoretical as well as application oriented research in machine learning in this letter the incorporation of recent results in reduced convex hulls rchs to a nearest point algorithm npa leads to an elegant and efficient solution to the svm classification task with encouraging practical results to real world classification problems i e linear or nonlinear and separable or nonseparable

the local gromov witten theory of mathbb c mathbb p 1 and integrable hierarchies in this paper we begin the study of the relationship between the local gromov witten theory of calabi yau rank two bundles over the projective line and the theory of integrable hierarchies we first of all construct explicitly in a large number of cases the hamiltonian dispersionless hierarchies that govern the full descendent genus zero theory our main tool is the application of dubrovin s formalism based on associativity equations to the known results on the genus zero theory from local mirror symmetry and localization the hierarchies we find are apparently new with the exception of the resolved conifold mathcal o mathbb p 1 1 bigoplus mathcal o mathbb p 1 1 in the equivariantly calabi yau case for this example the relevant dispersionless system turns out to be related to the long wave limit of the ablowitz ladik lattice this identification provides us with a complete procedure to reconstruct the dispersive hierarchy which should conjecturally be related to the higher genus theory of the resolved conifold we give a complete proof of this conjecture for genus g 1 our methods are based on establishing analogously to the case of kdv a quasi triviality property for the ablowitz ladik hierarchy at the leading order of the dispersive expansion we furthermore provide compelling evidence in favour of the resolved conifold ablowitz ladik correspondence at higher genus by testing it successfully in the primary sector for g 2

geometric algorithms to large margin classifier based on affine hulls the geometric framework for binary data classification problems provides an intuitive foundation for the comprehension and application of geometric optimization algorithms leading to practical solutions of real world classification problems in this paper some theoretical results on the candidate extreme points of the notion of reduced affine hull rah are introduced these results allow the existing nearest point algorithms to be directly applied to solve both separable and inseparable classification problems based on rahs successfully and efficiently as the practical applications of the new theoretical results the popular gilbert schlesinger kozinec and mitchell dem yanov malozemov algorithms are presented to solve binary classification problems in the context of the rah framework the theoretical analysis and some experiments show that the proposed methods successfully achieve significant performance

a polynomial approach to minimax frequency domain optimization of multivariable feedback systems the properties of a feedback system where the plant has rational transfer matrix h and the compensator has transfer matrix g can be characterized through the system functions s i hg and t g i hg 1 good disturbance attenuation robustness limited bandwidth and compensator roll off may be obtained by minimizing a criterion of the form z where z v s t w1 wls t w2 w2t v with respect to the compensator transfer matrix g here v w1 and w2 are suitable rational weighting matrices the solution of the problem can be reduced to a pair of matrix polynomial equations since the optimal solution is highly non unique special solutions with additional optimality properties are considered as well the paper includes a discussion of the numerical solution of the polynomial equations and of the question how to choose the weighting matrices to ensure required feedback system properties an example illustrates the results

nondefinite least squares and its relation to h sub infinity minimum error state estimation the problem of recursive nondefinite least squares state estimation of continuous time stationary processes is solved by applying pontryagin s maximum principle a comparison of the derived solution to the result that is obtained for the h sub infinity minimum error estimation suggests a new interpretation for the h sub infinity optimal estimation mechanism according to this interpretation the estimator tries to optimally estimate the required combination of the states in the l sub 2 norm sense against the worst disturbance signal that stems from a fictitious measurement of this combination

hirota equations for the extended bigraded toda hierarchy and the total descendent potential of cp1 orbifolds we prove that the hirota quadratic equations of milanov and tseng define an integrable hierarchy which is equivalent to the extended bigraded toda hierarchy ebth in particular this confirms a conjecture that relates the total descendent potential of the orbifold ck m with a tau function of the ebth

game theory approach to finite time horizon optimal estimation in this game a measurement record is given and the first player looks for the best estimate of a prespecified combination of the system states in the presence of a hostile process noise signal and system initial condition that are applied by his adversary it turns out that the game possesses a saddle point solution which leads to an optimal smoothed estimate that is identical to the corresponding l sub 2 optimal estimate a similar game in which the estimate is restricted to be causal is formulated and solved this game provides for the first time a saddle point equilibrium interpretation to finite time h sub infinity optimal filtered estimation the two games are very closely related it is shown that in the first game the first player s strategy which is the optimal smoothed estimate is a linear fractional transformation of the h sub infinity optimal filter which applies a nonzero free contracting q parameter it therefore achieves a unity h sub infinity norm bound for the operator that relates the exogeneous signals to the estimation error

tau function and hirota bilinear equations for the extended bigraded toda hierarchy in this paper we generalize the sato theory to the extended bigraded toda hierarchy we revise the definition of the lax equations give the sato equations wave operators hirota bilinear identities and show the existence of tau function t meanwhile we prove the validity of its fay like identities and hirota bilinear equations hbes in terms of vertex operators whose coefficients take values in the algebra of differential operators in contrast to hbes of the usual integrable system the current hbes are equations of the product of operators involving e x and t

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

an h sub spl infin optimization and its fast algorithm for time variant system identification in some estimation or identification techniques a forgetting factor spl rho has been used to improve the tracking performance for time varying systems however the value of spl rho has been typically determined empirically without any evidence of optimality in our previous work this open problem is solved using the framework of h sub spl infin optimization the resultant h sub spl infin filter enables the forgetting factor spl rho to be optimized through a process noise that is determined by the filter riccati equation this paper seeks to further explain the previously derived h sub spl infin filter giving an h sub spl infin interpretation of its tracking capability additionally a fast algorithm of the h sub spl infin filter called the fast h sub spl infin filter is presented when the observation matrix has a shifting property finally the effectiveness of the derived fast algorithm is illustrated for time variant system identification using several computer simulations here the fast h sub spl infin filter is shown to outperform the well known least mean square algorithm and the fast kalman filter in convergence rate

frequency domain conditions for the stability of perturbed polynomials new necessary and sufficient conditions for the stability of perturbed polynomials of continuous systems are given in the frequency domain the conditions are equivalent and in some respects more powerful than the well known kharitonov conditions the new conditions allow considerable freedom in distributing the available uncertainty margin among the different coefficients of a polynomial and provide an indication as to whether the maximum allowable margin of uncertainty for a given polynomial has been reached

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients