rigid symmetries and conservation laws in non lagrangian field theory making use of the lagrange anchor construction introduced earlier to quantize non lagrangian field theories we extend the noether theorem beyond the class of variational dynamics

boosting the margin a new explanation for the effectiveness of voting methods one of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large and often is observed to decrease even after the training error reaches zero in this paper we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label we show that techniques used in the analysis of vapnik s support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error we also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples finally we compare our explanation to those based on the bias variance decomposition

symmetrization approach to concentration inequalities for empirical processes we introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional as an application we give a new easy proof of talagrand s concentration inequality for empirical processes where besides symmetrization we use only talagrand s concentration inequality on the discrete cube 0 1 n as another application of this technique we prove new vapnik chervonenkis type inequalities for example for vc classes of functions we prove a classical inequality of vapnik and chervonenkis only with normalization by the sum of variance and sample variance

zeta functions for the spectrum of the non commutative harmonic oscillators this paper investigates the spectral zeta function of the non commutative harmonic oscillator studied in pw1 2 it is shown as one of the basic analytic properties that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at s 1 and further that it has a zero at all non positive even integers i e at s 0 and at those negative even integers where the riemann zeta function has the so called trivial zeros as a by product of the study both the upper and the lower bounds are also given for the first eigenvalue of the non commutative harmonic oscillator

fast rates for support vector machines using gaussian kernels for binary classification we establish learning rates up to the order of n 1 for support vector machines svms with hinge loss and gaussian rbf kernels these rates are in terms of two assumptions on the considered distributions tsybakov s noise assumption to establish a small estimation error and a new geometric noise condition which is used to bound the approximation error unlike previously proposed concepts for bounding the approximation error the geometric noise assumption does not employ any smoothness assumption 1 introduction in recent years support vector machines svms have been the subject of many theoretical considerations despite this effort their learning performance on restricted classes of distributions is still widely unknown in particular it is unknown under which nontrivial circumstances svms can guarantee fast learning rates the aim of this work is to use concepts like tsybakov s noise assumption and local rademacher averages to establish learning rates up to the order of n 1 for nontrivial distributions in addition to these concepts that are used to deal with the stochastic part of the analysis we also introduce a geometric assumption for distributions that allows us to estimate the approximation properties of gaussian rbf kernels unlike many other concepts introduced for bounding the approximation error our geometric assumption is not in terms of smoothness but describes the concentration and the noisiness of the data generating distribution near the decision boundary let us formally introduce the statistical classification problem to this end let us fix a subset x r d we write y 1 1 given a finite training set

asymptotics in empirical risk minimization in this paper we study a two category classification problem we indicate the categories by labels y 1 and y 1 we observe a covariate or feature x x d consider a collection ha of classifiers indexed by a finite dimensional parameter a and the classifier ha that minimizes the prediction error over this class the parameter a is estimated by the empirical risk minimizer n over the class where the empirical risk is calculated on a training sample of size n we apply the kim pollard theorem to show that under certain differentiability assumptions n converges to a with rate n 1 3 and also present the asymptotic distribution of the renormalized estimator for example let v0 denote the set of x on which given x x the label y 1 is more likely than the label y 1 if x is one dimensional the set v0 is the union of disjoint intervals the problem is then to estimate the thresholds of the intervals we obtain the asymptotic distribution of the empirical risk minimizer when the classifiers have k thresholds where k is fixed we furthermore consider an extension to higher dimensional x assuming basically that v0 has a smooth boundary in some given parametric class we also discuss various rates of convergence when the differentiability conditions are possibly violated here we again restrict ourselves to one dimensional x we show that the rate is n 1 in certain cases and then also obtain the asymptotic distribution for the empirical prediction error

learning by mirror averaging given a collection of m different estimators or classifiers we study the problem of model selection type aggregation i e we construct a new estimator or classifier called aggregate which is nearly as good as the best among them with respect to a given risk criterion we define our aggregate by a simple recursive procedure which solves an auxiliary stochastic linear programming problem related to the original non linear one and constitutes a special case of the mirror averaging algorithm we show that the aggregate satisfies sharp oracle inequalities under some general assumptions the results allow one to construct in an easy way sharp adaptive nonparametric estimators for several problems including regression classification and density estimation

concentration inequalities large and moderate deviations for self normalized empirical processes we consider the supremum w n of self normalized empirical processes indexed by unbounded classes of functions f such variables are of interest in various statistical applications for example the likelihood ratio tests of contamination using the herbst method we prove an exponential concentration inequality for w n under a second moment assumption on the envelope function of f this inequality is applied to obtain moderate deviations for w n we also provide large deviations results for some unbounded parametric classes f

rademacher penalties and structural risk minimization we suggest a penalty function to be used in various problems of structural risk minimization this penalty is data dependent and is based on the sup norm of the so called rademacher process indexed by the underlying class of functions sets the standard complexity penalties used in learning problems and based on the vc dimensions of the classes are conservative upper bounds in a probabilistic sense uniformly over the set of all underlying distributions for the penalty we suggest thus for a particular distribution of training examples one can expect better performance of learning algorithms with the data driven rademacher penalties we obtain oracle inequalities for the theoretical risk of estimators obtained by structural minimization of the empirical risk with rademacher penalties the inequalities imply some form of optimality of the empirical risk minimizers we also suggest an iterative approach to structural risk minimization with rademacher penalties in which the hierarchy of classes is not given in advance but is determined in the data driven iterative process of risk minimization we prove probabilistic oracle inequalities for the theoretical risk of the estimators based on this approach as well

gauge transformation in einstein yang mills theories we discuss the relation between space time diffeomorphisms and gauge transformations in theories of the yang mills type coupled with einstein s general relativity we show that local symmetries of the hamiltonian and lagrangian formalisms of these generally covariant gauge systems are equivalent when gauge transformations are required to induce transformations which are projectable under the legendre map although pure yang mills gauge transformations are projectable by themselves diffeomorphisms are not instead the projectable symmetry group arises from infinitesimal diffeomorphism inducing transformations which must depend on the lapse function and shift vector of the space time metric plus associated gauge transformations our results are generalizations of earlier results by ourselves and by salisbury and sundermeyer

aggregation via empirical risk minimization given a finite set f of estimators the problem of aggregation is to construct a new estimator whose risk is as close as possible to the risk of the best estimator in f it was conjectured that empirical minimization performed in the convex hull of f is an optimal aggregation method but we show that this conjecture is false despite that we prove that empirical minimization in the convex hull of a well chosen empirically determined subset of f is an optimal aggregation method

on the essential spectrum of certain non commutative oscillators we show here that the spectrum of the family of non commutative harmonic oscillators q w x d for r in the range 1 is 0 and is entirely essential spectrum the previous existing results concern the case 1 case in which q w x d is globally elliptic with a discrete spectrum whose qualitative properties are being extensively studied and ours therefore extend the picture to the range of parameters 1

equivalence classes of perturbations in cosmologies of bianchi types i and v interpretation in the preceding paper 26 gauge invariant variables were defined that characterize an almost bianchi type i or almost bianchi type v universe filled with a nonbarotropic perfect fluid one can think of these basic variables denoted collectively by d as having at least two aspects first d gives an explicit i e analytical representation of the equivalence class of perturbations in fact this equivalence class is uniquely determined from d and vice versa second any gauge invariant quantity with respect to a bianchi type i or bianchi type v background model is obtainable linearly from d through purely algebraic and differential operations among many other things the above properties of d facilitate new insights into the question of why a standard and well known formulation based on the stewart walker lemma does not enable one to describe the equivalence classes of perturbations and to find gauge invariant variables independent of d if we pose an analogous question in regard to almost robertson walker universe models a different but none the less instructive answer is obtained in this case the stewart walker lemma provides a complete framework both for constructing all gauge invariant variables and for determining the equivalence classes of perturbations because of the sometimes confusing statements in the literature nontrivial comparisons with other work on linear perturbations in anisotropic background models are made we also present what we believe are some interesting applications of our ideas to the subject of quantum field theory in curved space time

statistical performance of support vector machines the support vector machine svm algorithm is well known to the computer learning community for its very good practical results the goal of the present paper is to study this algorithm from a statistical perspective using tools of concentration theory and empirical processes our main result builds on the observation made by other authors that the svm can be viewed as a statistical regularization procedure from this point of view it can also be interpreted as a model selection principle using a penalized criterion it is then possible to adapt general methods related to model selection in this framework to study two important points 1 what is the minimum penalty and how does it compare to the penalty actually used in the svm algorithm 2 is it possible to obtain oracle inequalities in that setting for the specific loss function used in the svm algorithm we show that the answer to the latter question is positive and provides relevant insight to the former our result shows that it is possible to obtain fast rates of convergence for svms

optimal aggregation of classifiers in statistical learning classification can be considered as nonparametric estimation of sets where the risk is defined by means of a specific distance between sets associated with misclassification error it is shown that the rates of convergence of classifiers depend on two parameters the complexity of the class of candidate sets and the margin parameter the dependence is explicitly given indicating that optimal fast rates approaching o n 1 can be attained where n is the sample size and that the proposed classifiers have the property of robustness to the margin the main result of the paper concerns optimal aggregation of classifiers we suggest a classifier that automatically adapts both to the complexity and to the margin and attains the optimal fast rates up to a logarithmic factor

b r s algebras analysis of the consistency equations in gauge theory we compute all possible anomalous terms in quantum gauge theory in the natural class of polynomials of differential forms by using the appropriate cohomological and algebraic methods we do it for all dimensions of spacetime and all structure groups with reductive lie algebras

characteristic cohomology of p form gauge theories the characteristic cohomologyh k char d for an arbitrary set of freep form gauge fields is explicitly worked out in all form degreesk n 1 wheren is the spacetime dimension it is shown that this cohomology is finite dimensional and completely generated by the forms dual to the field strengths the gauge invariant characteristic cohomology is also computed the results are extended to interactingp form gauge theories with gauge invariant interactions implications for the brst cohomology are mentioned

surface charge algebra in gauge theories and thermodynamic integrability surface charges and their algebra in interacting lagrangian gauge field theories are constructed out of the underlying linearized theory using techniques from the variational calculus in the case of exact solutions and symmetries the surface charges are interpreted as a pfaff system integrability is governed by frobenius theorem and the charges associated with the derived symmetry algebra are shown to vanish in the asymptotic context we provide a generalized covariant derivation of the result that the representation of the asymptotic symmetry algebra through charges may be centrally extended comparison with hamiltonian and covariant phase space methods is made all approaches are shown to agree for exact solutions and symmetries while there are differences in the asymptotic context

spectral analysis of non commutative harmonic oscillators the lowest eigenvalue and no crossing abstract the lowest eigenvalue of non commutative harmonic oscillators q 0 0 1 is studied it is shown that q can be decomposed into four self adjoint operators q p 1 2 q p and all the eigenvalues of each operator q p are simple we show that the lowest eigenvalue of q is simple whenever furthermore a jacobi matrix representation of q p is given and spectrum of q p is considered numerically

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

homological perturbation theory and the algebraic structure of the antifield antibracket formalism for gauge theories the algebraic structure of the antifield antibracket formalism for both reducible and irreducible gauge theories is clarified this is done by using the methods of homological perturbation theory hpt a crucial ingredient of the construction is the koszul tate complex associated with the stationary surface of the classical extremals the koszul tate differential acts on the antifields and is graded by the antighost number it provides a resolution of the algebraa of functions defined on the stationary surface namely it is acyclic except at degree zero where its homology group reduces toa acyclicity only holds because of the introduction of the ghosts of ghosts and provides an alternative criterion for what is meant by a proper solution of the master equation the existence of the brst symmetry follows from the techniques of hpt the classical lagrangian brst cohomology is completely worked out and shown to be isomorphic with the cohomology of the exterior derivative along the gauge orbits on the stationary surface the algebraic structure of the formalism is identical with the structure of the hamiltonian brst construction the role played there by the constraint surface is played here by the stationary surface only elementary quantum questions general properties of the measure are addressed

svm soft margin classifiers linear programming versus quadratic programming support vector machine svm soft margin classifiers are important learning algorithms for classification problems they can be stated as convex optimization problems and are suitable for a large data setting linear programming svm classifiers are especially efficient for very large size samples but little is known about their convergence compared with the well understood quadratic programming svm classifier in this article we point out the difficulty and provide an error analysis our analysis shows that the convergence behavior of the linear programming svm is almost the same as that of the quadratic programming svm this is implemented by setting a stepping stone between the linear programming svm and the classical 1 norm soft margin classifier an upper bound for the misclassification error is presented for general probability distributions explicit learning rates are derived for deterministic and weakly separable distributions and for distributions satisfying some tsybakov noise condition

empirical margin distributions and bounding the generalization error of combined classifiers we prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifiers such combinations could be implemented by neural networks or by voting methods of combining the classifiers such as boosting and bagging the bounds are in terms of the empirical distribution of the margin of the combined classifier they are based on the methods of the theory of gaussian and empirical processes comparison inequalities symmetrization method concentration inequalities and they improve previous results of bartlett 1998 on bounding the generalization error of neural networks in terms of ell 1 norms of the weights of neurons and of schapire freund bartlett and lee 1998 on bounding the generalization error of boosting we also obtain rates of convergence in levy distance of empirical margin distribution to the true margin distribution uniformly over the classes of classifiers and prove the optimality of these rates

convexity classification and risk bounds many of the classification algorithms developed in the machine learning literature including the support vector machine and boosting can be viewed as minimum contrast methods that minimize a convex surrogate of the 0 1 loss function the convexity makes these algorithms computationally efficient the use of a surrogate however has statistical consequences that must be balanced against the computational virtues of convexity to study these issues we provide a general quantitative relationship between the risk as assessed using the 0 1 loss and the risk as assessed using any nonnegative surrogate loss function we show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function that it satisfies a pointwise form of fisher consistency for classification the relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications we also present a refined version of this result in the

lagrangian approach to the physical degree of freedom count in this paper we present a lagrangian method that allows the physical degree of freedom count for any lagrangian system without having to perform neither dirac nor covariant canonical analyses the essence of our method is to establish a map between the relevant lagrangian parameters of the current approach and the hamiltonian parameters that enter in the formula for the counting of the physical degrees of freedom as is given in dirac s method once the map is obtained the usual hamiltonian formula for the counting can be expressed in terms of lagrangian parameters only and therefore we can remain in the lagrangian side without having to go to the hamiltonian one using the map it is also possible to count the number of first and second class constraints within the lagrangian formalism only for the sake of completeness the geometric structure underlying the current approach developed for systems with a finite number of degrees of freedom is uncovered with the help of the covariant canonical formalism finally the method is illustrated in several examples including the relativistic free particle

selective rademacher penalization and reduced error pruning of decision trees rademacher penalization is a modern technique for obtaining data dependent bounds on the generalization error of classifiers it appears to be limited to relatively simple hypothesis classes because of computational complexity issues in this paper we nevertheless apply rademacher penalization to the in practice important hypothesis class of unrestricted decision trees by considering the prunings of a given decision tree rather than the tree growing phase this study constitutes the first application of rademacher penalization to hypothesis classes that have practical significance we present two variations of the approach one in which the hypothesis class consists of all prunings of the initial tree and another in which only the prunings that are accurate on growing data are taken into account moreover we generalize the error bounding approach from binary classification to multi class situations our empirical experiments indicate that the proposed new bounds outperform distribution independent bounds for decision tree prunings and provide non trivial error estimates on real world data sets

on the spectrum of certain non commutative harmonic oscillators and semiclassical analysis a localization and cardinality property along with a multiplicity result of the spectrum of certain 2 2 globally elliptic systems of ordinary differential operators a class of vector valued deformations of the classical harmonic oscillator called non commutative harmonic oscillators will be described here the basic tool is the study of a semiclassical reference system

spacetime locality of the brst formalism the spacetime locality of the brst formalism is investigated the analysis covers gauge theories with either closed or open algebras and is undertaken in the explicit context of the antifield formulation of the brst theory under appropriate conditions the homology of the koszul tate differential modulo the spacetime exterior derivative is shown to be trivial in the space of non integrated densities with positive antighost and pure ghost numbers as a result i the solution of the master equation can be taken to be a local functional ii the gauge fixed action is also a local functional provided one takes the gauge fixing fermion to be a local functional as well and iii the brst transformation is local

covariant differential identities and conservation laws in metric torsion theories of gravitation i general consideration arbitrary diffeomorphically invariant metric torsion theories of gravity are considered it is assumed that lagrangians of such theories contain derivatives of field variables tensor densities of arbitrary ranks and weights up to a second order only the generalized klein noether methods for constructing manifestly covariant identities and conserved quantities are developed manifestly covariant expressions are constructed without including auxiliary structures like a background metric in the riemann cartan space the following manifestly generally covariant results are presented a the complete generalized system of differential identities the klein noether identities is obtained b the generalized currents of three types depending on an arbitrary vector field displacements are constructed they are the canonical noether current symmetrized belinfante current and identically conserved hilbert bergmann current in particular it is stated that the symmetrized belinfante current does not depend

moment inequalities for functions of independent random variables a general method for obtaining moment inequalities for functions of independent random variables is presented it is a generalization of the entropy method which has been used to derive concentration inequalities for such functions boucheron lugosi and massart ann probab 31 2003 1583 1614 and is based on a generalized tensorization inequality due to latala and oleszkiewicz lecture notes in math 1745 2000 147 168 the new inequalities prove to be a versatile tool in a wide range of applications we illustrate the power of the method by showing how it can be used to effortlessly re derive classical inequalities including rosenthal and kahane khinchine type inequalities for sums of independent random variables moment inequalities for suprema of empirical processes and moment inequalities for rademacher chaos and u statistics some of these corollaries are apparently new in particular we generalize talagrand s exponential inequality for rademacher chaos of order 2 to any order we also discuss applications for other complex functions of independent random variables such as suprema of boolean polynomials which include as special cases subgraph counting problems in random graphs

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

existence uniqueness and cohomology of the classical brst charge with ghosts of ghosts a complete canonical formulation of the brst theory of systems with redundant gauge symmetries is presented these systems includep form gauge fields the superparticle and the superstring we first define the koszul tate differential and explicitly show how the introduction of the momenta conjugate to the ghosts of ghosts makes it acyclic the global existence of the brst generator is then demonstrated and the brst charge is proved to be unique up to canonical transformations in the extended phase space which includes the ghosts finally the brst cohomology in classical mechanics is investigated and shown to be equal to the cohomology of the exterior derivative along the gauge orbits as in the irreducible case this is done by re expressing the exterior algebra along the gauge orbits as a free differential algebra containing generators of higher degree which are identified with the ghosts of ghosts the quantum cohomology is not dealt with

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

on the performance of kernel classes we present sharp bounds on the localized rademacher averages of the unit ball in a reproducing kernel hilbert space in terms of the eigenvalues of the integral operator associated with the kernel we use this result to estimate the performance of the empirical minimization algorithm when the base class is the unit ball of the reproducing kernel hilbert space

local brst cohomology in the antifield formalism ii application to yang mills theory yang mills models with compact gauge group coupled to matter fields are considered the general tools developed in a companion paper are applied to compute the local cohomology of the brst differentials modulo the exterior space time derivatived for all values of the ghost number in the space of polynomials in the fields the ghosts the antifields sources for the brst variations and their derivatives new solutions to the consistency conditionssa db 0 depending non trivially on the antifields are exhibited for a semi simple gauge group however these new solutions arise only at ghost number two or higher thus at ghost number zero or one the inclusion of the antifields does not bring in new solutions to the consistency conditionsa db 0 besides the already known ones the analysis does not use power counting and is purely cohomological it can be easily extended to more general actions containing higher derivatives of the curvature or chern simons terms

natural symmetries of the yang mills equations a natural generalized symmetry of the yang mills equations is defined as an infinitesimal transformation of the yang mills field built in a local gauge invariant and poincare invariant fashion from the yang mills field strength and its derivatives to any order which maps solutions of the field equations to other solutions on the jet bundle of yang mills connections a spinorial coordinate system is introduced that is adapted to the solution subspace defined by the yang mills equations in terms of this coordinate system the complete classification of natural symmetries is carried out in a straightforward manner it is found that all natural symmetries of the yang mills equations stem from the gauge transformations admitted by the equations

noether currents and charges for maxwell like lagrangians the hilbert noether theorem states that a current associated with diffeomorphism invariance of a lagrangian vanishes on shell modulo a divergence of an arbitrary superpotential application of the noether procedure to physical lagrangians yields however meaningful and measurable currents the well known solution to this paradox is to involve the variation of the metric tensor such procedure for the field considered on a fixed flat background is sophisticated logically one needs to introduce the variation of a fixed field and formal we analyse the noether procedure for a generic diffeomorphism invariant p form field model we show that the noether current of the field considered on a variable background coincides with the current treated in a fixed geometry consistent description of the canonical energy momentum current is possible only if the dynamics of the geometry gravitation is taken into account however even the truncated consideration yields the proper expression we examine the examples of the free p form gauge field theory the gr in the coframe representation and the metric free electrodynamics although the variation of a metric tensor is not acceptable in the latter case the noether procedure yields the proper result

general quadratic gauge theory constraint structure symmetries and physical functions how can we relate the constraint structure and constraint dynamics of the general gauge theory in the hamiltonian formulation to specific features of the theory in the lagrangian formulation especially relate the constraint structure to the gauge transformation structure of the lagrangian action how can we construct the general expression for the gauge charge if the constraint structure in the hamiltonian formulation is known whether we can identify the physical functions defined as commuting with first class constraints in the hamiltonian formulation and the physical functions defined as gauge invariant functions in the lagrangian formulation the aim of the present paper is to consider the general quadratic gauge theory and to answer the above questions for such a theory in terms of strict assertions to fulfil such a programme we demonstrate the existence of the so called superspecial phase space variables in terms of which the quadratic hamiltonian action takes a simple canonical form on the basis of such a representation we analyse a functional arbitrariness in the solutions of the equations of motion of the quadratic gauge theory and derive the general structure of symmetries by analysing a symmetry equation we then use these results to identify the two definitions of physical functions and thus prove the dirac conjecture

local brst cohomology in the antifield formalism i general theorems we establish general theorems on the cohomologyh s d of the brst differential modulo the spacetime exterior derivative acting in the algebra of localp forms depending on the fields and the antifields sources for the brst variations it is shown thath k s d is isomorphic toh k d in negative ghost degree k k 0 where is the koszul tate differential associated with the stationary surface the cohomology grouph1 d in form degreen is proved to be isomorphic to the space of constants of the motion thereby providing a cohomological reformulation of noether s theorem more generally the grouph k d in form degreen is isomorphic to the space ofn k forms that are closed when the equations of motion hold the groupsh k d k 2 are shown to vanish for standard irreducible gauge theories the grouph2 d is then calculated explicitly for electromagnetism yang mills models and einstein gravity the invariance of the groupsh k s d under the introduction of non minimal variables and of auxiliary fields is also demonstrated in a companion paper the general formalism is applied to the calculation ofh k s d in yang mills theory which is carried out in detail for an arbitrary compact gauge group

presymplectic current and the inverse problem of the calculus of variations the inverse problem of the calculus of variations asks whether a given system of partial differential equations pdes admits a variational formulation we show that the existence of a presymplectic form in the variational bicomplex when horizontally closed on solutions allows us to construct a variational formulation for a subsystem of the given pde no constraints on the differential order or number of dependent or independent variables are assumed the proof follows a recent observation of bridges hydon and lawson math proc cambridge philos soc 148 01 159 178 2010 and generalizes an older result of henneaux ann phys 140 1 45 64 1982 from ordinary differential equations odes to pdes uniqueness of the variational formulation is also discussed

clustering for the spectrum ofh pseudodifferential operators with periodic flow on an energy surface abstract we study the repartition of the eigenvalues of anh pseudodifferential operator in a small neighbourhood of an energy where the hamiltonian flow associated with the principal symbol of the operator is periodic the proof is based on a geometric lemma and perturbation techniques to obtain the results from the known theorem in the case of a periodic flow in an energy band b helffer and d robert ann inst h poincare41 1984 291 331

non commutative harmonic oscillators and fuchsian ordinary differential operators the spectral problem for non commutative harmonic oscillators is shown to be equivalent to solve fuchsian ordinary differential equations with four regular singular points in a complex domain

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

calcul fonctionnel sur les operateurs admissibles et application abstract let h v h h 2 2 v lim x v x under suitable conditions we prove that h v h is a pseudodifferential operator whose symbol has an asymptotic a h j 0 h j a j more general pseudodifferential operator s classes are also considered we apply this result to study the semi classical behaviour of the spectrum of h v as h 0 so we improve recent results obtained by j chazarain and by the author in collaboration with b helffer furthermore we give a precise meaning to the formal development considered in b grammaticos and a voros work ann physics 123 1979 359 380

concentration inequalities using the entropy method we investigate a new methodology worked out by ledoux and massart to prove concentration of measure inequalities the method is based on certain modified logarithmic sobolev inequalities we provide some very simple and general ready to use inequalities one of these inequalities may be considered as an exponential version of the efron stein inequality the main purpose of this paper is to point out the simplicity and the generality of the approach we show how the new method can recover many of talagrand s revolutionary inequalities and provide new applications in a variety of problems including rademacher averages rademacher chaos the number of certain small subgraphs in a random graph and the minimum of the empirical risk in some statistical estimation problems

local rademacher complexities and oracle inequalities in risk minimization let be a class of measurable functions f s 0 1 defined on a probability space s mathcal a p given a sample x1 xn of i i d random variables taking values in s with common distribution p let pn denote the empirical measure based on x1 xn we study an empirical risk minimization problem pnf min f given a solution fn of this problem the goal is to obtain very general upper bounds on its excess risk mathcal e p hat f n p hat f n inf f in mathcal f pf expressed in terms of relevant geometric parameters of the class using concentration inequalities and other empirical processes tools we obtain both distribution dependent and data dependent upper bounds on the excess risk that are of asymptotically correct order in many examples the bounds involve localized sup norms of empirical and rademacher processes indexed by functions from the class we use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification

greedy function approximation a gradient boosting machine function estimation approximation is viewed from the perspective of numerical optimization in function space rather than parameter space a connection is made between stagewise additive expansions and steepest descent minimization a general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion specific algorithms are presented for least squares least absolute deviation and huber m loss functions for regression and multiclass logistic likelihood for classification special enhancements are derived for the particular case where the individual additive components are regression trees and tools for interpreting such treeboost models are presented gradient boosting of regression trees produces competitive highly robust interpretable procedures for both regression and classification especially appropriate for mining less than clean data connections between this approach and the boosting methods of freund and shapire and friedman hastie and tibshirani are discussed

local symmetries and constraints the general relationship between local symmetries occurring in a lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied first a prescription applicable to an arbitrary lagrangian field theory for the construction of phase space from the manifold of field configurations on space time is given next a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses as special cases the usual gauge transformations of yang mills theory and general relativity local symmetries on phase space are then defined via projection from field configuration space it is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space moreover the constraints thereby obtained are always first class and the poisson bracket algebra of the constraint functions is isomorphic to the lie bracket algebra of the local symmetries on the constraint submanifold of phase space the differences that occur in the structure of constraints in yang mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space in yang mills theory all the field independent local symmetries project to all of phase space whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are field dependent as by products of the present work definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary lagrangian field theory and the noether current density associated with an arbitrary local symmetry a number of properties of these currents are established and some relationships between them are obtained

boosting method for local learning in statistical pattern recognition we propose a local boosting method in classification problems borrowing from an idea of the local likelihood method our proposal local boosting includes a simple device for localization for computational feasibility we proved the bayes risk consistency of the local boosting in the framework of probably approximately correct learning inspection of the proof provides a useful viewpoint for comparing ordinary boosting and local boosting with respect to the estimation error and the approximation error both boosting methods have the bayes risk consistency if their approximation errors decrease to zero compared to ordinary boosting local boosting may perform better by controlling the trade off between the estimation error and the approximation error ordinary boosting with complicated base classifiers or other strong classification methods including kernel machines may have classification performance comparable to local boosting with simple base classifiers for example decision stumps local boosting however has an advantage with respect to interpretability local boosting with simple base classifiers offers a simple way to specify which features are informative and how their values contribute to a classification rule even though locally several numerical studies on real data sets confirm these advantages of local boosting

the quantum rabi model and lie algebra representations of mathfrak s mathfrak l _ 2 the aim of the present paper is to understand the spectral problem of the quantum rabi model in terms of lie algebra representations of we define a second order element of the universal enveloping algebra of which through the image of a principal series representation of provides a picture equivalent to the quantum rabi model drawn by confluent heun differential equations by this description in particular we give a representation theoretic interpretation of the degenerate part of the spectrum i e judd s eigenstates of the rabi hamiltonian due to ku in 1985 which is a part of the exceptional spectrum parameterized by integers we also discuss the non degenerate part of the exceptional spectrum of the model in addition to the judd eigenstates from a viewpoint of infinite dimensional irreducible submodules or subquotients of the non unitary principal series such as holomorphic discrete series representations of

reduced phase space quotienting procedure for gauge theories we present a reduction procedure for gauge theories based on quotienting out the kernel of the presymplectic form in configuration velocity space local expressions for a basis of this kernel are obtained using phase space procedures the obstructions to the formulation of the dynamics in the reduced phase space are identified and circumvented we show that this reduction procedure is equivalent to the standard dirac method as long as the dirac conjecture holds that the dirac hamiltonian containing the primary first class constraints with their lagrange multipliers can be enlarged to an extended dirac hamiltonian which includes all first class constraints without any change of the dynamics the quotienting procedure is always equivalent to the extended dirac theory even when it differs from the standard dirac theory the differences occur when there are ineffective constraints and in these situations we conclude that the standard dirac method is preferable at least for classical theories an example is given to illustrate these features as well as the possibility of having phase space formulations with an odd number of physical degrees of freedom

inegalites de concentration pour les processus empiriques de classes de parties we propose new concentration inequalities for maxima of set indexed empirical processes our approach is based either on entropy inequalities or on martingale methods the improvements we get concern the rate function which is exactly the large deviations rate function of a binomial law in most of the cases

covariant hamiltonian boundary conditions in general relativity for spatially bounded space time regions we investigate the covariant hamiltonian symplectic structure of general relativity for spatially bounded regions of space time with a fixed time flow vector for existence of a well defined hamiltonian variational principle taking into account a spatial boundary it is necessary to modify the standard arnowitt deser misner hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field the most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obtained from the symplectic current pulled back to the hypersurface boundary of the space time region a main result is that we obtain a covariant derivation of dirichlet neumann and mixed type boundary conditions on the gravitational field at a fixed boundary hypersurface together with the associated hamiltonian boundary terms as well we establish uniqueness of these boundary conditions under certain assumptions motivated by the form of the symplectic current our analysis uses a noether charge method which extends and unifies several results developed in recent literature for general relativity as an illustration of the method we apply it to the maxwell field equations to derive allowed boundary conditions and boundary terms for the existence of a well defined hamiltonian variational principle for an electromagnetic field in a fixed spatially bounded region of minkowski space time

adaptive model selection using empirical complexities given n independent replicates of a jointly distributed pair x y rd r we wish to select from a fixed sequence of model classes f1 f2 a deterministic prediction rule f rd r whose risk is small we investigate the possibility of empirically assessing the complexity of each model class that is the actual difficulty of the estimation problem within each class the estimated complexities are in turn used to define an adaptive model selection procedure which is based on complexity penalized empirical risk the available data are divided into two parts the first is used to form an empirical cover of each model class and the second is used to select a candidate rule from each cover based on empirical risk the covering radii are determined empirically to optimize a tight upper bound on the estimation error an estimate is chosen from the list of candidates in order to minimize the sum of class complexity and empirical risk a distinguishing feature of the approach is that the complexity of each model class is assessed empirically based on the size of its empirical cover finite sample performance bounds are established for the estimates and these bounds are applied to several non parametric estimation problems the estimates are shown to achieve a favorable tradeoff between approximation and estimation error and to perform as well as if the distribution dependent complexities of the model classes were known beforehand in addition it is shown that the estimate can be consistent and even possess near optimal rates of convergence when each model class has an infinite vc or pseudo dimension for regression estimation with squared loss we modify our estimate to achieve a faster rate of convergence appears in annals of statistics vol 27 1830 1864 ams 1991 subject classifications primary 62g07 62g20 secondary 62h30

simultaneous adaptation to the margin and to complexity in classification we consider the problem of adaptation to the margin and to complexity in binary classification we suggest a learning method with a numerically easy aggregation step adaptivity both to the margin and complexity in classification usually involves empirical risk minimization or rademacher complexities which lead to numerical difficulties on the other hand there exist classifiers that are easy to compute and that converge with fast rates but are not adaptive combining these classifiers by our aggregation procedure we get numerically realizable adaptive classifiers that converge with fast rates

complexity regularization via localized random penalties in this article model selection via penalized empirical loss minimization in nonparametric classification problems is studied data dependent penalties are constructed which are based on estimates of the complexity of a small subclass of each model class containing only those functions with small empirical loss the penalties are novel since those considered in the literature are typically based on the entire model class oracle inequalities using these penalties are established and the advantage of the new penalties over those based on the complexity of the whole model class is demonstrated

a high dimensional wilks phenomenon a theorem by wilks asserts that in smooth parametric density estimation the difference between the maximum likelihood and the likelihood of the sampling distribution converges toward a chi square distribution where the number of degrees of freedom coincides with the model dimension this observation is at the core of some goodness of fit testing procedures and of some classical model selection methods this paper describes a non asymptotic version of the wilks phenomenon in bounded contrast optimization procedures using concentration inequalities for general functions of independent random variables it proves that in bounded contrast minimization as for example in statistical learning theory the difference between the empirical risk of the minimizer of the true risk in the model and the minimum of the empirical risk the excess empirical risk satisfies a bernstein like inequality where the variance term reflects the dimension of the model and the scale term reflects the noise conditions from a mathematical statistics viewpoint the significance of this result comes from the recent observation that when using model selection via penalization the excess empirical risk represents a minimum penalty if non asymptotic guarantees concerning prediction error are to be provided from the perspective of empirical process theory this paper describes a concentration inequality for the supremum of a bounded non centered actually non positive empirical process combining the now classical analysis of m estimation building on talagrand s inequality for suprema of empirical processes and versatile moment inequalities for functions of independent random variables this paper develops a genuine bernstein like inequality that seems beyond the reach of traditional tools

local rademacher complexities we propose new bounds on the error of learning algorithms in terms of a data dependent notion of complexity the estimates we establish give optimal rates and are based on a local and empirical version of rademacher averages in the sense that the rademacher averages are computed from the data on a subset of functions with small empirical error we present some applications to classification and prediction with convex function classes and with kernel classes in particular