empirical and poisson processes on classes of sets or functions too large for central limit theorems let p be the uniform probability law on the unit cube i d in d dimensions and p n the corresponding empirical measure for various classes of sets a i d upper and lower bounds are found for the probable size of sup p n p a a if is the collection of lower layers in i 2 or of convex sets in i 3 an asymptotic lower bound is left left log n right mathord left vphantom left log n right n right n 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right left log log n right delta 1 mathord left vphantom 1 2 right 2 quad rm for rm any delta 0 thus the law of the iterated logarithm fails for these classes if 0 is the greatest integer 0 when d 1 the same lower bound is obtained as for the lower layers in i 2 or convex sets in i 3 for 0 alpha underline le d 1 there is also an upper bound equal to a power of log n times the lower bound so the powers of n are sharp

the relative value of labeled and unlabeled samples in pattern recognition with an unknown mixing parameter we observe a training set q composed of l labeled samples x sub 1 spl theta sub 1 x sub l spl theta sub l and u unlabeled samples x sub 1 x sub u the labels spl theta sub i are independent random variables satisfying pr spl theta sub i 1 spl eta pr spl theta sub i 2 1 spl eta the labeled observations x sub i are independently distributed with conditional density f sub spl theta i spl middot given spl theta sub i let x sub 0 spl theta sub 0 be a new sample independently distributed as the samples in the training set we observe x sub 0 and we wish to infer the classification spl theta sub 0 in this paper we first assume that the distributions f sub 1 spl middot and f sub 2 spl middot are given and that the mixing parameter is unknown we show that the relative value of labeled and unlabeled samples in reducing the risk of optimal classifiers is the ratio of the fisher informations they carry about the parameter spl eta we then assume that two densities g sub 1 spl middot and g sub 2 spl middot are given but we do not know whether g sub 1 spl middot f sub 1 spl middot and g sub 2 spl middot f sub 2 spl middot or if the opposite holds nor do we know spl eta thus the learning problem consists of both estimating the optimum partition of the observation space and assigning the classifications to the decision regions here we show that labeled samples are necessary to construct a classification rule and that they are exponentially more valuable than unlabeled samples

learning by canonical smooth estimation i simultaneous estimation this paper examines the problem of learning from examples in a framework that is based on but more general than valiant s probably approximately correct pac model for learning in our framework the learner observes examples that consist of sample points drawn and labeled according to a fixed unknown probability distribution based on this empirical data the learner must select from a set of candidate functions a particular function or hypothesis that will accurately predict the labels of future sample points the expected mismatch between a hypothesis prediction and the label of a new sample point is called the hypothesis generalization error following the pioneering work of vapnik and chervonenkis others have attacked this sort of learning problem by finding hypotheses that minimize the relative frequency based empirical error estimate we generalize this approach by examining the simultaneous estimation problem we demonstrate how one can learn from such a simultaneous error estimate and propose a new class of estimators called smooth estimators we characterize the class of simultaneous estimation problems solvable by a smooth estimator

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

on exponential bounds on the bayes risk of the kernel classification rule the exponential distribution free bounds for the kernel classification rule are derived the equivalence of all modes of the global convergence of the rule is established under optimal assumptions on the smoothing sequence also derived is the optimal global rate of convergence of the kernel regression estimate within the class of lipschitz distributions the rate is optimal for the nonparametric regression but not for classifications it is shown using the martingale device that weak strong and complete l sub 1 bayes risk consistencies are equivalent consequently the conditions on the smoothing sequence h sub n to 0 and nh sub n to infinity are necessary and sufficient for bayes risk consistency of the kernel classification rule the rate of convergence of the kernel classification rule is also given

minimax nonparametric classification ii model selection for adaptation for pt i see ibid vol 45 no 7 p 2271 84 1999 we study nonparametric estimation of a conditional probability for classification based on a collection of finite dimensional models for the sake of flexibility different types of models linear or nonlinear are allowed as long as each satisfies a dimensionality assumption we show that with a suitable model selection criterion the penalized maximum likelihood estimator has a risk bounded by an index of resolvability expressing a good tradeoff among approximation error estimation error and model complexity the bound does not require any assumption on the target conditional probability and can be used to demonstrate the adaptivity of estimators based on model selection examples are given with both splines and neural nets and problems of high dimensional estimation are considered the resulting adaptive estimator is shown to behave optimally or near optimally over sobolev classes with unknown orders of interaction and smoothness and classes of integrable fourier transform of gradient in terms of rates of convergence the performance is the same as if one knew which of them contains the true conditional probability in advance the corresponding classifier also converges optimally or nearly optimally simultaneously over these classes

an optimal global nearest neighbor metric a quadratic metric dao x y x y t ao x y is proposed which minimizes the mean squared error between the nearest neighbor asymptotic risk and the finite sample risk under linearity assumptions a heuristic argument is given which indicates that this metric produces lower mean squared error than the euclidean metric a nonparametric estimate of ao is developed if samples appear to come from a gaussian mixture an alternative parametrically directed distance measure is suggested for nearness decisions within a limited region of space examples of some two class gaussian mixture distributions are included

correlations equalities and some upper bounds for the critical temperature for spin one systems abstract starting from correlation identities for the blume capel spin 1 systems and using correlation inequalities we obtain rigorous upper bounds for the critical temperature the obtained results improve over effective field type results

probably approximately correct learning in fuzzy classification systems an efficient method for learning trapezoidal membership functions for fuzzy predicates is presented positive and negative examples of one class are given together with a system of classification rules the learned membership functions can be used for the fuzzy predicates occurring in the given rules to classify further examples we show that the obtained classification is approximately correct with high probability this justifies the obtained fuzzy sets within one particular classification problem instead of relying on a subjective meaning of fuzzy predicates as normally done by a domain expert

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

monte carlo study of the triangular blume capel model under bond randomness the effects of bond randomness on the universality aspects of a two dimensional d 2 blume capel model embedded in the triangular lattice are discussed the system is studied numerically in both its first and second order phase transition regimes by a comprehensive finite size scaling analysis for a particularly suitable value of the disorder strength we find that our data for the second order phase transition emerging under random bonds from the second order regime of the pure model are compatible with the universality class of the two dimensional 2d random ising model furthermore we find evidence that the second order transition emerging under bond randomness from the first order regime of the pure model belongs again to the same universality class although the first finding reinforces the scenario of strong universality in the 2d ising model with quenched disorder the second is in difference from the critical behavior emerging under randomness in the cases of the ex first order transitions of the potts model finally our results verify previous renormalization group calculations on the blume capel model with disorder in the crystal field coupling

smooth discrimination analysis ties f and g can be based on the estimation of the set g x fx gx we consider applications where it is appropriate to assume that the region g has a smooth boundary or belongs to another nonparametric class of sets in particular this assumption makes sense if discrimination is used as a data analytic tool decision rules based on minimization of empirical risk over the whole class of sets and over sieves are considered their rates of convergence are obtained we show that these rules achieve optimal rates for estimation of g and optimal rates of convergence for bayes risks an interesting conclusion is that the optimal rates for bayes risks can be very fast in particular faster than the parametric root n rate these fast rates cannot be guaranteed for plug in rules

the optimal distance measure for nearest neighbor classification a local distance measure is shown to optimize the performance of the nearest neighbor two class classifier for a finite number of samples the difference between the finite sample error and the asymptotic error is used as the criterion of improvement this new distance measure is compared to the well known euclidean distance an algorithm for practical implementation is introduced this algorithm is shown to be computationally competitive with the present nearest neighbor procedures and is illustrated experimentally a closed form for the corresponding second order moment of this criterion is found finally the above results are extended to

consistency of data driven histogram methods for density estimation and classification we present general sufficient conditions for the almost surel1 consistency of histogram density estimates based on data dependent partitions analogous conditions guarantee the almost sure risk consistency of histogram classification schemes based on data dependent partitions multivariate data is considered throughout in each case the desired consistency requires shrinking cells subexponential growth of a combinatorial complexity measure and sub linear growth of the number of cells it is not required that the cells of every partition be rectangles with sides paralles to the coordinate axis or that each cell contain a minimum number of points no assumptions are made concerning the common distribution of the training vectors we apply the results to establish the consistency of several known partitioning estimates including the kn spacing density estimate classifiers based on statistically equivalent blocks and classifiers based on multivariate clustering schemes

on the minimum probability of error of classification with incomplete patterns abstract some expected features of sample patterns in a classification system may be missing immeasurable or obscured when a sample pattern with such an incomplete set of features is presented to a classifier how reliable is the decision thus made by the classifier what is the effect of those absent features to the accuracy and correctness of the classification how to evaluate or estimate the probability of error of such a decision compared to that made on the use of more features or a supposedly complete set of features we derive the solutions of above problems from a point of view of bayes classifier the traditional decision theory is extended to the classifications with incomplete feature sets it shows that in case of classifying incomplete patterns the bayes classifier constructed on the marginal distributions of presented features also minimizes the probability of error we then use bhattacharyya bound to obtain an analytic form for estimating the error probabilities of classifications made on the partially available features with respect to that using more or a complete set of features directions are given to decision making on problems of in which situation the effect of missing or obscured features can be neglected or must be seriously considered experimental results have shown consistence with the theoretical analysis

the random field ising model with an asymmetric and anisotropic bimodal probability distribution abstract the ising model in the presence of a random field drawn from the asymmetric and anisotropic trimodal probability distribution p h i p h i h 0 q h i h 0 r h i is investigated the partial probabilities p q r take on values within the interval 0 1 consistent with the constraint p q r 1 asymmetric distribution h i is the random field variable with basic absolute value h 0 strength is the competition parameter which is the ratio between the respective strength of the random magnetic field in the two principal directions z and z and is positive so that the random fields are competing anisotropic distribution this probability distribution is an extension of the bimodal one allowing for the existence in the lattice of non magnetic particles or vacant sites the current random field ising system displays mainly second order phase transitions which for some values of p q and h 0 are followed by first order phase transitions joined smoothly by a tricritical point occasionally two tricritical points appear implying another second order phase transition in addition to these points re entrant phenomena can be seen for appropriate ranges of the temperature and random field for specific values of p and q using the variational principle we write down the equilibrium equation for the magnetization and solve it for both phase transitions and at the tricritical point in order to determine the magnetization profile with respect to h 0 considered as an independent variable in addition to the temperature

global convergence of the recursive kernel regression estimates with applications in classification and nonlinear system estimation an improved exponential bound on the l sub 1 error for the recursive kernel regression estimates is derived it is shown using the martingale device that weak strong and complete l sub 1 consistencies are equivalent consequently the conditions on a certain smoothing sequence are necessary and sufficient for strong l sub 1 consistency of the recursive kernel regression estimate the rates of global convergence are also given obtained results are applied to recursive classification rules and to nonlinear time series estimation

nonparametric estimation and classification using radial basis function nets and empirical risk minimization studies convergence properties of radial basis function rbf networks for a large class of basis functions and reviews the methods and results related to this topic the authors obtain the network parameters through empirical risk minimization the authors show the optimal nets to be consistent in the problem of nonlinear function approximation and in nonparametric classification for the classification problem the authors consider two approaches the selection of the rbf classifier via nonlinear function estimation and the direct method of minimizing the empirical error probability the tools used in the analysis include distribution free nonasymptotic probability inequalities and covering numbers for classes of functions

asymptotic efficiency of classifying procedures using the hermite series estimate of multivariate probability densities corresp pattern recognition procedures derived from a nonparametric estimate of multivariate probability density functions using the orthogonal hermite system are examined for sufficiently regular densities the convergence rate of the mean integrated square error mise is o n l epsilon epsilon 0 where n is the number of observations and is independent of the dimension as a consequence the rate at which the probability of misclassification converges to the bayes probability of error as the length n of the learning sequence tends to infinity is also independent of the dimension of the class densities and equals o n 1 2 delta delta o

learning by canonical smooth estimation ii learning and choice of model complexity this paper analyzes the properties of a procedure for learning from examples this canonical learner is based on a canonical error estimator developed in part i in learning problems one can observe data that consists of labeled sample points and the goal is to find a model or hypothesis from a set of candidates that will accurately predict the labels of new sample points the expected mismatch between a hypothesis prediction and the actual label of a new sample point is called the hypothesis generalization error we compare the canonical learner with the traditional technique of finding hypotheses that minimize the relative frequency based empirical error estimate it is shown that for a broad class of learning problems the set of cases for which such empirical error minimization works is a proper subset of the cases for which the canonical learner works we derive bounds to show that the number of samples required by these two methods is comparable we also address the issue of how to determine the appropriate complexity for the class of candidate hypotheses

structure constants related to symmetric hopf algebras abstract let k be an algebraically closed field of characteristic 0 in this paper we continue our study of structure constants for semisimple hopf algebras h whose character algebra is commutative and for non semisimple factorizable ribbon hopf algebras this is done from the point of view of symmetric algebras such as group algebras in particular we consider general fusion rules which are structure constants associated to products of irreducible characters and structure constants associated to generalizations of class sums and conjugacy classes our methods are reminiscent on one hand of the methods employed when analyzing tilting modules for certain quantum groups and on the other hand of splitting modules for certain drinfeld doubles the family of irreducible characters is divided in two according to the vanishing of their quantum dimension the fusion rules on the character algebra are computed with respect to this division

automatic classification of cervical cells using a binary tree classifier abstract a k means clustering algorithm for designing binary tree classifiers is introduced for the classification of cervical cells at each nonterminal node of the designed binary tree classifier two sets of effective feature are selected one is based on the bhattacharyya distance a measure of separability between two classes the other is based on the merits of classification accuracy the classification result has shown the effectiveness of the features and the binary tree classifier used

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

solvability for semisimple hopf algebras via integrals abstract we use integrals of left coideal subalgebras to develop harmonic analysis for semisimple hopf algebras we show how n the space of functional on n is embedded in h we define a bilinear form on n and show that irreducible n characters are orthogonal with respect to that form we then give an explicit formula for induced characters of n and show how the induced characters are embedded in r h in the second part we give an intrinsic definition for solvable semisimple hopf algebras via left coideal subalgebras and their integrals we show how this definition generalizes solvability for finite groups in particular commutative and nilpotent hopf algebras are solvable we finally prove an analogue of burnside theorem a semisimple quasitriangular hopf algebras of dimension p a q b is solvable

combinations of weak classifiers to obtain classification systems with both good generalization performance and efficiency in space and time we propose a learning method based on combinations of weak classifiers where weak classifiers are linear classifiers perceptrons which can do a little better than making random guesses a randomized algorithm is proposed to find the weak classifiers they are then combined through a majority vote as demonstrated through systematic experiments the method developed is able to obtain combinations of weak classifiers with good generalization performance and a fast training time on a variety of test problems and real applications theoretical analysis on one of the test problems investigated in our experiments provides insights on when and why the proposed method works in particular when the strength of weak classifiers is properly chosen combinations of weak classifiers can achieve a good generalization performance with polynomial space and time complexity

the optimal strategy of a tree classifier abstract this paper deals with the decision rules of a tree classifier for performing the classification at each nonterminal node under the assumption of complete probabilistic information for given tree structure and feature subsets to be used the optimal decision rules strategy are derived which minimize the overall probability of misclassification the primary result is illustrated by an example

on nonparametric identification of wiener systems in this paper a new method for the identification of the wiener nonlinear system is proposed the system being a cascade connection of a linear dynamic subsystem and a nonlinear memoryless element is identified by a two step semiparametric approach the impulse response function of the linear part is identified via the nonlinear least squares approach with the system nonlinearity estimated by a pilot nonparametric kernel regression estimate the obtained estimate of the linear part is then used to form a nonparametric kernel estimate of the nonlinear element of the wiener system the proposed method permits recovery of a wide class of nonlinearities which need not be invertible as a result the proposed algorithm is computationally very efficient since it does not require a numerical procedure to calculate the inverse of the estimate furthermore our approach allows non gaussian input signals and the presence of additive measurement noise however only linear systems with a finite memory are admissible the conditions for the convergence of the proposed estimates are given computer simulations are included to verify the basic theory

position space renormalization group study of the spin 1 random semi infinite blume capel model abstract we study the spin 1 blume capel model under a random crystal field in the tridimensional semi infinite case this has been done by using the real space renormalization group approximation and specifically the migdal kadanoff technique interesting results are obtained which tell us that the randomness destroys the first order phase transitions and only those of the second order occur we give the list of nine fixed points and their topology describing the surface critical behavior five new types of phase diagram are found with a rich variety of phase transitions in accordance with the values of the bulk and surface probabilities and the ratios linking bulk and surface interactions

machine learning with data dependent hypothesis classes we extend the vc theory of statistical learning to data dependent spaces of classifiers this theory can be viewed as a decomposition of classifier design into two components the first component is a restriction to a data dependent hypothesis class and the second is empirical risk minimization within that class we define a measure of complexity for data dependent hypothesis classes and provide data dependent versions of bounds on error deviance and estimation error we also provide a structural risk minimization procedure over data dependent hierarchies and prove consistency we use this theory to provide a framework for studying the trade offs between performance and computational complexity in classifier design as a consequence we obtain a new family of classifiers with dimension independent performance bounds and efficient learning procedures

optimal bandwidth selection in nonparametric regression function estimation kernel estimators of an unknown multivariate regression function are investigated a bandwidth selection rule is considered which can be formulated in terms of cross validation under mild assumptions on the kernel and the unknown regression function it is seen that this rule is asymptotically optimal

the two and three dimensional spin 3 2 random blume capel model by the position space renormalization group we use the migdal kadanoff renormalization group technique to study the spin 3 2 blume capel model under a random crystal field in the two and three dimensional cases studying the fixed points and the phase diagrams established we find interesting results allowing us to understand the critical behavior of the system in the two dimensional case the randomness even in small amounts removes completely the first order transition between the two ferromagnetic phases present replacing it by a smooth continuation only the second order phase transitions occur in the three dimensional case the first order phase transition disappears only at a certain threshold of randomness below this threshold we observe the presence of an end point where finishes the first order transition line inside the ferromagnetic phases this end point reaches t 0k at a critical value of probability beyond which only the second order transitions occur

on an asymptotically optimal adaptive classifier design criterion a new approach for estimating classification errors is presented in the model there are two types of classification error empirical and generalization error the first is the error observed over the training samples and the second is the discrepancy between the error probability and empirical error in this research the vapnik and chervonenkis dimension vcdim is used as a measure for classifier complexity based on this complexity measure an estimate for generalization error is developed an optimal classifier design criterion the generalized minimum empirical error criterion gmee is used the gmee criterion consists of two terms the empirical and the estimate of generalization error as an application the criterion is used to design the optimal neural network classifier a corollary to the gamma optimality of neural network based classifiers is proven thus the approach provides a theoretic foundation for the connectionist approach to optimal classifier design experimental results to validate this approach

estimation of a regression function by maxima of minima of linear functions in this paper estimation of a regression function from independent and identically distributed random variables is considered estimates are defined by minimization of the empirical l2 risk over a class of functions which are defined as maxima of minima of linear functions results concerning the rate of convergence of the estimates are derived in particular it is shown that for smooth regression functions satisfying the assumption of single index models the estimate is able to achieve up to some logarithmic factor the corresponding optimal one dimensional rate of convergence hence under these assumptions the estimate is able to circumvent the so called curse of dimensionality the small sample behavior of the estimates is illustrated by applying them to simulated data

automatic pattern recognition a study of the probability of error a test sequence is used to select the best rule from a class of discrimination rules defined in terms of the training sequence the vapnik chervonenkis and related inequalities are used to obtain distribution free bounds on the difference between the probability of error of the selected rule and the probability of error of the best rule in the given class the bounds are used to prove the consistency and asymptotic optimality for several popular classes including linear discriminators nearest neighbor rules kernel based rules histogram rules binary tree classifiers and fourier series classifiers in particular the method can be used to choose the smoothing parameter in kernel based rules to choose k in the k nearest neighbor rule and to choose between parametric and nonparametric rules

are we counting or measuring something let h be a semisimple hopf algebras over an algebraically closed field k of characteristic 0 we define hopf algebraic analogues of commutators and their generalizations and show how they are related to h the hopf algebraic analogue of the commutator subgroup we introduce a family of central elements of h which on one hand generate h and on the other hand give rise to a family of functionals on h when h kg g a finite group these functionals are counting functions on g it is not clear yet to what extent they measure any specific invariant of the hopf algebra however when h is quasitriangular they are at least characters on h

multiple binary decision tree classifiers abstract binary decision trees based on nonparametric statistical models of the data provide a solution to difficult decision problems where there are many classes and many available features related in a complex manner unfortunately the technique requires a very large training set and is often limited by the size of the training set rather than by the discriminatory power of the features this paper demonstrates that higher classification accuracies can be obtained from the same training set by using a combination of decision trees and by reaching a consensus using dempster and shafer s theory of evidence

a classification procedure using the multiple fourier series abstract a nonparametric procedure using the multiple fourier series is presented and examined conditions for the weak and strong bayes risk consistency are given and the rates at which the probability of misclassification converges to the bayes probability of error are presented the rates for the convergence in mean as well as w p 1 are independent of the dimension

nonparametric estimation via empirical risk minimization a general notion of universal consistency of nonparametric estimators is introduced that applies to regression estimation conditional median estimation curve fitting pattern recognition and learning concepts general methods for proving consistency of estimators based on minimizing the empirical error are shown in particular distribution free almost sure consistency of neural network estimates and generalized linear estimators is established

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

strong violation of critical phenomena universality wang landau study of the two dimensional blume capel model under bond randomness we study the pure and random bond versions of the square lattice ferromagnetic blume capel model in both the first order and second order phase transition regimes of the pure model phase transition temperatures thermal and magnetic critical exponents are determined for lattice sizes in the range l 20 100 via a sophisticated two stage numerical strategy of entropic sampling in dominant energy subspaces using mainly the wang landau algorithm the second order phase transition emerging under random bonds from the second order regime of the pure model has the same values of critical exponents as the two dimensional ising universality class with the effect of the bond disorder on the specific heat being well described by double logarithmic corrections our findings thus supporting the marginal irrelevance of quenched bond randomness on the other hand the second order transition emerging under bond randomness from the first order regime of the pure model has a distinctive universality class with nu 1 30 6 and beta nu 0 128 5 these results amount to a strong violation of universality principle of critical phenomena since these two second order transitions with different sets of critical exponents are between the same ferromagnetic and paramagnetic phases furthermore the latter of these two sets of results supports an extensive but weak universality since it has the same magnetic critical exponent but a different thermal critical exponent as a wide variety of two dimensional systems with and without quenched disorder in the conversion by bond randomness of the first order transition of the pure system to second order we detect by introducing and evaluating connectivity spin densities a microsegregation that also explains the increase we find in the phase transition temperature under bond randomness

criticality in the randomness induced second order phase transition of the triangular ising antiferromagnet with nearest and next nearest neighbor interactions abstract using a wang landau entropic sampling scheme we investigate the effects of quenched bond randomness on a particular case of a triangular ising model with nearest j n n and next nearest neighbor j n n n antiferromagnetic interactions we consider the case r j n n n j n n 1 for which the pure model is known to have a columnar ground state where rows of nearest neighbor spins up and down alternate and undergo a weak first order phase transition from the ordered to the paramagnetic state with the introduction of quenched bond randomness we observe the effects signaling the expected conversion of the first order phase transition to a second order phase transition and using the lee kosterlitz method we quantitatively verify this conversion the emerging under random bonds continuous transition shows a strongly saturating specific heat behavior corresponding to a negative exponent and belongs to a new distinctive universality class with 1 135 11 1 744 9 and 0 124 8 thus our results for the critical exponents support an extensive but weak universality and the emerged continuous transition has the same magnetic critical exponent but a different thermal critical exponent as a wide variety of two dimensional 2d systems without and with quenched disorder

tree classifier design with a permutation statistic abstract this paper proposes a permutation statistic as a basis for designing binary tree classifiers of d dimensional patterns the statistic permits an objective and easily interpreted evaluation of a feature threshold pair that is the core of the design paradigm it has a known distribution that is independent of the number of training patterns after defining the permutation statistic and explaining its properties the paper proposes the design procedure whereby each non terminal node is assigned a feature and a threshold on that feature the design procedure is demonstrated on six test sets and compared to designs based on mutual information

coupled samples in simulation assume that we wish to generate two samples of n independent identically distributed random variables x1 xn and y1 yn where x1 and y1 have densities f and g respectively if these samples are used in a simulation and f is close to g it is sometimes desirable to have close simulation results this can be achieved by insisting that both samples agree in most of their components that is xi yi for as many i as possible under the given distributional constraints samples with this property are said to be optimally coupled in this paper we propose and study various methods of coupling two samples a sequence of samples and an infinite family of samples

random hyperplane search trees a hyperplane search tree is a binary tree used to store a set s of n d dimensional data points in a random hyperplane search tree for s the root represents a hyperplane defined by d data points drawn uniformly at random from s the remaining data points are split by the hyperplane and the definition is used recursively on each subset we assume that the data are points in general position in mathbb r d we show that uniformly over all such data sets s the expected height of the hyperplane tree is not worse than that of the k d tree or the ordinary one dimensional random binary search tree and that for any fixed d ge3 the expected height improves over that of the standard random binary search tree by an asymptotic factor strictly greater than one

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

an iterative growing and pruning algorithm for classification tree design a critical issue in classification tree design obtaining right sized trees i e trees which neither underfit nor overfit the data is addressed instead of stopping rules to halt partitioning the approach of growing a large tree with pure terminal nodes and selectively pruning it back is used a new efficient iterative method is proposed to grow and prune classification trees this method divides the data sample into two subsets and iteratively grows a tree with one subset and prunes it with the other subset successively interchanging the roles of the two subsets the convergence and other properties of the algorithm are established theoretical and practical considerations suggest that the iterative free growing and pruning algorithm should perform better and require less computation than other widely used tree growing and pruning algorithms numerical results on a waveform recognition problem are presented to support this view

semisimple hopf algebras via geometric invariant theory abstract we study hopf algebras via tools from geometric invariant theory we show that all the invariants we get can be constructed using the integrals of the hopf algebra and its dual together with the multiplication and the comultiplication and that these invariants determine the isomorphism class of the hopf algebra we then define certain canonical subspaces i n v i j of tensor powers of h and h and use the invariant theory to prove that these subspaces satisfy a certain non degeneracy condition using this non degeneracy condition together with results on symmetric monoidal categories we prove that the spaces i n v i j can also be described as h i h j a where a is the group of hopf automorphisms of h as a result we prove that the number of possible hopf orders of any semisimple hopf algebra over a given number ring is finite we give some examples of these invariants arising from the theory of frobenius schur indicators and from reshetikhin turaev invariants of three manifolds we give a complete description of the invariants for a group algebra proving that they all encode the number of homomorphisms from some finitely presented group to the group we also show that if all the invariants are algebraic integers then the hopf algebra satisfies kaplansky s sixth conjecture the dimensions of the irreducible representations of h divide the dimension of h

universality of the ising and the s 1 model on archimedean lattices a monte carlo determination the ising models s 1 2 and s 1 are studied by efficient monte carlo schemes on the 3 4 6 4 and the 3 3 3 3 6 archimedean lattices the algorithms used a hybrid metropolis wolff algorithm and a parallel tempering protocol are briefly described and compared with the simple metropolis algorithm accurate monte carlo data are produced at the exact critical temperatures of the ising model for these lattices their finite size analysis provide with high accuracy all critical exponents which as expected are the same with the well known 2d ising model exact values a detailed finite size scaling analysis of our monte carlo data for the s 1 model on the same lattices provides very clear evidence that this model obeys also very well the 2d ising model critical exponents as a result we find that recent monte carlo simulations and attempts to define effective dimensionality for the s 1 model on these lattices are misleading accurate estimates are obtained for the critical amplitudes of the logarithmic expansions of the specific heat for both models on the two archimedean lattices

binary partitioning perceptual grouping and restoration with semidefinite programming we introduce a novel optimization method based on semidefinite programming relaxations to the field of computer vision and apply it to the combinatorial problem of minimizing quadratic functionals in binary decision variables subject to linear constraints the approach is tuning parameter free and computes high quality combinatorial solutions using interior point methods convex programming and a randomized hyperplane technique apart from a symmetry condition no assumptions such as metric pairwise interactions are made with respect to the objective criterion as a consequence the approach can be applied to a wide range of problems applications to unsupervised partitioning figure ground discrimination and binary restoration are presented along with extensive ground truth experiments from the viewpoint of relaxation of the underlying combinatorial problem we show the superiority of our approach to relaxations based on spectral graph theory and prove performance bounds

estimation of extreme risk regions under multivariate regular variation when considering d possibly dependent random variables one is often interested in extreme risk regions with very small probability p we consider risk regions of the form z rd f z where f is the joint density and a small number estimation of such an extreme risk region is difficult since it contains hardly any or no data using extreme value theory we construct a natural estimator of an extreme risk region and prove a refined form of consistency given a random sample of multivariate regularly varying random vectors in a detailed simulation and comparison study the good performance of the procedure is demonstrated we also apply our estimator to financial data

lower bounds in pattern recognition and learning lower bounds are derived for the performance of any pattern recognition algorithm which using training data selects a discrimination rule from a certain class of rules the bounds involve the vapnik chervonenkis dimension of the class and l the minimal error probability within the class we provide lower bounds when l 0 the usual assumption in valiant s theory of learning and l 0

histogram regression estimation using data dependent partitions we establish general sufficient conditions for the l 2 conoistency of multivariate histogram regression estimates based on data dependent partitions these same conditions insure the consistency of partitioning regression estimates based on local polynomial fits and with an additional regularity assumption the consistency of histogram estimates for conditional medians our conditions require shrinking cells subexponential growth of a combinatorial complexity measure and sublinear growth of restricted cell counts it is not assumed that the cells of every partition be rectangles with sides parallel to the coordinate axis or that each cell contain a minimum number of points response variables are assumed to be bounded throughout our results may be applied to a variety of partitioning schemes we established the consistency of histograms regression estimates based on cubic partitions with data dependent offsets k thresholding in one dimension and empirically optimal nearest neighbor clustering schemes in addition it is shown that empirically optimal regression trees are consistent when the size of the trees grows with the number of samples at an appropriate rate

combining classifiers via discretization abstract i consider a method for combining different classifiers to develop more effective classification rules the proposed combined classifier which turns out to be strongly consistent is quite simple to use in real applications it is also shown that this combined classifier is strongly asymptotically at least as good as any one of the individual classifiers in addition if one of the individual classifiers is already bayes optimal asymptotically then so is the combined classifier

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

the rates of convergence of kernel regression estimates and classification rules both nonrecursive and recursive nonparametric regression estimates are studied the rates of weak and strong convergence of kernel estimates as well as corresponding multiple classification errors are derived without assuming the existence of the density of the measurements an application of the obtained results to a nonparametric bayes predication is presented

invariance principles for sums of banach space valued random elements and empirical processes almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable banach space it is then shown that empirical processes readily fit into this general framework thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes

braided groups and quantum fourier transform abstract we show that acting on every finite dimensional factorizable ribbon hopf algebra h there are invertible operators s t obeying the modular identities s t 3 s 2 where is a constant the class includes the finite dimensional quantum groups uq g associated to complex simple lie algebras we give the example of uq sl 2 at a root of unity in detail as well as an example relating to anyons the operator s plays the role of quantum fourier transform and acts naturally on h viewed by transmutation as a braided group h a braided cocommutative hopf algebra in a braided category it obeys s 2 s 1 where s is the antipode of h the results follow as an application of previous category theoretical constructions

static critical behavior of the q states potts model high resolution entropic study abstract here we report a precise computer simulation study of the static critical properties of the two dimensional q states potts model using very accurate data obtained from a modified wang landau wl scheme proposed by caparica and cunha netto 2012 this algorithm is an extension of the conventional wl sampling but the authors changed the criterion to update the density of states during the random walk and established a new procedure to windup the simulation run these few changes have allowed a more precise microcanonical averaging which is essential to a reliable finite size scaling analysis in this work we used this new technique to determine the static critical exponents and in an unambiguous fashion the static critical exponents were determined as 0 10811 77 1 4459 31 and 0 8197 17 for the q 3 case and 0 0877 37 1 3161 69 and 0 7076 10 for the q 4 potts model a comparison of the present results with conjectured values and with those obtained from other well established approaches strengthens this new way of performing wl simulations