optimal grouping for group minimax hypothesis testing bayesian hypothesis testing and minimax hypothesis testing represent extreme instances of detection in which the prior probabilities of the hypotheses are either completely and precisely known or are completely unknown group minimax also known as gamma minimax is a robust intermediary between bayesian and minimax hypothesis testing that allows for coarse or partial advance knowledge of the hypothesis priors by using information on sets in which the prior lies existing work on group minimax however does not consider the question of how to define the sets or groups of priors it is assumed that the groups are given in this paper we propose a novel intermediate detection scheme formulated through the quantization of the space of prior probabilities that optimally determines groups and also representative priors within the groups we show that when viewed from a quantization perspective group minimax amounts to determining centroids with a minimax bayes risk error divergence distortion criterion the appropriate bregman divergence for this task in addition the optimal partitioning of the space of prior probabilities is a bregman voronoi diagram together the optimal grouping and representation points are an epsilon net with respect to bayes risk error divergence and permit a rate distortion type asymptotic analysis of detection performance with the number of groups examples of detecting signals corrupted by additive white gaussian noise and of distinguishing exponentially distributed signals are presented

square root penalty adaptation to the margin in classification and in edge estimation we consider the problem of adaptation to the margin in binary classification we suggest a penalized empirical risk minimization classifier that adaptively attains up to a logarithmic factor fast optimal rates of convergence for the excess risk that is rates that can be faster than n 1 2 where n is the sample size we show that our method also gives adaptive estimators for the problem of edge estimation

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

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

the resultant on compact riemann surfaces we introduce a notion of the resultant of two meromorphic functions on a compact riemann surface and demonstrate its usefulness in several respects for example we exhibit several integral formulas for the resultant relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact riemann surface as a particular application the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the schottky double of the domain

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

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

global several parameter bifurcation and continuation theorems a unified approach via complementing maps we use the concept of complementing map in order to present a unified approach to the continuation principle global implicit function theorem and global bifurcation results for compact vector fields depending on several parameters

on the topological structure of the set of generalized solutions of the catenary problem the global topological structure of the space of configurations of a non rotating elastic string under compression and tension is studied the part of the string under tension is specified by a measurable subset of the interval the set of such intervals with the hausdorff topology is considered a parameter space for the equation satisfied by the string and the solutions are shown to form an infinitedimensional continuum over this parameter space a new global topological theorem is needed since the parameter space is not euclidean the topological theorem is based on the fixed point transfer

nonorientability of the index bundle and several parameter bifurcation the universal parameter space for sturm liouville boundary value problems is a two torus we show that the index bundle of the universal family of linear s l boundary value problems is non orientable using this we find a one dimensional set of bifurcation points for nonlinear bvp of sturm liouville type

selected topics on quadrature domains this is a selection of facts old and recent about quadrature domains the text written in the form of a survey is addressed to non experts and covers a variety of phenomena related to quadratur

alexander invariant for perturbations of fredholm operators abstract the alexander type invariant for the bifurcation problems involving perturbations of linear fredholm operators of nonnegative index is discussed and compared with another homotopy invariant called the bifurcation index detecting global bifurcations of solutions to multiparameter equations

bifurcation of heteroclinic orbits via an index theory heteroclinic orbits for one parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications inspired by a recent bifurcation result for homoclinic trajectories of nonautonomous vectorfield proved by author in 13 we define a new mathbf z 2 index and we construct a index theory for heteroclinic orbits of nonautonomous vectorfield we prove an index theorem by showing that under some standard transversality assumptions the mathbf z 2 index is equal to the parity a homotopy invariant for paths of fredholm operators of index 0 as a direct consequence of the index theory developed in this paper we get a new bifurcation result for heteroclinic orbits

on the connectivity properties of the solution set of parametrized families of compact vector fields a useful tool in the study of global continua of solutions of nonlinear partial differential equations is a connectivity result on the fixed points set of a l parameter family of maps that goes back to leray and schauder 171 and it was proved in its full generality by browder b in describing it we shall restrict ourselves to the case of parametrized families of compact vector fields defined in open subsets of banach spaces

ullemar s formula for the moment map ii we prove the complex analogue of ullemar s formula for the jacobian of the complex moment mapping this formula was previously established in the real case

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

fast learning rates for plug in classifiers it has been recently shown that under the margin or low noise assumption there exist classifiers attaining fast rates of convergence of the excess bayes risk that is rates faster than n 1 2 the work on this subject has suggested the following two conjectures i the best achievable fast rate is of the order n 1 and ii the plug in classifiers generally converge more slowly than the classifiers based on empirical risk minimization we show that both conjectures are not correct in particular we construct plug in classifiers that can achieve not only fast but also super fast rates that is rates faster than n 1 we establish minimax lower bounds showing that the obtained rates cannot be improved

a global bifurcation index for set valued perturbations of fredholm operators abstract in the paper the homotopy invariant detecting global bifurcations of solutions to multi parameter equations involving general set valued perturbations of linear fredholm operators of non negative index is introduced some applications to the existence problems for differential inclusions are provided

linear analysis of quadrature domains iii the linear data of a quadrature domain introduced in the first paper in this series is further investigated the matricial structure of all hyponormal operators with finite rank self commutators which is inductively constructed from such linear data is discussed in particular a conservation law of a determinantal function in the iterative process of defining these matrices is exploited the spectrum of the determining left upper corner of some of these operators is analyzed in connection with the existence of generalized quadrature formulas on planar domains

minimax optimal bounds for detectors based on estimated prior probabilities in many signal detection and classification problems we have knowledge of the distribution under each hypothesis but not the prior probabilities this paper is aimed at providing theory to quantify the performance of detection via estimating prior probabilities from either labeled or unlabeled training data the error or risk is considered as a function of the prior probabilities we show that the risk function is locally lipschitz in the vicinity of the true prior probabilities and the error of detectors based on estimated prior probabilities depends on the behavior of the risk function in this locality in general we show that the error of detectors based on the maximum likelihood estimate mle of the prior probabilities converges to the bayes error at a rate of n 1 2 where n is the number of training data if the behavior of the risk function is more favorable then detectors based on the mle have errors converging to the corresponding bayes errors at optimal rates of the form n 1 alpha 2 where alpha 0 is a parameter governing the behavior of the risk function with a typical value alpha 1 the limit alpha rightarrow infty corresponds to a situation where the risk function is flat near the true probabilities and thus insensitive to small errors in the mle in this case the error of the detector based on the mle converges to the bayes error exponentially fast with n we show that the bounds are achievable no matter given labeled or unlabeled training data and are minimax optimal in the labeled case

laplacian growth and whitham equations of soliton theory the laplacian growth the hele shaw problem of multiply connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of whitham equations known in soliton theory the whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations through this connection the laplacian growth is understood as a flow in the moduli space of riemann surfaces