observers and initial state recovering for a class of hyperbolic systems via lyapunov method a b s t r a c t recently the problem of estimating the initial state of some linear infinite dimensional systems from measurements on a finite interval was solved by using the sequence of forward and backward observers ramdani tucsnak and weiss 2010 in the present paper we introduce a direct lyapunov approach to the problem and extend the results to the class of semilinear systems governed by wave and beam equations with boundary measurements from a finite interval we first design forward observers and derive linear matrix inequalities lmis for the exponential stability of the estimation errors further we obtain simple finite dimensional conditions in terms of lmis for an upper bound t on the minimal time that guarantees the convergence of the sequence of forward and backward observers on 0 t for the initial state recovering this t represents also an upper bound on the observability time for observation times bigger than t these lmis give upper bounds on the convergence rate of the iterative algorithm in the norm defined by the lyapunov functions in our approach t is found as the minimal dwelling time for the switched exponentially stable forward and backward estimation error systems with the different lyapunov functions liberzon 2003 the efficiency of the results is illustrated by numerical examples 2013 elsevier ltd all rights reserved

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

empirical risk minimization for heavy tailed losses the purpose of this paper is to discuss empirical risk minimization when the losses are not necessarily bounded and may have a distribution with heavy tails in such situations usual empirical averages may fail to provide reliable estimates and empirical risk minimization may provide large excess risk however some robust mean estimators proposed in the literature may be used to replace empirical means in this paper we investigate empirical risk minimization based on a robust estimate proposed by catoni we develop performance bounds based on chaining arguments tailored to catoni s mean estimator

posterior concentration and fast convergence rates for generalized bayesian learning abstract in this paper we study the learning rate of generalized bayes estimators in a general setting where the hypothesis class can be uncountable and have an irregular shape the loss function can have heavy tails and the optimal hypothesis may not be unique we prove that under the multi scale bernstein s condition the generalized posterior distribution concentrates around the set of optimal hypotheses and the generalized bayes estimator can achieve fast learning rate our results are applied to show that the standard bayesian linear regression is robust to heavy tailed distributions

from ɛ entropy to kl entropy analysis of minimum information complexity density estimation we consider an extension of e entropy to a kl divergence based complexity measure for randomized density estimation methods based on this extension we develop a general information theoretical inequality that measures the statistical complexity of some deterministic and randomized density estimators consequences of the new inequality will be presented in particular we show that this technique can lead to improvements of some classical results concerning the convergence of minimum description length and bayesian posterior distributions moreover we are able to derive clean finite sample convergence bounds that are not obtainable using previous approaches

asymptotic behaviour of the posterior distribution in overfitted mixture models abstract summary we study the asymptotic behaviour of the posterior distribution in a mixture model when the number of components in the mixture is larger than the true number of components a situation which is commonly referred to as an overfitted mixture we prove in particular that quite generally the posterior distribution has a stable and interesting behaviour since it tends to empty the extra components this stability is achieved under some restriction on the prior which can be used as a guideline for choosing the prior some simulations are presented to illustrate this behaviour

feedbacks for nonautonomous regular linear systems we introduce nonautonomous well posed and absolutely regular linear systems as quadruples consisting of an evolution family and output input and input output maps subject to natural hypotheses in the spirit of weiss work these maps are represented in terms of admissible observation and control operators the latter in an approximate sense in the time domain in this setting the closed loop system exists for a canonical class of admissible feedbacks and it inherits the absolute regularity and other properties of the given system in particular we can iterate feedbacks

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

on aggregation for heavy tailed classes we introduce an alternative to the notion of fast rate in learning theory which coincides with the optimal error rate when the given class happens to be convex and regular in some sense while it is well known that such a rate cannot always be attained by a learning procedure i e a procedure that selects a function in the given class we introduce an aggregation procedure that attains that rate under rather minimal assumptions for example that the l q and l 2 norms are equivalent on the linear span of the class for some q 2 and the target random variable is square integrable the key components in the proof include a two sided isomorphic estimator on distances between class members which is based on the median of means and an almost isometric lower bound of the form n 1 sum i 1 n f 2 x i ge 1 zeta mathbb e f 2 which holds uniformly in the class both results only require that the l q and l 2 norms are equivalent on the linear span of the class for some q 2

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

fast rates in statistical and online learning the speed with which a learning algorithm converges as it is presented with more data is a central problem in machine learning a fast rate of convergence means less data is needed for the same level of performance the pursuit of fast rates in online and statistical learning has led to the discovery of many conditions in learning theory under which fast learning is possible we show that most of these conditions are special cases of a single unifying condition that comes in two forms the central condition for proper learning algorithms that always output a hypothesis in the given model and stochastic mixability for online algorithms that may make predictions outside of the model we show that under surprisingly weak assumptions both conditions are in a certain sense equivalent the central condition has a re interpretation in terms of convexity of a set of pseudoprobabilities linking it to density estimation under misspecification for bounded losses we show how the central condition enables a direct proof of fast rates and we prove its equivalence to the bernstein condition itself a generalization of the tsybakov margin condition both of which have played a central role in obtaining fast rates in statistical learning yet while the bernstein condition is two sided the central condition is one sided making it more suitable to deal with unbounded losses in its stochastic mixability form our condition generalizes both a stochastic exp concavity condition identified by juditsky rigollet and tsybakov and vovk s notion of mixability our unifying conditions thus provide a substantial step towards a characterization of fast rates in statistical learning similar to how classical mixability characterizes constant regret in the sequential prediction with expert advice setting

minimum phase infinite dimensional second order systems in general better performance can be achieved with a controlled minimum phase system than a controlled nonminimum phase system we show that a wide class of second order infinite dimensional systems with either velocity or position measurements are minimum phase the results are illustrated by two examples

convergence rates of least squares regression estimators with heavy tailed errors we study the performance of the least squares estimator lse in a general nonparametric regression model when the errors are independent of the covariates but may only have a p th moment p geq1 in such a heavy tailed regression setting we show that if the model satisfies a standard entropy condition with exponent alpha in 0 2 then the l 2 loss of the lse converges at a rate mathcal o mathbf p bigl n frac 1 2 alpha vee n frac 1 2 frac 1 2p bigr such a rate cannot be improved under the entropy condition alone this rate quantifies both some positive and negative aspects of the lse in a heavy tailed regression setting on the positive side as long as the errors have p geq1 2 alpha moments the l 2 loss of the lse converges at the same rate as if the errors are gaussian on the negative side if p 1 2 alpha there are many hard models at any entropy level alpha for which the l 2 loss of the lse converges at a strictly slower rate than other robust estimators the validity of the above rate relies crucially on the independence of the covariates and the errors in fact the l 2 loss of the lse can converge arbitrarily slowly when the independence fails the key technical ingredient is a new multiplier inequality that gives sharp bounds for the multiplier empirical process associated with the lse we further give an application to the sparse linear regression model with heavy tailed covariates and errors to demonstrate the scope of this new inequality

information theoretic upper and lower bounds for statistical estimation in this paper we establish upper and lower bounds for some statistical estimation problems through concise information theoretic arguments our upper bound analysis is based on a simple yet general inequality which we call the information exponential inequality we show that this inequality naturally leads to a general randomized estimation method for which performance upper bounds can be obtained the lower bounds applicable for all statistical estimators are obtained by original applications of some well known information theoretic inequalities and approximately match the obtained upper bounds for various important problems moreover our framework can be regarded as a natural generalization of the standard minimax framework in that we allow the performance of the estimator to vary for different possible underlying distributions according to a predefined prior

generalization bounds for averaged classifiers we study a simple learning algorithm for binary classification instead of predicting with the best hypothesis in the hypothesis class that is the hypothesis that minimizes the training error our algorithm predicts with a weighted average of all hypotheses weighted exponentially with respect to their training error we show that the prediction of this algorithm is much more stable than the prediction of an algorithm that predicts with the best hypothesis by allowing the algorithm to abstain from predicting on some examples we show that the predictions it makes when it does not abstain are very reliable finally we show that the probability that the algorithm abstains is comparable to the generalization error of the best hypothesis in the class

determinantal characterization of canonical curves and combinatorial theta identities we characterize genus g canonical curves by the vanishing of combinatorial products of g 1 determinants of brill noether matrices this also implies the characterization of canonical curves in terms of g 2 g 3 2 theta identities a remarkable mechanism based on a basis of h 0 k c expressed in terms of szego kernels reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions such a basis together with the fay trisecant identity also leads to the solution of the question of expressing the determinant of brill noether matrices in terms of theta functions without using the problematic klein fay section

semilinear observation systems abstract in this paper we introduce locally lipschitz observation systems for nonlinear semigroups and show that they can be represented by an admissible nonlinear output operator defined on a suitable subspace in the semilinear case this concept fits well to the lebesgue extension known from linear system theory for semilinear systems we show robustness of exact observability near equilibria under locally small lipschitz perturbations finally we apply our results to a damped nonlinear plate equation and a semilinear thermo elastic system