mdl denoising the so called denoising problem relative to normal models for noise is formalized such that noise is defined as the incompressible part in the data while the compressible part defines the meaningful information bearing signal such a decomposition is effected by minimization of the ideal code length called for by the minimum description length mdl principle and obtained by an application of the normalized maximum likelihood technique to the primary parameters their range and their number for any orthonormal regression matrix such as defined by wavelet transforms the minimization can be done with a threshold for the squared coefficients resulting from the expansion of the data sequence in the basis vectors defined by the matrix

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

on the structure of stationary solutions of the navier stokes equations this paper is a supplementary section to 1 we show that without any additional hypothesis the main result in 1 theorem 1 can be considerably strengthened

a center manifold for second order semilinear differential equations on the real line and applications to the existence of wave trains for the gurtin mccamy equation

fisher information and stochastic complexity by taking into account the fisher information and removing an inherent redundancy in earlier two part codes a sharper code length as the stochastic complexity and the associated universal process are derived for a class of parametric processes the main condition required is that the maximum likelihood estimates satisfy the central limit theorem the same code length is also obtained from the so called maximum likelihood code

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

smoothness of bounded solutions of nonlinear evolution equations abstract it is shown that in many cases globally defined bounded solutions of evolution equations are as smooth in time as the corresponding operator even if a general solution of the initial value problem is much less smooth i e initial values for bounded solutions are selected in such a way that optimal smoothness is attained in particular solutions which bifurcate from certain steady states such as periodic orbits almost periodic orbits and also homo and heteroclinic orbits have this property as examples a neutral functional differential equation a slightly damped non linear wave equation and a heat equation are considered in the latter case the space variable is included into the discussion of smoothness finally generalized hopf bifurcation in infinite dimensions is considered here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized

analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near ideal estimation performance in various asymptotic frameworks however image processing practice has shown that universal thresholding methods are outperformed by simple bayesian estimators assuming independent wavelet coefficients and heavy tailed priors such as generalized gaussian distributions ggds in this paper we investigate various connections between shrinkage methods and maximum a posteriori map estimation using such priors in particular we state a simple condition under which map estimates are sparse we also introduce a new family of complexity priors based upon rissanen s universal prior on integers one particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length mdl principle we develop analytical expressions for the shrinkage rules implied by ggd and complexity priors this allows us to show the equivalence between universal hard thresholding map estimation using a very heavy tailed ggd and mdl estimation using one of the new complexity priors theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis specifications of the prior a basic concern in image processing applications

on ground states of superlinear p laplacian equations in rn abstract we consider superlinear p laplacian equations in r n with a potential which is periodic or has a bounded potential well without assuming the ambrosetti rabinowitz type condition and the monotonicity of the function t f x t t p 1 we prove the existence of ground states even for the case p 2 our results extend the recent results of li wang and zeng ann inst h poincare anal non lineaire 23 2006 829 837

understanding waveshrink variance and bias estimation summary donoho johnstone s waveshrink procedure has proved valuable for function estimation and nonparametric regression waveshrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise waveshrink has very broad asymptotic near optimality properties and achieves the optimal risk to within a factor of log n in this paper we derive computationally efficient formulae for computing the exact bias variance and l2 risk of waveshrink estimates in finite sample situations we use these formulae to understand the behaviour of waveshrink estimators construct approximate confidence intervals and bias estimates for waveshrink and compute ideal thresholds for a given function we show that hard shrinkage has smaller bias but larger variance than soft shrinkage and that significantly smaller thresholds should be used for soft shrinkage we also compute minimax thresholds for waveshrink estimators and demonstrate that the minimax thresholds can nearly achieve the ideal rank for a range of functions

thresholding estimators for linear inverse problems and deconvolutions thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded for signals in a set sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence deconvolutions with kernels having a fourier transform which vanishes at high frequencies are examples of unstable inverse problems where a thresholding in a wavelet basis is a suboptimal estimator a new mirror wavelet basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals this thresholding estimator is used to restore blurred satellite images

deconvolution by thresholding in mirror wavelet bases the deconvolution of signals is studied with thresholding estimators that decompose signals in an orthonormal basis and threshold the resulting coefficients a general criterion is established to choose the orthonormal basis in order to minimize the estimation risk wavelet bases are highly sub optimal to restore signals and images blurred by a low pass filter whose transfer function vanishes at high frequencies a new orthonormal basis called mirror wavelet basis is constructed to minimize the risk for such deconvolutions an application to the restoration of satellite images is shown

model selection and the principle of minimum description length this article reviews the principle of minimum description length mdl for problems of model selection by viewing statistical modeling as a means of generating descriptions of observed data the mdl framework discriminates between competing models based on the complexity of each description this approach began with kolmogorov s theory of algorithmic complexity matured in the literature on information theory and has recently received renewed attention within the statistics community here we review both the practical and the theoretical aspects of mdl as a tool for model selection emphasizing the rich connections between information theory and statistics at the boundary between these two disciplines we find many interesting interpretations of popular frequentist and bayesian procedures as we show mdl provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared we illustrate the mdl principle by considering problems in regression nonpar

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

propagating fronts and the center manifold theorem we prove the existence of propagating front solutions for the swift hohenberg equation sh using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations they describe stationary solutions and front solutions of the partial differential equation sh we identify the well known amplitude equation as the lowest order approximation to the equation of motion on the center manifold

wavelet based texture retrieval using generalized gaussian density and kullback leibler distance we present a statistical view of the texture retrieval problem by combining the two related tasks namely feature extraction fe and similarity measurement sm into a joint modeling and classification scheme we show that using a consistent estimator of texture model parameters for the fe step followed by computing the kullback leibler distance kld between estimated models for the sm step is asymptotically optimal in term of retrieval error probability the statistical scheme leads to a new wavelet based texture retrieval method that is based on the accurate modeling of the marginal distribution of wavelet coefficients using generalized gaussian density ggd and on the existence a closed form for the kld between ggds the proposed method provides greater accuracy and flexibility in capturing texture information while its simplified form has a close resemblance with the existing methods which uses energy distribution in the frequency domain to identify textures experimental results on a database of 640 texture images indicate that the new method significantly improves retrieval rates e g from 65 to 77 compared with traditional approaches while it retains comparable levels of computational complexity

small amplitude steady water waves with critical layers non symmetric waves abstract the problem for two dimensional steady water waves with vorticity is considered using methods of spatial dynamics we reduce the problem to a finite dimensional hamiltonian system the reduced system describes all small amplitude solutions of the problem and as an application we give a proof of the existence of non symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one

center manifolds under nonuniform hyperbolicity maximal regularity abstract we establish the existence of invariant center manifolds with maximal c r regularity for a nonautonomous dynamics with discrete time we consider the general case of perturbations of a nonuniform exponential trichotomy our proof uses the fiber contraction principle and allows linear perturbations without any further effort

new stochastic mode reduction strategy for dissipative systems we present a new methodology for studying non hamiltonian nonlinear systems based on an information theoretical extension of a renormalization group technique using a modified maximum entropy principle we obtain a rigorous dimensionally reduced description for such systems the neglected degrees of freedom by this reduction are replaced by a systematically defined stochastic process under a constraint on the second moment this then forms the basis of a computationally efficient method numerical computations for the generalized kuramoto sivashinsky equation support our method and reveal that the long time underlying stochastic process of the fast unresolved modes obeys a universal distribution that does not depend on the initial conditions and which we rigorously derive by the maximum entropy principle

iterative wiener filters for image restoration the iterative wiener filter which successively uses the wiener filtered signal as an improved prototype to update the covariance estimates is investigated the convergence properties of this iterative filter are analyzed it has been shown that this iterative process converges to a signal which does not correspond to the minimum mean squared error solution based on the analysis an alternate iterative filter is proposed to correct for the convergence error the theoretical performance of the filter has been shown to give minimum mean squared error in practical implementation when there is unavoidable error in the covariance computation the filter may still result in undesirable restoration its performance has been investigated and a number of experiments in a practical setting were conducted to demonstrate its effectiveness

wavelet thresholding via mdl for natural images we study the application of rissanen s 1989 principle of minimum description length mdl to the problem of wavelet denoising and compression for natural images after making a connection between thresholding and model selection we derive an mdl criterion based on a laplacian model for noiseless wavelet coefficients we find that this approach leads to an adaptive thresholding rule while achieving mean squared error performance comparable with other popular thresholding schemes the mdl procedure tends to keep far fewer coefficients from this property we demonstrate that our method is an excellent tool for simultaneous denoising and compression we make this claim precise by analyzing mdl thresholding in two optimality frameworks one in which we measure rate and distortion based on quantized coefficients and one in which we do not quantize but instead record rate simply as the number of nonzero coefficients

nonlinear wavelet shrinkage with bayes rules and bayes factors abstract wavelet shrinkage the method proposed by the seminal work of donoho and johnstone is a disarmingly simple and efficient way of denoising data shrinking wavelet coefficients was proposed from several optimality criteria in this article a wavelet shrinkage by coherent bayesian inference in the wavelet domain is proposed the methods are tested on standard donoho johnstone test functions

smooth center manifolds for nonuniformly partially hyperbolic trajectories abstract we establish the existence of unique smooth center manifolds for ordinary differential equations v a t v f t v in banach spaces assuming that v a t v admits a nonuniform exponential trichotomy this allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories in addition we prove that the center manifolds are as regular as the vector field our proof of the c k smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space to the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting

the dichotomy theorem for evolution bi families we prove that the operator g the closure of the rst order dierential operator d dt d t on l2 r x is fredholm if and only if the not well posed equation u 0 t d t u t t 2 r has exponential dichotomies on r and r and the ranges of the dichotomy projections form a fredholm pair moreover the index of this pair is equal to the fredholm index of g here x is a hilbert space d t a b t a is the generator of a bi semigroup b is a bounded piecewise strongly continuous operator valued function also we prove some perturbations results and consider various examples of not well posed problems

strong optimality of the normalized ml models as universal codes and information in data we show that the normalized maximum likelihood nml distribution as a universal code for a parametric class of models is closest to the negative logarithm of the maximized likelihood in the mean code length distance where the mean is taken with respect to the worst case model inside or outside the parametric class we strengthen this result by showing that when the data generating models are restricted to be the most benevolent ones in that they incorporate all the constraints in the data and no more the bound cannot be beaten in essence by any code except when the mean is taken with respect to the data generating models in a set of vanishing size these results allow us to decompose the code of the data into two parts the first having all the useful information in the data that can be extracted with the family in question and the rest which has none and we obtain a measure for the useful information in data

steady flows of inviscid fluids under localized perturbations on essaie d obtenir tous les ecoulements stationnaires dans un canal avec un petit obstacle pour des nombres de froude proches de la valeur critique fo

a center stable manifold theorem for differential equations in banach spaces we prove a center stable manifold theorem for a class of differential equations in infinite dimensional banach spaces

abundance of entire solutions to nonlinear elliptic equations by the variational method abstract we study entire bounded solutions to the equation u u u 3 0 in r 2 our approach is purely variational and is based on concentration arguments and symmetry considerations this method allows us to construct in an unified way several types of solutions with various symmetries radial breather type rectangular triangular hexagonal etc both positive and sign changing it is also applicable for more general equations in any dimension

robustness of nonuniform exponential trichotomies in banach spaces abstract we establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x a t x in banach spaces we also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy we consider both trichotomies in semi infinite intervals and trichotomies in r

existence of weak solutions for a quasilinear equation in rn abstract this paper studies the p laplacian equation p u v x u p 2 u f x u in r n where 1 p n 1 and v x is a nonnegative continuous function under some conditions on f x u and v x we prove the existence of nontrivial solutions for sufficiently large

dissipation induced instability phenomena in infinite dimensional systems this paper develops a rigorous notion of dissipation induced instability in infinite dimensions as an extension of the classical concept implicitly introduced by thomson and tait for finite degree of freedom mechanical systems over a century ago here we restrict ourselves to a particular form of infinite dimensional systems partial differential equations whose inherent function analytic differences from finite dimensional systems make uncovering this notion more intricate in building the concept of dissipation induced instability in infinite dimensions we found arnold s and yudovich s nonlinear stability methods for conservative and dissipative systems respectively along with some new existence theory for solutions to be the essential foundation however when proving the results for classical solutions as motivated by their direct physical significance we had to overcome a number of fundamental difficulties associated with existing stability analysis methods which has led to new techniques in particular in this work we establish the connection of existence and general stability theories in strong and weak topologies and provide new insights into the physics and geometry of the dissipation induced instability phenomena in infinite dimensional systems as a paradigm and the first infinite dimensional example to be rigorously analyzed we use a two layer quasi geostrophic beta plane model which describes the fundamental baroclinic instability in atmospheric and ocean dynamics early formal linear approximate studies suggested that this system can be destabilized after the introduction of dissipation

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

fully exponential laplace approximations to expectations and variances of nonpositive functions abstract tierney and kadane 1986 presented a simple second order approximation for posterior expectations of positive functions they used laplace s method for asymptotic evaluation of integrals in which the integrand is written as f exp nh and the function h is approximated by a quadratic the form in which they applied laplace s method however was fully exponential the integrand was written instead as exp nh log f this allowed first order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second order expansion for the ratio other second order expansions hartigan 1965 johnson 1970 lindley 1961 1980 mosteller and wallace 1964 require computation of more derivatives of the log likelihood function in this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions to obtain a second order approximation to an expectation e g we use the fully exponential method to approximate

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

polynomial contractions and lyapunov regularity abstract the purpose of this note is twofold to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time and to show that it persists under sufficiently small linear and nonlinear perturbations the notion of polynomial contraction mimics the notion of exponential contraction but with the exponential decay replaced by a polynomial decay we show that this behavior is exhibited by a large class of dynamics by giving necessary conditions in terms of polynomial lyapunov exponents finally we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations we also consider the case of nonuniform polynomial contractions for which the lyapunov stability is not uniform

nonconvergent bounded solutions of semilinear heat equations on arbitrary domains abstract we consider the dirichlet problem for the semilinear heat equation 0 1 u t u g x u x where is an arbitrary bounded domain in r n n 2 with c 2 boundary we find a c function g x u such that 0 1 has a bounded solution whose limit set is a continuum of equilibria this extends and improves an earlier result of the first author with rybakowski in which is a disk in r 2 and g is of finite differentiability class we also show that 0 1 can have an infinite dimensional manifold of nonconvergent bounded trajectories

adaptive wavelet thresholding for image denoising and compression the first part of this paper proposes an adaptive data driven threshold for image denoising via wavelet soft thresholding the threshold is derived in a bayesian framework and the prior used on the wavelet coefficients is the generalized gaussian distribution ggd widely used in image processing applications the proposed threshold is simple and closed form and it is adaptive to each subband because it depends on data driven estimates of the parameters experimental results show that the proposed method called bayesshrink is typically within 5 of the mse of the best soft thresholding benchmark with the image assumed known it also outperforms sureshrink donoho and johnstone 1994 1995 donoho 1995 most of the time the second part of the paper attempts to further validate claims that lossy compression can be used for denoising the bayesshrink threshold can aid in the parameter selection of a coder designed with the intention of denoising and thus achieving simultaneous denoising and compression specifically the zero zone in the quantization step of compression is analogous to the threshold value in the thresholding function the remaining coder design parameters are chosen based on a criterion derived from rissanen s minimum description length mdl principle experiments show that this compression method does indeed remove noise significantly especially for large noise power however it introduces quantization noise and should be used only if bitrate were an additional concern to denoising

smooth conjugacy of centre manifolds in this paper we prove that for a system of ordinary differential equations of class c r 1 1 r 0 and two arbitrary c r 1 1 local centre manifolds of a given equilibrium point the equations when restricted to the centre manifolds are c r conjugate the same result is proved for similinear parabolic equations the method is based on the geometric theory of invariant foliations for centre stable and centre unstable manifolds

exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders in applications solitary wave solutions of semilinear elliptic equations u g u u 0 x y r in infinite cylinders frequently arise as travelling waves of parabolic equations as such their bifurcations are an interesting issue interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful still there are major obstacles in obtaining for instance bifurcation results similar to those for ordinary differential equations in this article persistence and continuation of exponential dichotomies for linear elliptic equations is proved with this technique at hands lyapunov schmidt reduction near solitary waves can be applied as an example existence of shift dynamics near solitary waves is shown if a perturbation h x u u periodic inxis added

the minimum description length principle in coding and modeling we review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling stochastic complexity is formulated as the solution to optimum universal coding problems extending shannon s basic source coding theorem the normalized maximized likelihood mixture and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms we assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference context tree modeling density estimation and model selection in gaussian linear regression serve as examples

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

on multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in rn abstract in this work we study the existence multiplicity and concentration of positive solutions for the following class of quasilinear problems n n u v x u n 2 u f u in r n u x 0 x r n where n u is the n laplacian operator is a positive parameter n 2 v r n r is a continuous function and f r r is a c 1 function having critical exponential growth

an invariant bayesian model selection principle for gaussian data in a sparse representation in this paper we develop a code length principle which is invariant to the choice of parameterization on the model distributions that is the code length remains the same under smooth transformations on the likelihood parameters an invariant approximation formula for easy computation of the marginal distribution is provided for gaussian likelihood models we provide invariant estimators of the model parameters and formulate conditions under which these estimators are essentially posteriori unbiased for gaussian models an upper bound on the coarseness of discretization on the model parameters is deduced we introduce a discrimination measure between probability distributions and use it to construct probability distributions on model classes and show how this may induce an additional code length term k 4log2k for a k parameter model the total code length is shown to be closely related to the normalized maximum likelihood nml code length of rissanen when choosing jeffreys prior distribution on the model parameters together with a uniform prior distribution on the model classes our model selection principle is applied to a gaussian estimation problem for data in a wavelet representation and its performance is tested and compared to alternative wavelet based estimation methods in numerical experiments

inferring decision trees using the minimum description length principle abstract we explore the use of rissanen s minimum description length principle for the construction of decision trees empirical results comparing this approach to other methods are given

geometric stability analysis for periodic solutions of the swift hohenberg equation in this paper we describe invariant geometrical structures in the phase space of the swift hohenberg equation in a neighborhood of its periodic stationary states we show that in spite of the fact that these states are only marginally stable i e the linearized problem about these states has continuous spectrum extending all the way up to zero there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long time behavior of solutions near these stationary solutions in particular using this point of view we obtain a new demonstration of schneider s recent proof that these states are nonlinearly stable

statistical inference occam s razor and statistical mechanics on the space of probability distributions the task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions using the techniques of low temperature expansions i arrive at a systematic series for the bayesian posterior probability of a model family that significantly extends known results in the literature in particular i arrive at a precise understanding of how occam s razor the principle that simpler models should be preferred until the data justify more complex models is automatically embodied by probability theory these results require a measure on the space of model parameters and i derive and discuss an interpretation of jeffreys prior distribution as a uniform prior over the distributions indexed by a family finally i derive a theoretical index of the complexity of a parametric family relative to some true distribution that i call the razor of the model the form of the razor immediately suggests several interesting questions in the theory of learning that can be studied using the techniques of statistical mechanics

extensive properties of the complex ginzburg landau equation we study the set of solutions of the complex ginzburg landau equation in r d d 3 we consider the global attracting set i e the forward map of the set of bounded initial data and restrict it to a cube q l of side l we cover this set by a minimal number n q l of balls of radius in l infin q l we show that the kolmogorov entropy per unit length exists in particular we bound by which shows that the attracting set is smaller than the set of bounded analytic functions in a strip we finally give a positive lower bound

minimax description length for signal denoising and optimized representation approaches to wavelet based denoising or signal enhancement have generally relied on the assumption of normally distributed perturbations to relax this assumption which is often violated in practice we derive a robust wavelet thresholding technique based on the minimax description length mmdl principle we first determine the least favorable distribution in the spl epsiv contaminated normal family as the member that maximizes the entropy we show that this distribution and the best estimate based upon it namely the maximum likelihood estimate together constitute a saddle point the mmdl approach results in a thresholding scheme that is resistant to heavy tailed noise we further extend this framework and propose a novel approach to selecting an adapted or best basis bb that results in optimal signal reconstruction finally we address the practical case where the underlying signal is known to be bounded and derive a two sided thresholding technique that is resistant to outliers and has bounded error