infinitely divisible cascades to model the statistics of natural images we propose to model the statistics of natural images thanks to the large class of stochastic processes called infinitely divisible cascades idcs idcs were first introduced in one dimension to provide multifractal time series to model the so called intermittency phenomenon in hydrodynamical turbulence we have extended the definition of scalar idcs from one to n dimensions and commented on the relevance of such a model in fully developed turbulence in 1 in this paper we focus on the particular 2d case idcs appear as good candidates to model the statistics of natural images they share most of their usual properties and appear to be consistent with several independent theoretical and experimental approaches of the literature we point out the interest of idcs for applications to procedural texture synthesis

contraction conditions for average and α discount optimality in countable state markov games with unbounded rewards the goal of this paper is to provide a theory of n person markov games with unbounded cost for a countable state space and compact action spaces we investigate both the finite and infinite horizon problems for the latter we consider the discounted cost as well as the expected average cost we present conditions for the infinite horizon problems for which equilibrium policies exist for all players within the stationary policies and show that the costs in equilibrium policies exist for all players within the stationary policies and show that the costs in equilibrium satisfy the optimality equations similar results are obtained for the finite horizon costs for which equilibrium policies are shown to exist for all players within the markov policies as special case of n person games we investigate the zero sum 2 players game for which we establish the convergence of the value iteration algorithm we conclude by studying an application of a zero sum markov game in a queueing model

learning multiple tasks with kernel methods we study the problem of learning many related tasks simultaneously using kernel methods and regularization the standard single task kernel methods such as support vector machines and regularization networks are extended to the case of multi task learning our analysis shows that the problem of estimating many task functions with regularization can be cast as a single task learning problem if a family of multi task kernel functions we define is used these kernels model relations among the tasks and are derived from a novel form of regularizers specific kernels that can be used for multi task learning are provided and experimentally tested on two real data sets in agreement with past empirical work on multi task learning the experiments show that learning multiple related tasks simultaneously using the proposed approach can significantly outperform standard single task learning particularly when there are many related tasks but few data per task

mixed risk neutral minimax control of discrete time finite state markov decision processes addresses the control design problem for discrete time finite state markov decision processes when both risk neutral and minimax objectives are of interest we introduce the mixed risk neutral minimax objective and utilize results from risk neutral and minimax control to derive an information state process and dynamic programming equations for the value function we synthesize optimal control laws both on the finite and the infinite horizons we study the effectiveness of both the mixed risk neutral minimax family and the risk sensitive family of controllers as tools to tradeoff risk neutral and minimax objectives we conclude that the mixed risk neutral minimax family is more effective at the cost of increased controller complexity

what is the goal of sensory coding a number of recent attempts have been made to describe early sensory coding in terms of a general information processing strategy in this paper two strategies are contrasted both strategies take advantage of the redundancy in the environment to produce more effective representations the first is described as a compact coding scheme a compact code performs a transform that allows the input to be represented with a reduced number of vectors cells with minimal rms error this approach has recently become popular in the neural network literature and is related to a process called principal components analysis pca a number of recent papers have suggested that the optimal compact code for representing natural scenes will have units with receptive field profiles much like those found in the retina and primary visual cortex however in this paper it is proposed that compact coding schemes are insufficient to account for the receptive field properties of cells in the mammalian visual pathway in contrast it is proposed that the visual system is near to optimal in representing natural scenes only if optimality is defined in terms of sparse distributed coding in a sparse distributed code all cells in the code have an equal response probability across the class of images but have a low response probability for any single image in such a code the dimensionality is not reduced rather the redundancy of the input is transformed into the redundancy of the firing pattern of cells it is proposed that the signature for a sparse code is found in the fourth moment of the response distribution i e the kurtosis in measurements with 55 calibrated natural scenes the kurtosis was found to peak when the bandwidths of the visual code matched those of cells in the mammalian visual cortex codes resembling wavelet transforms are proposed to be effective because the response histograms of such codes are sparse i e show high kurtosis when presented with natural scenes it is proposed that the structure of the image that allows sparse coding is found in the phase spectrum of the image it is suggested that natural scenes to a first approximation can be considered as a sum of self similar local functions the inverse of a wavelet possible reasons for why sensory systems would evolve toward sparse coding are presented

robust control of partially observable failing systems this paper is concerned with optimal maintenance decision making in the presence of model misspecification specifically we are interested in the situation where the decision maker fears that a nominal bayesian model may be miss specified or unrealistic and would like to find policies that work well even when the underlying model is flawed to this end we formulate a robust dynamic optimization model for condition based maintenance in which the decision maker explicitly accounts for distrust in the nominal bayesian model by solving a worst case problem against a second agent nature who has the ability to alter the underlying model distributions in an adversarial manner the primary focus of our analysis is on establishing structural properties and insights that hold in the face of model miss specification in particular we prove i an explicit characterization of nature s optimal response through an analysis of the robust dynamic programming equation ii convexity results for both the robust va

occlusion models for natural images a statistical study of a scale invariant dead leaves model we develop a scale invariant version of matheron s dead leaves model for the statistics of natural images the model takes occlusions into account and resembles the image formation process by randomly adding independent elementary shapes such as disks in layers we compare the empirical statistics of two large databases of natural images with the statistics of the occlusion model and find an excellent qualitative and good quantitative agreement at this point this is the only image model which comes close to duplicating the simplest elementary statistics of natural images such as the scale invariance property of marginal distributions of filter responses the full co occurrence statistics of two pixels and the joint statistics of pairs of haar wavelet responses

spaces of distributions interpolation by translates of a basis function and error estimates interpolation with translates of a basis function is a common process in approximation theory the most elementary form of the interpolant consists of a linear combination of all translates by interpolation points of a single basis function frequently low degree polynomials are added to the interpolant one of the significant features of this type of interpolant is that it is often the solution of a variational problem in this paper we concentrate on developing a wide variety of spaces for which a variational theory is available for each of these spaces we show that there is a natural choice of basis function we also show how the theory leads to efficient ways of calculating the interpolant and to new error estimates

universal analytical forms for modeling image probabilities seeking probability models for images we employ a spectral approach where the images are decomposed using bandpass filters and probability models are imposed on the filter outputs also called spectral components we employ a two parameter family of probability densities called bessel k forms for modeling the marginal densities of the spectral components and demonstrate their fit to the observed histograms for video infrared and range images motivated by object based models for image analysis a relationship between the bessel parameters and the imaged objects is established using l sup 2 metric on the set of bessel k forms we propose a pseudometric on the image space for quantifying image similarities differences some applications including clutter classification and pruning of hypotheses for target recognition are presented

robust stochastic maximum principle for multi model worst case optimization this paper develops a version of the robust maximum principle applied to the minimax mayer problem formulated for stochastic differential equations with a control dependent diffusion term the parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each possible value of the parameter the cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term a terminal condition given by a vector function is also considered the optimal control strategies adapted for available information for the wide class of multi model systems given by a stochastic differential equation with parameters from a given finite set are constructed this problem belongs to the class of minimax stochastic optimization problems the p

a dynamic game approach to distributionally robust safety specifications for stochastic systems abstract this paper presents a new safety specification method that is robust against errors in the probability distribution of disturbances our proposed distributionally robust safe policy maximizes the probability of a system remaining in a desired set for all times subject to the worst possible disturbance distribution in an ambiguity set we propose a dynamic game formulation of constructing such policies and identify conditions under which a non randomized markov policy is optimal based on this existence result we develop a practical design approach to safety oriented stochastic controllers with limited information about disturbance distributions however an associated bellman equation involves infinite dimensional minimax optimization problems since the disturbance distribution may have a continuous density to alleviate computational issues we propose a duality based reformulation method that converts the infinite dimensional minimax problem into a semi infinite program that can be solved using existing convergent algorithms we prove that there is no duality gap and that this approach thus preserves optimality the results of numerical tests confirm that the proposed method is robust against distributional errors in disturbances while a standard stochastic safety verification tool is not

measurable selection theorems for minimax stochastic optimization problems this paper resolves the measurability questions which arise in the analysis of various minimax stochastic optimization models posed in borel spaces for a large class of such models we provide a set of sufficient conditions which ensure that the questions have positive answers these conditions are essentially weaker than those described in the existing literature moreover they allow us to conclude a fundamental theorem yielding some new results on the existence of universally measurable selections of extrema for both zero sum stochastic games and minimax stochastic optimal control problems in particular a random version of the well known minimax theorem of fan is in this way established the fundamental theorem yields also a descriptive set theoretic fact concerning the projections of borel sets the paper indicates also some counterexamples to possibly more general problems

self similarity properties of natural images resemble those of turbulent flows we show that the statistics of an edge type variable in natural images exhibits self similarity properties which resemble those of local energy dissipation in turbulent flows our results show that selfsimilarity and extended self similarity hold remarkably for the statistics of the local edge variance and that the very same models can be used to predict all of the associated exponents these results suggest using natural images as a laboratory for testing more elaborate scaling models of interest for the statistical description of turbulent flows the properties we have exhibited are relevant for the modeling of the early visual system they should be included in models designed for the prediction of receptive fields s0031 9007 97 05190 9

on the optimality equation for average cost markov control processes with feller transition probabilities we consider markov control processes with borel state space and feller transition probabilities satisfying some generalized geometric ergodicity conditions we provide a new theorem on the existence of a solution to the average cost optimality equation

zero sum stochastic games in borel spaces average payoff criteria this paper is concerned with two person zero sum dynamic stochastic games in borel spaces with possibly unbounded payoff function and several average or ergodic payoff criteria we give conditions under which the long run expected average payoff criterion the sample path average criterion the existence of solutions to the average payoff shapley equations and a certain martingale condition are all equivalent

minimax control of discrete time stochastic systems this paper gives a unified self contained presentation of minimax control problems for discrete time stochastic systems on borel spaces with possibly unbounded costs the main results include conditions for the existence of minimax strategies for finite horizon problems and infinite horizon discounted and undiscounted average cost criteria the results are specialized to control systems with unknown disturbance distributions also known as games against nature two examples illustrate the theory one of them on the mold level control problem which is a key problem in the steelmaking industry

on the use of stochastic differential games against nature to ergodic control problems with unknown parameters in this paper we propose an application of the so called games against nature for giving solution to an ergodic control problem governed by a general class of markov diffusion processes whose coefficients depend on an unknown and non observable parameter to this end we assume that the values of the parameter are taken by means of actions made by some opposite player of the controller the nature then the problem reduces to finding optimality for the controller given that the nature has chosen its best strategy such a control is also known as the worst case optimal control our analysis is based on the use of the dynamic programming technique by showing among other facts the existence of classical twice differentiable solutions of the so called hamilton jacobi bellman equation we also provide an example on economic welfare to illustrate our results

borel stochastic games with lim sup payoff we consider two person zero sum stochastic games with limit superior payoff function and borel measurable state and action spaces the games are shown to have a value and the value function is calculated by transfinite iteration of an operator and proved to be upper analytic the paper extends results of our earlier article 17 in which the same class of games was considered for countable state spaces and finite action sets

a parametric texture model based on joint statistics of complex wavelet coefficients we present a universal statistical model for texture images in the context of an overcomplete complex wavelet transform the model is parameterized by a set of statistics computed on pairs of coefficients corresponding to basis functions at adjacent spatial locations orientations and scales we develop an efficient algorithm for synthesizing random images subject to these constraints by iteratively projecting onto the set of images satisfying each constraint and we use this to test the perceptual validity of the model in particular we demonstrate the necessity of subgroups of the parameter set by showing examples of texture synthesis that fail when those parameters are removed from the set we also demonstrate the power of our model by successfully synthesizing examples drawn from a diverse collection of artificial and natural textures

robust optimal control for minimax stochastic linear quadratic problem the robust maximum principle applied to the minimax linear quadratic problem is derived for stochastic differential equations containing a control dependent diffusion term the parametric families of the first and second order adjoint stochastic processes are obtained to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each value of the uncertain parameter from a given finite set the cost function is considered on a finite horizon contains the mathematical expectation of both an integral and a terminal term and on an infinite one a time averaged losses function these problems belong to the class of minimax stochastic optimization problems it is shown that the construction of the minimax optimal controller can be reduced to an optimization problem on a finitedimensional simplex and consists in the analysis of the dependence of ric

robust maximum principle for multi model lq problem this paper presents the version of the robust maximum principle in the context of multi model control formulated as the minimax bolza problem the cost function contains a terminal term as well as an integral one a fixed horizon and terminal set are considered the necessary conditions of the optimality are derived for the class of uncertain systems given by an ordinary differential equation with parameters from a given finite set this problem consists in the control design providing a good behaviour for a given class of multi model system it is shown that the design of the minimax optimal controller is reduced to a finite dimensional optimization problem given at the corresponding simplex set containing the weight parameters to be found the robust optimal control may be interpreted as a mixture with the optimal weights of the controls which are optimal for each fixed parameter value the proof is based on the recent results obtained for minimax mayer problem boltyanski and poznyak 1999a the mini

statistics of natural images scaling in the woods we study the statistics of an ensemble of images taken in the woods distributions of local quantities such as contrast are scale invariant and have nearly exponential tails power spectra exhibit scaling with a nontrivial exponent these data limit the information content of natural images and point to the importance of gain control strategies in visual processing

a unified theory of radial basis functions native hilbert spaces for radial basis functions ii abstract this contribution provides a new formulation of the theory of radial basis functions in the context of integral operators instead of fourier transforms the most important tools now are expansions into eigenfunctions this unifies the theory of radial basis functions in r d with the theory of zonal functions on the sphere sd 1 and the theory of kernel functions on riemannian manifolds new characterizations of native spaces and positive definite functions are included the paper is a self contained continuation of an earlier survey r schaback international series of numerical mathematics vol 132 birkhauser basel 1999 pp 255 282 over the native spaces associated to not necessarily radial basis functions

vectorial interpolation using radial basis like functions this paper deals with vector field interpolation i e the data are r3 values located in scattered r3 points while the interpolating function is a function from r3 into r3 in order to take into account possible connections between the components of the interpolant we derive it by solving a variational spline problem involving the rotational and the divergence of the interpolant and depending on a parameter significative of the balance of the rotational part and of the divergence part and on the order m of derivatives of the rotational and divergence involved in the minimized seminorm we so obtain interpolants whose expression is x ni 1 x xi ai pm 1 x where is some 3 3 matrix function pm 1 is a degree m 1 vectorial polynomial and where the ai are r3 vectors besides the ai meet a relation generalizing the usual orthogonality to all polynomials of degree at most m 1 for 1 we find the usual m harmonic splines in each component of numerical examples show the interest of the method and we compare the so obtained functions with the ones obtained by matlab s procedures

random cascades on wavelet dyadic trees we introduce a new class of random fractal functions using the orthogonal wavelet transform these functions are built recursively in the space scale half plane of the orthogonal wavelet transform cascading from an arbitrary given large scale towards small scales to each random fractal function corresponds a random cascading process referred to as a w cascade on the dyadic tree of its orthogonal wavelet coefficients we discuss the convergence of these cascades and the regularity of the so obtained random functions by studying the support of their singularity spectra then we show that very different statistical quantities such as correlation functions on the wavelet coefficients or the wavelet based multifractal formalism partition functions can be used to characterize very precisely the underlying cascading process we illustrate all our results on various numerical examples

minimax selection theorems on considere un jeu a somme nulle et a 2 joueurs et on donne certaines conditions generales nouvelles qui assurent l existence et la mesurabilite de la fonction valeur du jeu et l existence de strategies mesurables optimales pour les deux joueurs

on non scale invariant infinitely divisible cascades multiplicative processes multifractals and more recently also infinitely divisible cascades have seen increased popularity in a host of applications requiring versatile multiscale models ranging from hydrodynamic turbulence to computer network traffic from image processing to economics the methodologies prevalent as of today rely to a large extent on iterative schemes used to produce infinite detail and repetitive structure across scales while appealing due to their simplicity these constructions have limited applicability as they lead by default to power law progression of moments through scales to nonstationary increments and often to inherent log periodic scaling which favors an exponential set of scales this paper studies and develops a wide class of infinitely divisible cascades idc thereby establishing the first reported cases of controllable scaling of moments in non power law form embedded in the framework of idc these processes exhibit stationary increments and scaling over a continuous range of scales criteria for convergence further statistical properties as well as matlab routines are provided

a liapounov bound for solutions of the poisson equation in this paper we consider psi irreducible markov processes evolving in discrete or continuous time on a general state space we develop a liapounov function criterion that permits one to obtain explicit bounds on the solution to the poisson equation and in particular obtain conditions under which the solution is square integrable these results are applied to obtain sufficient conditions that guarantee the validity of a functional central limit theorem for the markov process as a second consequence of the bounds obtained a perturbation theory for markov processes is developed which gives conditions under which both the solution to the poisson equation and the invariant probability for the process are continuous functions of its transition kernel the techniques are illustrated with applications to queueing theory and autoregressive processes

the multifractal structure of contrast changes in natural images from sharp edges to textures we present a formalism that leads naturally to a hierarchical description of the different contrast structures in images providing precise definitions of sharp edges and other texture components within this formalism we achieve a decomposition of pixels of the image in sets the fractal components of the image such that each set contains only points characterized by a fixed strength of the singularity of the contrast gradient in its neighborhood a crucial role in this description of images is played by the behavior of contrast differences under changes in scale contrary to naive scaling ideas where the image is thought to have uniform transformation properties field 1987 each of these fractal components has its own transformation law and scaling exponents a conjecture on their biological relevance is also given

multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws we define a large class of continuous time multifractal random measures and processes with arbitrary log infinitely divisible exact or asymptotic scaling law these processes generalize within a unified framework both the recently defined log normal multifractal random walk j f muzy j delour and e bacry eur j phys b 17 537 2000 e bacry j delour and j f muzy phys rev e 64 026103 2001 and the log poisson product of cylindrical pulses j barral and b b mandelbrot cowles foundation discussion paper no 1287 2001 unpublished our construction is based on some continuous stochastic multiplication as introduced in f schmitt and d marsan eur j phys b 20 3 2001 from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades we prove the stochastic convergence of the defined processes and study their main statistical properties the question of genericity universality of limit multifractal processes is addressed within this new framework we finally provide a method for numerical simulations and discuss some specific examples

advertising models stochastic games and myopic strategies this paper analyzes dynamic advertising models we present conditions that imply that the payoffs under myopic strategies are linear in the state variable which represents the number of customers for each firm these conditions which determine a subclass of stochastic games are satisfied by several dynamic advertising models we give a criterion to check whether this model contains an equilibrium point within the class of myopic strategies that is also an equilibrium point with regard to all strategies

zero sum average semi markov games fixed point solutions of the shapley equation this paper deals with zero sum average semi markov games with borel state and action spaces unbounded payoffs and mean holding times a solution to the shapley equation is obtained via the banach fixed point theorem assuming that the model satisfies a lyapunov like condition a growth hypothesis on the payoff function and the mean holding times besides standard continuity and compactness requirements

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

risk averse dynamic programming for markov decision processes we introduce the concept of a markov risk measure and we use it to formulate risk averse control problems for two markov decision models a finite horizon model and a discounted infinite horizon model for both models we derive risk averse dynamic programming equations and a value iteration method for the infinite horizon problem we develop a risk averse policy iteration method and we prove its convergence we also propose a version of the newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence finally we discuss relations to min max markov decision models

modeling and control of stochastic systems with poorly known dynamics this paper is concerned with controlling poorly known systems for which only a simplified and rough model is available for control design there are many systems that cannot be reasonably probed for the sake of identification yet they are important for areas such as economy populations or medicine the ideas are developed around an alternative way to account for the bare modeling in a stochastic based setting and to heighten the control features for such a modified model the mathematical framework for the optimal control reveals important features such as the raising of a precautionary feedback policy of keep the action unchanged inaction for short on a certain state space region this feature is not seen in the robust approach but has been pointed out and permeates part of the economics literature the control problem relies on the viscosity solution for the hamilton jacobi bellman equation and the value of the problem is shown to be convex when specialized to the quadratic problem with discounted cost the exact solution inside the inaction region is given by a lyapunov type of equation and asymptotically for large state values by a riccati like equation this scenario bridges to the stochastic stability analysis for the controlled model the single control input is developed in full part analytically part numerically for the scalar case and an approximation is tested for the multidimensional case the advantage of the precautionary policy is substantial in some situations

analytical form for a bayesian wavelet estimator of images using the bessel k form densities a novel bayesian nonparametric estimator in the wavelet domain is presented in this approach a prior model is imposed on the wavelet coefficients designed to capture the sparseness of the wavelet expansion seeking probability models for the marginal densities of the wavelet coefficients the new family of bessel k forms bkf densities are shown to fit very well to the observed histograms exploiting this prior we designed a bayesian nonlinear denoiser and we derived a closed form for its expression we then compared it to other priors that have been introduced in the literature such as the generalized gaussian density ggd or the spl alpha stable models where no analytical form is available for the corresponding bayesian denoisers specifically the bkf model turns out to be a good compromise between these two extreme cases hyperbolic tails for the spl alpha stable and exponential tails for the ggd moreover we demonstrate a high degree of match between observed and estimated prior densities using the bkf model finally a comparative study is carried out to show the effectiveness of our denoiser which clearly outperforms the classical shrinkage or thresholding wavelet based techniques