the information lost in erasures we consider sources and channels with memory observed through erasure channels in particular we examine the impact of sporadic erasures on the fundamental limits of lossless data compression lossy data compression channel coding and denoising we define the erasure entropy of a collection of random variables as the sum of entropies of the individual variables conditioned on all the rest the erasure entropy measures the information content carried by each symbol knowing its context the erasure entropy rate is shown to be the minimal amount of bits per erasure required to recover the lost information in the limit of small erasure probability when we allow recovery of the erased symbols within a prescribed degree of distortion the fundamental tradeoff is described by the erasure rate distortion function which we characterize we show that in the regime of sporadic erasures knowledge at the encoder of the erasure locations does not lower the rate required to achieve a given distortion when no additional encoded information is available the erased information is reconstructed solely on the basis of its context by a denoiser connections between erasure entropy and discrete denoising are developed the decrease of the capacity of channels with memory due to sporadic memoryless erasures is also characterized in wide generality

the true self repelling motion we construct and study a continuous real valued random process which is of a new type it is self interacting self repelling but only in a local sense it only feels the self repellance due to its occu pation time measure density in the immediate neighbourhood of the point it is just visiting we focus on the most natural process with these properties that we call true self repelling motion this is the continuous counterpart to the integer valued true self avoiding walk which had been studied among others by the first author one of the striking properties of true self repelling motion is that although the coupleoxt occupation time measure of x at time tu is a continuous markov process x is not driven by a stochastic diaerential equation and is not a semi martingale it turns out for instance that it has a finite variation of order 3 2 which contrasts with the finite quadratic variation of semi martingales one of the key tools in the construction of x is a continuous system of coalescing brownian motions similar to those that have been constructed by arratia a1 a2 we derive various properties of x existence and properties of the occupation time densities ltoxu local variation etc and an identity that shows that the dynamics of x can be very loosely speaking described as follows dxt is equal to the gradient in space of ltoxu in a gener alized sense even though x7 ltoxu is not diaerentiable

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

the brownian web arratia arratia r 1979 ph d thesis university of wisconsin madison and unpublished work and later toth and werner toth b werner w 1998 probab theory relat fields111 375 452 constructed random processes that formally correspond to coalescing one dimensional brownian motions starting from every space time point we extend their work by constructing and characterizing what we call the brownian web as a random variable taking values in an appropriate metric space whose points are compact sets of paths this leads to general convergence criteria and in particular to convergence in distribution of coalescing random walks in the scaling limit to the brownian web

expansions and contractions of isotropic stochastic flows of homeomorphisms a sequence of piecewise constant approximations to rescaled isotropic homeomorphic stochastic flows is shown to converge weakly in skorohod metric to the coalescing brownian flow intermittent behavior of isotropic flows is exposed and the clustering properties of isotropic flows are studied by the means of this convergence we obtain qualitative and quantitative description of expansions and contractions of an arbitrary isotropic homeomorphic flow on large time and space scales

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

asymptotical statistics of misspecified hidden markov models this paper deals with the problem of modeling data generated by an ergodic stochastic process as the output of a hidden markov model hmm more specifically we consider the problem of fitting a parametric family of hmm with continuous output to an ergodic stochastic process with continuous values which does not necessarily belong to the family in this context we derive the main asymptotic results almost sure consistency of the maximum likelihood estimator asymptotic normality of the estimation error and the exact rates of almost sure convergence

universal fir mmse filtering we consider the problem of causal estimation i e filtering of a real valued signal corrupted by zero mean time independent real valued additive noise under the mean squared error mse criterion we build a universal filter whose per symbol squared error for every bounded underlying signal is essentially as small as that of the best finite duration impulse response fir filter of a given order we do not assume a stochastic mechanism generating the underlying signal and assume only that the variance of the noise is known to the filter the regret of the expected mse of our scheme is shown to decay as o logn n where n is the length of the signal moreover we present a stronger concentration result which guarantees the performance of our scheme not only in expectation but also with high probability our result implies a conventional stochastic setting result i e when the underlying signal is a stationary process our filter achieves the performance of the optimal fir filter we back our theoretical findings with several experiments showcasing the potential merits of our universal filter in practice our analysis combines tools from the problems of universal filtering and competitive online regression

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

locally adaptive bandwidth choice for kernel regression estimators abstract kernel estimators with a global bandwidth are commonly used to estimate regression functions on the other hand it is obvious that the choice of a local bandwidth can lead to better results because a larger class of kernel estimators is available evidently this may in turn affect variability the optimal bandwidths depend essentially on the regression function itself and on the residual variance and it is desirable to estimate them from the data in this article a local bandwidth estimator is studied a comparison with its global bandwidth equivalent is performed both in theory and in simulations as the main result it is shown that the possible gain in mean integrated squared error of the resulting regression estimator must be paid for by a larger variability of the estimator this may lead to worse results if the sample size is small an algorithm has been devised that puts special weight on stability aspects our simulation study shows that improvements over a global bandwidth estimator

asymptotic normality of the maximum likelihood estimator for general hidden markov models hidden markov models hmms have during the last decade become a widespread tool for modeling sequences of dependent random variables inference for such models is usually based on the maximum likelihood estimator mle and consistency of the mle for general hmms was recently proved by leroux in this paper we show that under mild conditions the mle is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the fisher information

linear minimax regret estimation of deterministic parameters with bounded data uncertainties we develop a new linear estimator for estimating an unknown parameter vector x in a linear model in the presence of bounded data uncertainties the estimator is designed to minimize the worst case regret over all bounded data vectors namely the worst case difference between the mean squared error mse attainable using a linear estimator that does not know the true parameters x and the optimal mse attained using a linear estimator that knows x we demonstrate through several examples that the minimax regret estimator can significantly increase the performance over the conventional least squares estimator as well as several other least squares alternatives

modified massive arratia flow and wasserstein diffusion extending previous work arxiv 1408 0628 by the first author we present a variant of the arratia flow which consists of a collection of coalescing brownian motions starting from every point of the unit interval the important new feature of the model is that individual particles carry mass which aggregates upon coalescence and which scales the diffusivity of each particle in an inverse proportional way in this work we relate the induced measure valued process to the wasserstein diffusion of arxiv 0704 0704 first we present the process as a martingale solution to a spde similar to arxiv 0704 0704 second as our main result we show a varadhan formula for short times which is governed by the quadratic wasserstein distance

state classification for a class of measure valued branching diffusions in a brownian medium the spatial structure of a new class of measure valued diffusions which arise as limits in distribution of a sequence of interacting branching particle systems is investigated we obtain the following criterion of state classification for these superprocesses their effective state space is contained in the set of purely atomic measures or the set of absolutely continuous measures according as e 0 or e 0 when the coefficient of the motion generator is a smooth function

discrete denoising with shifts we introduce s dude a new algorithm for denoising discrete memoryless channel dmc corrupted data the algorithm which generalizes the recently introduced dude discrete universal denoiser aims to compete with a genie that has access in addition to the noisy data also to the underlying clean data and that can choose to switch up to m times between sliding window denoisers in a way that minimizes the overall loss when the underlying data form an individual sequence we show that the s dude performs essentially as well as this genie provided that m is sublinear in the size of the data when the clean data are emitted by a piecewise stationary process we show that the s dude achieves the optimum distribution dependent performance provided that the same sublinearity condition is imposed on the number of switches to further substantiate the universal optimality of the s dude we show that when the number of switches is allowed to grow linearly with the size of the data any sequence of scheme s fails to compete in the above sense using dynamic programming we derive an efficient implementation of the s dude which has complexity time and memory growing linearly with the data size and the number of switches m preliminary experimental results are presented suggesting that s dude has the capacity to improve on the performance attained by the original dude in applications where the nature of the data abruptly changes in time or space as is often the case in practice

estimating the number of states of a finite state source the problem of estimating the number of states of a finite alphabet finite state source is investigated an estimator is developed that asymptotically attains the minimum probability of understanding the number of states among all estimators with a prescribed exponential decay rate of overestimation probability the proposed estimator relies on the lempel ziv data compression algorithm in an intuitively appealing manner

polynomial boundary treatment for wavelet regression summary to overcome boundary problems with wavelet regression we propose a simple method that reduces bias at the boundaries it is based on a combination of wavelet functions and low order polynomials the utility of the method is illustrated with simulation studies and a real example asymptotic results show that the estimators are competitive with other nonparametric procedures and wavelet regression suffer from boundary or edge effects to overcome these problems when dealing with smooth functions eubank speckman 1990 1991a suggested a simple method called polynomial trigonometric regression in which they write the estimator of f as a sum of trigonometric functions and a low order polynomial the latter is expected to account for the boundary problem they illustrated their approach with examples and showed that the rates of convergence for their estimators over a particular smoothness class of functions are optimal whether or not the regression function is periodic however as noted by these authors sharp peaks in the regression function can cause a ringing phenomenon and other wiggles and this limits the usefulness of polynomial trigonometric regression to extend this procedure to functions with singularities a possible solution is to replace the trigonometric part of the estimator by an estimator based on wavelets while keeping the polynomial part to model the boundaries the choice of wavelets as a replacement for the trigonometric part is a natural one since wavelets are well known to be much more efficient than trigonometric functions when fitting curves with sharp features the main goal of this paper is to study these new estimators that are combinations of low order polynomial terms and wavelet terms hereafter referred to as polynomial wavelet regression in 2 wavelet regression is briefly reviewed and some of the existing methods to correct boundary problems in wavelet regression are recalled a detailed description of the polynomial wavelet regression method is presented in 3 the asymptotic properties of the method are discussed in

a competitive minimax approach to robust estimation of random parameters we consider the problem of estimating in the presence of model uncertainties a random vector x that is observed through a linear transformation h and corrupted by additive noise we first assume that both the covariance matrix of x and the transformation h are not completely specified and develop the linear estimator that minimizes the worst case mean squared error mse across all possible covariance matrices and transformations h in the region of uncertainty although the minimax approach has enjoyed widespread use in the design of robust methods we show that its performance is often unsatisfactory to improve the performance over the minimax mse estimator we develop a competitive minimax approach for the case where h is known but the covariance of x is subject to uncertainties and seek the linear estimator that minimizes the worst case regret namely the worst case difference between the mse attainable using a linear estimator ignorant of the signal covariance and the optimal mse attained using a linear estimator that knows the signal covariance the linear minimax regret estimator is shown to be equal to a minimum mse mmse estimator corresponding to a certain choice of signal covariance that depends explicitly on the uncertainty region we demonstrate through examples that the minimax regret approach can improve the performance over both the minimax mse approach and a plug in approach in which the estimator is chosen to be equal to the mmse estimator with an estimated covariance matrix replacing the true unknown covariance we then show that although the optimal minimax regret estimator in the case in which the signal and noise are jointly gaussian is nonlinear we often do not lose much by restricting attention to linear estimators

fluctuation results for hastings levitov planar growth we study the fluctuations of the outer domain of hastings levitov clusters in the small particle limit these are shown to be given by a continuous gaussian process mathcal f taking values in the space of holomorphic functions on z 1 of which we provide an explicit construction the boundary values mathcal w of mathcal f are shown to perform an ornstein uhlenbeck process on the space of distributions on the unit circle mathbb t which can be described as the solution to the stochastic fractional heat equation begin aligned frac partial partial t mathcal w t vartheta delta 1 2 mathcal w t vartheta sqrt 2 xi t vartheta end aligned where delta denotes the laplace operator acting on the spatial component and xi t vartheta is a space time white noise as a consequence we find that when the cluster is left to grow indefinitely the boundary process mathcal w converges to a log correlated fractional gaussian field which can be realised as delta 1 4 w for w complex white noise on mathbb t

jacobian nullwerte and algebraic equations abstract we present two applications of jacobian nullwerte both related with the resolution of algebraic equations of any degree we give a very simple expression of the roots of a polynomial of arbitrary degree in terms of derivatives of hyperelliptic theta functions this expression can be understood as an explicit proof of torelli s theorem in the hyperelliptic case we also give geometrical expressions of the discriminant of a polynomial both applications are based on a jacobian version of thomae s formula

universal switching linear least squares prediction in this paper we consider sequential regression of individual sequences under the square error loss we focus on the class of switching linear predictors that can segment a given individual sequence into an arbitrary number of blocks within each of which a fixed linear regressor is applied using a competitive algorithm framework we construct sequential algorithms that are competitive with the best linear regression algorithms for any segmenting of the data as well as the best partitioning of the data into any fixed number of segments where both the segmenting of the data and the linear predictors within each segment can be tuned to the underlying individual sequence the algorithms do not require knowledge of the data length or the number of piecewise linear segments used by the members of the competing class yet can achieve the performance of the best member that can choose both the partitioning of the sequence as well as the best regressor within each segment we use a transition diagram f m j willems 1996 to compete with an exponential number of algorithms in the class using complexity that is linear in the data length the regret with respect to the best member is o ln n per transition for not knowing the best transition times and o ln n for not knowing the best regressor within each segment where n is the data length we construct lower bounds on the performance of any sequential algorithm demonstrating a form of min max optimality under certain settings we also consider the case where the members are restricted to choose the best algorithm in each segment from a finite collection of candidate algorithms performance on synthetic and real data are given along with a matlab implementation of the universal switching linear predictor

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

weak convergence of the localized disturbance flow to the coalescing brownian flow we define a new state space for the coalescing brownian flow also known as the brownian web on the circle the elements of this space are families of order preserving maps of the circle depending continuously on two time parameters and having a certain weak flow property the space is equipped with a complete separable metric a larger state space allowing jumps in time is also introduced and equipped with a skorokhod type metric also complete and separable we prove that the coalescing brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps each jump arising from a small localized disturbance of the circle a local version of this result is also obtained in which the weak limit law is that of the coalescing brownian flow on the line our set up is well adapted to time reversal and our weak limit result provides a new proof of time reversibility of the coalescing brownian flow we also identify a martingale associated with the coalescing brownian flow on the circle and use this to make a direct calculation of the laplace transform of the time to complete coalescence

isotropic stochastic flows on considere les flots stochastiques isotropes dans un espace euclidien de d dimensions d 2 on relie la tendance des distances entre deux points et des vecteurs tangents a se borner ou s etendre a la dimension et a la proportion du flot solenoidal ou potentiel

shearlet based total variation diffusion for denoising we propose a shearlet formulation of the total variation tv method for denoising images shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization common approaches in combining wavelet like representations such as curvelets with tv or diffusion methods aim at reducing gibbs type artifacts after obtaining a nearly optimal estimate we show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain we also analyze the performance of a shearlet based diffusion method numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform furthermore the shearlet tv scheme requires far fewer iterations than similar competitors

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

a system of coalescing heavy diffusion particles on the real line we construct a modified arratia flow with mass and energy conservation we suppose that particles have a mass obeying the conservation law and their diffusion is inversely proportional to the mass our main result asserts that such a system exists under the assumption of the uniform mass distribution on an interval at the starting moment we introduce a stochastic integral with respect to such a flow and obtain the total local time as the density of the occupation measure for all particles

universal prediction of individual binary sequences in the presence of noise the problem of predicting the next outcome of an individual binary sequence based on noisy observations of the past is considered the goal of the predictor is to perform for each individual sequence almost as well as the best in a set of experts where performance is evaluated using a general loss function a comprehensive approach to prediction in this noisy setting is presented and proven generally efficient under appropriate conditions as an illustration of the applicability of the approach suggested for concrete situations two important special cases are explicitly treated the first is the case where the data corrupting noise process is binary valued where the observed bit is the bitwise xor of the clean bit and the noise bit the second case is that of real valued additive noise it is shown that even in this more challenging situation where the information available to the predictor regarding the past sequence is incomplete a predictor can be guaranteed to successfully compete with a whole set of experts in considerably strong senses

universal filtering via prediction we consider the filtering problem where a finite alphabet individual sequence is corrupted by a discrete memoryless channel and the goal is to causally estimate each sequence component based on the past and present noisy observations we establish a correspondence between the filtering problem and the problem of prediction of individual sequences which leads to the following result given an arbitrary finite set of filters there exists a filter which performs with high probability essentially as well as the best in the set regardless of the underlying noiseless individual sequence we use this relationship between the problems to derive a filter guaranteed of attaining the finite state filterability of any individual sequence by leveraging results from the prediction problem

strongly consistent code based identification and order estimation for constrained finite state model classes observations are made of data generated by a stationary ergodic finite alphabet information source according to an unknown statistical model two modeling problems the identification problem and the order estimation problem are considered it is required that the given model class in the identification problem and each given model class in the order estimation problem be a constrained finite state model class which is a type that includes many model classes of information theoretic interest strongly consistent decision rules are exhibited in both the identification problem and the order estimation problem the decision rules are code based in that a model class is chosen based on how well a certain code for that class encodes the observed data the code used for a model class is based on the maximum likelihood code for that class and asymptotic code performance is gauged by means of a key property of divergence rate distance

universal linear least squares prediction upper and lower bounds we consider the problem of sequential linear prediction of real valued sequences under the square error loss function for this problem a prediction algorithm has been demonstrated whose accumulated squared prediction error for every bounded sequence is asymptotically as small as the best fixed linear predictor for that sequence taken from the class of all linear predictors of a given order p the redundancy or excess prediction error above that of the best predictor for that sequence is upper bounded by a sup 2 p ln n n where n is the data length and the sequence is assumed to be bounded by some a we provide an alternative proof of this result by connecting it with universal probability assignment we then show that this predictor is optimal in a min max sense by deriving a corresponding lower bound such that no sequential predictor can ever do better than a redundancy of a sup 2 p ln n n

multiscale poisson data smoothing summary the paper introduces a framework for non linear multiscale decompositions of poisson data that have piecewise smooth intensity curves the key concept is conditioning on the sum of the observations that are involved in the computation of a given multiscale coefficient within this framework most classical wavelet thresholding schemes for data with additive homoscedastic noise can be used any family of wavelet transforms orthogonal biorthogonal or second generation can be incorporated in this framework our second contribution is to propose a bayesian shrinkage approach with an original prior for coefficients of this decomposition as such the method combines the advantages of the haar fisz transform with wavelet smoothing and bayesian multiscale likelihood models with additional benefits such as extendability towards arbitrary wavelet families simulations show an important reduction in average squared error of the output compared with the present techniques of anscombe or fisz variance stabilization or multiscale likelihood modelling

intermittency of the tracer gradient the problem of stirring a passive scalar tracer by a random velocity field is considered for gaussian velocity fields with infinitely small time and space scales it is shown that the tracer gradient is concentrated in a discrete set of points the points that form a random point process independent of the initial tracer distribution a complete description of this point process is given if the initial tracer field is a random function with homogeneous increments a full statistics of the jumps is also given

excited brownian motions as limits of excited random walks we obtain the convergence in law of a sequence of excited also called cookies random walks toward an excited brownian motion this last process is a continuous semi martingale whose drift is a function say of its local time it was introduced by norris rogers and williams as a simplified version of brownian polymers and then recently further studied by the authors to get our results we need to renormalize together the sequence of cookies the time and the space in a convenient way the proof follows a general approach already taken by toth and his coauthors in multiple occasions which goes through ray knight type results namely we first prove when is bounded and lipschitz that the convergence holds at the level of the local time processes this is done via a careful study of the transition kernel of an auxiliary markov chain which describes the local time at a given level then we prove a tightness result and deduce the convergence at the level of the full processes

universal piecewise linear prediction via context trees this paper considers the problem of piecewise linear prediction from a competitive algorithm approach in prior work prediction algorithms have been developed that are universal with respect to the class of all linear predictors such that they perform nearly as well in terms of total squared prediction error as the best linear predictor that is able to observe the entire sequence in advance in this paper we introduce the use of a context tree to compete against a doubly exponential number of piecewise linear affine models we use the context tree to achieve the total squared prediction error performance of the best piecewise linear model that can choose both its partitioning of the regressor space and its real valued prediction parameters within each region of the partition based on observing the entire sequence in advance uniformly for every bounded individual sequence this performance is achieved with a prediction algorithm whose complexity is only linear in the depth of the context tree per prediction upper bounds on the regret with respect to the best piecewise linear predictor are given for both the scalar and higher order case and lower bounds on the regret are given for the scalar case an explicit algorithmic description and examples demonstrating the performance of the algorithm are given

universal denoising of discrete time continuous amplitude signals we consider the problem of reconstructing a discrete time signal sequence with continuous valued components corrupted by a known memoryless channel when performance is measured using a per symbol loss function satisfying mild regularity conditions we develop a sequence of denoisers that although independent of the distribution of the underlying ldquocleanrdquo sequence is universally optimal in the limit of large sequence length this sequence of denoisers is universal in the sense of performing as well as any sliding window denoising scheme which may be optimized for the underlying clean signal our results are initially developed in a ldquosemi stochasticrdquo setting where the noiseless signal is an unknown individual sequence and the only source of randomness is due to the channel noise it is subsequently shown that in the fully stochastic setting where the noiseless sequence is a stationary stochastic process our schemes universally attain optimum performance the proposed schemes draw from nonparametric density estimation techniques and are practically implementable we demonstrate efficacy of the proposed schemes in denoising gray scale images in the conventional additive white gaussian noise awgn setting with additional promising results for less conventional noise distributions

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

the brownian web characterization and convergence the brownian web bw is the random network formally consist ing of the paths of coalescing one dimensional brownian motions start ing from every space time point in r r we extend the earlier work of arratia and of toth and werner by providing characterization and convergence results for the bw distribution including convergence of the system of all coalescing random walks to the bw under diffusive space time scaling we also provide characterization and convergence results for the double brownian web which combines the bw with its dual process of coalescing brownian motions moving backwards in time with forward and backward paths reflecting off each other for the bw deterministic space time points are almost surely of type 0 1 zero paths into the point from the past and exactly one path out of the point to the future we determine the hausdorff dimension for all types that actually occur dimension 2 for type 0 1 3 2 for 1 1 and 0 2 1 for 1 2 and 0 for 2 1 and 0 3

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

universal denoising for the finite input general output channel we consider the problem of reconstructing a finite alphabet signal corrupted by a known memoryless channel with a general output alphabet the goodness of the reconstruction is measured by a given loss function we constructively establish the existence of a universal sequence of denoiser s attaining asymptotically the optimum distribution dependent performance for any stationary source that may be generating the noiseless signal we show in fact that there is a whole family of denoiser sequences with this property these schemes are shown to be universal also in a semistochastic setting where the only randomness assumed is that associated with the channel noise the scheme is practical requiring o n sup 1 spl epsiv operations for any spl epsiv 0 and working storage size sublinear in the input data length this extends recent work that presented a discrete universal denoiser for recovering a discrete source corrupted by a discrete memoryless channel dmc

coalescing brownian flows a new approach the coalescing brownian flow on r is a process which was introduced by arratia coalescing brownian motions on the line 1979 univ wisconsin madison and toth and werner probab theory related fields 111 1998 375 452 and which formally corresponds to starting coalescing brownian motions from every space time point we provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks this result holds under a finite variance assumption and is thus optimal in previous works by fontes et al ann probab 32 2004 2857 2883 newman et al electron j probab 10 2005 21 60 the topology and state space required a moment of order 3 e for this convergence to hold the proof relies crucially on recent work of schramm and smirnov on scaling limits of critical percolation in the plane our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work in particular similar results are obtained in the case of coalescing brownian motions on the sierpinski gasket this is the first such result where the limiting paths do not enjoy the noncrossing property

on isotropic brownian motions this paper includes a detailed study of isotropic brownian motion on matrices and brownian flows associated with isotropic gaussian fields characteristic exponents stability asymptotic behaviour statistical equilibrium are the topics on which the main results are obtained

on the variation of the spectra of matrices abstract several distances between the spectra of two matrices are discussed and compared optimal bounds are given which enable us to reduce certain bounds on the eigenvalue variation of matrices by a factor of about two the results of bhatia and mukherjea and of bhatia and friedland on the eigenvalue variation are derived in an elementary way using results of henrici on the spectral variation

wavelet based data reduction techniques for process fault detection this article presents new data reduction methods based on the discrete wavelet transform to handle potentially large and complicated nonstationary data curves the methods minimize objective functions to balance the trade off between data reduction and modeling accuracy theoretic investigations provide the optimality of the methods and the large sample distribution of a closed form estimate of the thresholding parameter an upper bound of errors in signal approximation or estimation is derived based on evaluation studies with popular testing curves and real life datasets the proposed methods demonstrate their competitiveness with the existing engineering data compression and statistical data denoising methods for achieving the data reduction goals further experimentation with a tree based classification procedure for identifying process fault classes illustrates the potential of the data reduction tools extension of the engineering scalogram to the reduced size semiconductor fabrication data leads

from stein s unbiased risk estimates to the method of generalized cross validation this paper concerns the method of generalized cross validation gcv a promising way of choosing between linear estimates based on stein estimates and the associated unbiased risk estimates stein 1981 a new approach to gcv is developed many consistency results are obtained for the cross validated steinized estimates in the contexts of nearest neighbor nonparametric regression model selection ridge regression and smoothing splines moreover the associated stein s unbiased risk estimate is shown to be uniformly consistent in assessing the true loss not the risk consistency properties are examined as well when the sampling error is unknown finally we propose a variant of gcv to handle the case that the dimension of the raw data is known to be greater than that of their expected values

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

quaternionic loci in siegel s modular threefold let mathcal q d be the set of moduli points on siegel s modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order mathcal o in an indefinite quaternion algebra of discriminant d over mathbb q such that the rosati involution coincides with a positive involution of the form alpha mapsto mu 1 overline alpha mu on mathcal o for some mu in mathcal o with mu 2 d 0 in this paper we first give a formula for the number of irreducible components in mathcal q d strengthening an earlier result of rotger then for each irreducible component of genus 0 we determine its rational parameterization in terms of a hauptmodul of the associated shimura curve

universal filtering via hidden markov modeling the problem of discrete universal filtering in which the components of a discrete signal emitted by an unknown source and corrupted by a known discrete memoryless channel dmc are to be causally estimated is considered a family of filters are derived and are shown to be universally asymptotically optimal in the sense of achieving the optimum filtering performance when the clean signal is stationary ergodic and satisfies an additional mild positivity condition our schemes are comprised of approximating the noisy signal using a hidden markov process hmp via maximum likelihood ml estimation followed by the use of the forward recursions for hmp state estimation it is shown that as the data length increases and as the number of states in the hmp approximation increases our family of filters attains the performance of the optimal distribution dependent filter an extension to the case of channels with memory is also established

sequential prediction over hierarchical structures we study sequential compound decision problems in the context of sequential prediction of real valued sequences in particular we consider finite state fs predictors that are constructed based on a hierarchical structure such as the order preserving patterns of the sequence history we define hierarchical equivalence classes by tying certain models at a hierarchy level in a recursive manner in order to mitigate undertraining problems these equivalence classes defined on a hierarchical structure are then used to construct a super exponential number of sequential fs predictors based on their combinations and permutations we then introduce truly sequential algorithms with computational complexity only linear in the pattern length that 1 asymptotically achieve the performance of the best fs predictor or the best linear combination of all the fs predictors in an individual sequence manner without any stochastic assumptions over any data length n under a wide range of loss functions 2 achieve the mean square error of the best linear combination of all fs filters or predictors in the steady state for certain nonstationary models we illustrate the superior convergence and tracking capabilities of our algorithm with respect to several state of the art methods in the literature through simulations over synthetic and real benchmark data

flows coalescence and noise we are interested in stationary fluid random evolutions with independent increments under some mild assumptions we show they are solutions of a stochastic differential equation sde there are situations where these evolutions are not described by flows of diffeomorphisms but by coalescing flows or by flows of probability kernels in an intermediate phase for which there exist a coalescing flow and a flow of kernels solution of the sde a classification is given all solutions of the sde can be obtained by filtering a coalescing motion with respect to a subnoise containing the gaussian part of its noise thus the coalescing motion cannot be described by a white noise

minimax risk over l p balls for l p error consider estimating the mean vector from datan n 2 i withl q norm loss q 1 when is known to lie in ann dimensionall p ball p 0 for largen the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp q obvious exceptions aside the limiting ratio equals 1 only ifp q 2 our arguments are mostly indirect involving a reduction to a univariate bayes minimax problem whenp q simple non linear co ordinatewise threshold rules are asymptotically minimax at small signal to noise ratios and within a bounded factor of asymptotic minimaxity in general we also give asymptotic evaluations of the minimax linear risk our results are basic to a theory of estimation in besov spaces using wavelet bases to appear elsewhere

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

a global sparse gradient based coupled system for image denoising abstract we consider a coupled system for image denoising which consists of an anisotropic diffusion equation and a global sparse gradient model gsg the global sparse gradient model is used to estimate the edge map from the noisy image accurately and robustly the anisotropic diffusion process is guided by the edge map furthermore we prove the existence and uniqueness of the solutions of the coupled system with dirichlet initial boundary value problem finally comparing with the edge indicators based on local methods experimental results demonstrate that our proposed method has the state of the art performance among the pde based anisotropic diffusion approaches both in objective measurement and visual evaluation

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

universal switching fir filtering we revisit recently considered universal finite impulse response fir filtering problem and devise a scheme that asymptotically attains the expected mean square error mse of the best switching fir filters for every underlying bounded real valued signal provided that the switch rate of the best filters are sufficiently slow as a performance metric we consider adaptive expected regret the maximum difference between the expected mse of our filter and that of the best fir filter over any contiguous time interval our algorithm is shown to have o log2n bound on the adaptive expected regret with o n2 time complexity where n is the length of the signal the bound implies that regardless of the underlying signal the expected mse of our filter universally converges to that of the best switching fir filters if the number of switches is o n log2n the experimental results show that our filter outperforms its stationary counterpart particularly when the underlying signal has time varying characteristics we also show a heuristic scheme with o n time complexity works well without losing too much of the filtering performance

generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics we study systems of conservation laws arising in two models of adhesion particle dynamics the first is the system of free particles which stick under collision the second is a system of gravitationally interacting particles which also stick under collision in both cases mass and momentum are conserved at the collisions so the dynamics is described by 2 2 systems of conservations laws we show that for these systems global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the lax oleinik variational principle for scalar conservation laws however this weak solution is not unique among weak solutions satisfying the standard entropy condition we also study a modified gravitational model in which instead of momentum some other weighted velocity is conserved at collisions for this model we prove both existence and uniqueness of global weak solutions we then study the qualitative behavior of the solutions with random initial data we show that for continuous but nowhere differentiable random initial velocities all masses immediately concentrate on points even though they were continuously distributed initially and the set of shock locations is dense

adaptive universal linear filtering we consider the problem of online estimation of an arbitrary real valued signal corrupted by zero mean noise using linear estimators the estimator is required to iteratively predict the underlying signal based on the current and several last noisy observations and its performance is measured by the mean square error we design and analyze an algorithm for this task whose total square error on any interval of the signal is equal to that of the best fixed filter in hindsight with respect to the interval plus an additional term whose dependence on the total signal length is only logarithmic this bound is asymptotically tight and resolves the question of moon and wiessman universal fir mmse filtering ieee trans signal process vol 57 no 3 pp 1068 1083 2009 furthermore the algorithm runs in linear time in terms of the number of filter coefficients previous constructions required at least quadratic time

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

coalescing systems of non brownian particles a well known result of arratia shows that one can make rigorous the notion of starting an independent brownian motion at every point of an arbitrary closed subset of the real line and then building a set valued process by requiring particles to coalesce when they collide arratia noted that the value of this process will be almost surely a locally finite set at all positive times and a finite set almost surely if the initial value is compact the key to both of these facts is the observation that because of the topology of the real line and the continuity of brownian sample paths at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them we investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous we give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of brownian motions on the sierpinski gasket and stable processes on the real line with stable index greater than one

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

markov processes on time like graphs we study markov processes where the time parameter is replaced by paths in a directed graph from an initial vertex to a terminal one along each directed path the process is markov and has the same distribution as the one along any other directed path if two directed paths do not interact in a suitable sense then the distributions of the processes on the two paths are conditionally independent given their values at the common endpoint of the two paths conditions on graphs that support such processes e g hexagonal lattice are established next we analyze a particularly suitable family of markov processes called harnesses which includes brownian motion and other l evy processes on such time like graphs finally we investigate continuum limits of harnesses on a sequence of time like graphs that admits a limit in a suitable sense

order estimation and sequential universal data compression of a hidden markov source by the method of mixtures we consider first the estimation of the order i e the number of states of a discrete time finite alphabet stationary ergodic hidden markov source hms our estimator uses a description of the observed data in terms of a uniquely decodable code with respect to a mixture distribution obtained by suitably mixing a parametric family of distributions on the observation space this procedure avoids maximum likelihood calculations the order estimator is shown to be strongly consistent with the probability of underestimation decaying exponentially fast in the number n of observations while the probability of overestimation does not exceed cn sup 3 where c is a constant next we present a sequential algorithm for the uniquely decodable universal data compression of the hms which performs an on line estimation of source order followed by arithmetic coding this code asymptotically attains optimum average redundancy

universal minimax discrete denoising under channel uncertainty the goal of a denoising algorithm is to recover a signal from its noise corrupted observations perfect recovery is seldom possible and performance is measured under a given single letter fidelity criterion for discrete signals corrupted by a known discrete memoryless channel dmc the discrete universal denoiser dude was recently shown to perform this task asymptotically optimally without knowledge of the statistical properties of the source in the present work we address the scenario where in addition to the lack of knowledge of the source statistics there is also uncertainty in the channel characteristics we propose a family of discrete denoisers and establish their asymptotic optimality under a minimax performance criterion which we argue is appropriate for this setting as we show elsewhere the proposed schemes can also be implemented computationally efficiently

the brownian net the standard brownian web is a collection of coalescing one dimensional brownian mo tions starting from each point in space and time it arises as the diffusive scaling limit of a collection of coalescing random walks we show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability we call the limiting object the brownian net and study some of its elementary properties

a geometrical structure for an infinite oriented cluster and its uniqueness we consider the supercritical oriented percolation model let k be all the percolation points for each u k we write u as its rightmost path let g u u in this paper we show that g is a single tree with only one topological end we also present a relationship between k and g and construct a bijection between k and z using the preorder traversal algorithm through applications of this fundamental graph property we show the uniqueness of an infinite oriented cluster by ignoring finite vertices

the brownian fan we provide a mathematical study of the modified diffusion monte carlo dmc algorithm introduced in the companion article 3 dmc is a simulation technique that uses branching particle systems to represent expectations associated with feynman kac formulae we provide a detailed heuristic explanation of why in cases in which a stochastic integral appears in the feynman kac formula e g in rare event simulation continuous time filtering and other settings the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0 the situation studied here stands in stark contrast to the nai ve generalization of the dmc algorithm which would lead to an exponential explosion of the number of particles thus precluding the existence of any finite limiting object convergence is shown rigorously in the simplest possible situation of a random walk biased by a linear potential the resulting limiting object which we call the brownian fan is a very natural new mathematical object of independent interest 2014 wiley periodicals inc

universal prediction of individual sequences the problem of predicting the next outcome of an individual binary sequence using finite memory is considered the finite state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite state fs predictor it is proven that this fs predictability can be achieved by universal sequential prediction schemes an efficient prediction procedure based on the incremental parsing procedure of the lempel ziv data compression algorithm is shown to achieve asymptotically the fs predictability some relations between compressibility and predictability are discussed and the predictability is proposed as an additional measure of the complexity of a sequence

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed