weak approachability in this paper we study repeated games with vector payoffs following blackwell 2 we define weak approachability and its dual property weak excludability we use results from differential games with fixed duration to prove that every set is either weakly approachable or weakly excludable

image enhancement based on signal subspace approach this paper describes a block by block basis image enhancement algorithm which uses the signal subspace method to enhance images corrupted by uncorrelated additive noise the enhancement is performed by eliminating the noise components in the noise subspace and estimating the clean image from the remaining components in the signal subspace

trace pursuit a general framework for model free variable selection abstractwe propose trace pursuit for model free variable selection under the sufficient dimension reduction paradigm two distinct algorithms are proposed stepwise trace pursuit and forward trace pursuit stepwise trace pursuit achieves selection consistency with fixed p forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality the screening consistency property of forward trace pursuit based on sliced inverse regression is established finite sample performances of trace pursuit and other model free variable selection methods are compared through numerical studies supplementary materials for this article are available online

a simple adaptive procedure leading to correlated equilibrium we propose a new and simple adaptive procedure for playing a game regret matching in this procedure players may depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies in the past it is shown that our adaptive procedure guarantees that with probability one the empirical distributions of play converge to the set of correlated equilibria of the game

a necessary and sufficient condition for approachability in the framework of two person repeated games with vector payoffs blackwell defined the property of approachability and introduced a sufficient condition for it he also proved that this condition is necessary in the case of convex sets we generalize this result a set is approachable if and only if it contains a set that satisfies blackwell s condition

opportunistic approachability and generalized no regret problems blackwell s theory of approachability introduced in 1956 has since proved a useful tool in the study of a range of repeated multiagent decision problems given a repeated matrix game with vector payoffs a target set s is approachable by a certain player if he can ensure that the average payoff vector converges to that set for any strategy of the opponent in this paper we consider the case where a set need not be approachable in general but may be approached if the opponent played favorably in some sense in particular we consider nonconvex sets that satisfy blackwell s dual condition namely can be approached when the opponent plays a stationary strategy whereas the convex hull of such a set is approachable this is not generally the case for the original nonconvex set itself we start by defining a sense of restricted play of the opponent with stationary strategies being a special case and then formulate appropriate goals for an opportunistic approachability algorithm that can take advantage

comments on theoretical analysis of the max median filter by g r arce and m p mclaughlin in the original paper ibid vol assp 35 no 1 p 60 9 jan 1987 an approach the max median filter to remove noise from images without distorting geometrical information was introduced the proposed filter structure due to its asymmetrical nature is able to preserve only the subtle signal structures that are larger in intensity than the local background level the commenters suggest another filter structure that has the ability to preserve both positive and negative geometrical information with respect to the background level

response based approachability with applications to generalized no regret problems blackwell s theory of approachability provides fundamental results for repeated games with vector valued payoffs which have been usefully applied in the theory of learning in games and in devising online learning algorithms in the adversarial setup a target set s is approachable by a player the agent in such a game if he can ensure that the average payoff vector converges to s no matter what the opponent does blackwell provided two equivalent conditions for a convex set to be approachable standard approachability algorithms rely on the primal condition which is a geometric separation condition and essentially require to compute at each stage a projection direction from a certain point to s here we introduce an approachability algorithm that relies on blackwell s dual condition which requires the agent to have a feasible response to each mixed action of the opponent namely a mixed action such that the expected payoff vector belongs to s thus rather than projections the proposed algorithm relies on computing the response to a certain action of the opponent at each stage we demonstrate the utility of the proposed approach by applying it to certain generalizations of the classical regret minimization problem which incorporate side constraints reward to cost criteria and so called global cost functions in these extensions computation of the projection is generally complex while the response is readily obtainable

a new algorithm for image edge extraction using a statistical classifier approach this correspondence describes a new algorithm for extracting edges from natural images starting from a simple image model the algorithm poses the problem of edge extraction as a statistical classifier problem the algorithm is capable of extracting and detecting edges from images even in the presence of noise a step by step mathematical derivation of the algorithm reveals the flexibility of the algorithm with pertinent parameters that can be varied for the specific need of the user finally the proposed edge operator is compared to the well known marr hildreth s edge operator

online learning with sample path constraints we study online learning where a decision maker interacts with nature with the objective of maximizing her long term average reward subject to some sample path average constraints we define the reward in hindsight as the highest reward the decision maker could have achieved while satisfying the constraints had she known nature s choices in advance we show that in general the reward in hindsight is not attainable the convex hull of the reward in hindsight function is however attainable for the important case of a single constraint the convex hull turns out to be the highest attainable function using a calibrated forecasting rule we provide an explicit strategy that attains this convex hull we also measure the performance of heuristic methods based on non calibrated forecasters in experiments involving a cpu power management problem

adaptive postprocessing algorithms for low bit rate video signals a new adaptive postprocessing algorithm to enhance the quality of a noisy video sequence is presented the algorithm recognizes that the visibility of noise depends on local signal characteristics it therefore classifies the video signal into different classes and uses separate nonlinear filters matched to each class the most general version of the algorithm employs motion compensated frame averaging to improve picture quality in a first stage a classification algorithm subsequently divides subblocks of pixels in the averaged frame into four classes edge smooth nonsmooth with motion and nonsmooth without motion spatial algorithms that perform multilevel median filtering double median filtering and median filtering are used for pixels belonging to edge smooth and nonsmooth with motion categories pixels in the nonsmooth unmoving category are left unfiltered to preserve corresponding image texture in a simpler version of this four class system the motion cues and motion compensated frame averaging are eliminated and the purely spatial filtering is based on a three class algorithm when used at the output of a 3 d subband coder at 384 kbps the spatial postfilter was shown to provide a consistent gain in subjectively evaluated picture quality twenty five viewers participated in an experiment involving three coded sequences in a pairwise comparison of postfiltered and unfiltered sequences the postfiltered version was judged to be better in 63 out of 75 instances

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

optimal sufficient dimension reduction in regressions with categorical predictors though partial sliced inverse regression partial sir chiaromonte et al 2002 sufficient dimension reduction in regressions with categorical predictors ann statist 30 475 497 extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice when this condition fails partial sir may provide misleading results in this article we propose a new estimation method via a minimum discrepancy approach without this restriction our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi squared distribution it also gives us the ability to test predictor effects an asymptotic chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained

efficient integration of sufficient dimension reduction and prediction in discriminant analysis abstractsufficient dimension reduction sdr methods are popular model free tools for preprocessing and data visualization in regression problems where the number of variables is large unfortunate

a geometric approach to multi criterion reinforcement learning we consider the problem of reinforcement learning in a controlled markov environment with multiple objective functions of the long term average reward type the environment is initially unknown and furthermore may be affected by the actions of other agents actions that are observed but cannot be predicted beforehand we capture this situation using a stochastic game model where the learning agent is facing an adversary whose policy is arbitrary and unknown and where the reward function is vector valued state recurrence conditions are imposed throughout in our basic problem formulation a desired target set is specified in the vector reward space and the objective of the learning agent is to approach the target set in the sense that the long term average reward vector will belong to this set we devise appropriate learning algorithms that essentially use multiple reinforcement learning algorithms for the standard scalar reward problem which are combined using the geometric insight from the theory of approachability for vector valued stochastic games we then address the more general and optimization related problem where a nested class of possible target sets is prescribed and the goal of the learning agent is to approach the smallest possible target set which will generally depend on the unknown system parameters a particular case which falls into this framework is that of stochastic games with average reward constraints and further specialization provides a reinforcement learning algorithm for constrained markov decision processes some basic examples are provided to illustrate these results

kernel dimension reduction in regression we present a new methodology for sufficient dimension reduction sdr our methodology derives directly from the formulation of sdr in terms of the conditional independence of the covariate x from the response y given the projection of x on the central subspace cf j amer statist assoc 86 1991 316 342 and regression graphics 1998 wiley we show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel hilbert spaces and we show how this characterization leads to an m estimator for the central subspace the resulting estimator is shown to be consistent under weak conditions in particular we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for sdr methods we also present empirical results showing that the new methodology is competitive in practice

three dimensional subband coding of video we describe and show the results of video coding based on a three dimensional 3 d spatio temporal subband decomposition the results include a 1 mbps coder based on a new adaptive differential pulse code modulation scheme adpcm and adaptive bit allocation this rate is useful for video storage on cd rom coding results are also shown for a 384 kbps rate that are based on adpcm for the lowest frequency band and a new form of vector quantization geometric vector quantization gvq for the data in the higher frequency bands gvq takes advantage of the inherent structure and sparseness of the data in the higher bands results are also shown for a 128 kbps coder that is based on an unbalanced tree structured vector quantizer utsvq for the lowest frequency band and gvq for the higher frequency bands the results are competitive with traditional video coding techniques and provide the motivation for investigating the 3 d subband framework for different coding schemes and various applications

diagnostic studies in sufficient dimension reduction sufficient dimension reduction in regression aims to reduce the predictor dimension by replacing the original predictors with some set of linear combinations of them without loss of information numerous dimension reduction methods have been developed based on this paradigm however little effort has been devoted to diagnostic studies within the context of dimension reduction in this paper we introduce methods to check goodness of fit for a given dimension reduction subspace the key idea is to extend the so called distance correlation to measure the conditional dependence relationship between the covariates and the response given a reduction subspace our methods require minimal assumptions which are usually much less restrictive than the conditions needed to justify the original methods asymptotic properties of the test statistic are studied numerical examples demonstrate the effectiveness of the proposed approach

guaranteed performance regions in markovian systems with competing decision makers a decision maker is facing a dynamic system which is being controlled by himself herself as well as by other decision makers he she considers a vector of performance measures acceptable performance is defined through a set in the space of performance vectors whether this decision maker can guarantee a time averaged performance vector which approaches this desired set is considered for the worst case scenario in which other decision makers may for selfish reasons try to exclude his her vector from the desired set for a controlled markov model of the system a sufficient condition for approachability is given and appropriate control strategies are constructed under certain recurrence conditions a complete characterization of approachability is then provided for convex sets the mathematical formulation leads to a theory of approachability for stochastic games with vector payoffs a simple queuing example illustrates this approach

sufficient dimension reduction via inverse regression a family of dimension reduction methods the inverse regression ir family is developed by minimizing a quadratic objective function an optimal member of this family the inverse regression estimator ire is proposed along with inference methods and a computational algorithm the ire has at least three desirable properties 1 its estimated basis of the central dimension reduction subspace is asymptotically efficient 2 its test statistic for dimension has an asymptotic chi squared distribution and 3 it provides a chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of predictors given the remaining predictors current methods like sliced inverse regression belong to a suboptimal class of the ir family comparisons of these methods are reported through simulation studies the approach developed here also allows a relatively straightforward derivation of the asymptotic null distribution of the test statistic for dimension used in slic

double window hodges lehman d filter and hybrid d median filter for robust image smoothing the double window hodges lehman filter dwd filter and a hybrid d median filter hdm filter for robust image smoothing are proposed an adaptive mixture of the median and the d filter the hdm filter first makes decisions about the presence of edges on the basis of a two way classification of pixels near and around the pixel to be filtered subsequently straightforward d filtering is used in the absence of edges and median filtering is used in the presence of edges the dwd filter uses two windows and d filtering the smaller window is used to preserve the details then the larger window to provide for sufficient smoothing detailed simulation results show that the hdm filter while retaining all the good properties of the dwd filter consistently performs better in terms of signal to noise ratio than the dwd filter and a number of other filters including the median filter the dwd filter is shown to have simpler structure although not necessarily lesser computational complexity

sparse sliced inverse regression sliced inverse regression sir is an innovative and effective method for dimension reduction and data visualization of high dimensional problems it replaces the original variables with low dimensional linear combinations of predictors without any loss of regression information and without the need to prespecify a model or an error distribution however it suffers from the fact that each sir component is a linear combination of all the original predictors thus it is often difficult to interpret the extracted components by representing sir as a regression type optimization problem we propose in this article a new method called sparse sir that combines the shrinkage idea of the lasso with sir to produce both accurate and sparse solutions the efficacy of the proposed method is verified by simulation and a real data example is given

the empirical bayes envelope and regret minimization in competitive markov decision processes this paper proposes an extension of the regret minimizing framework from repeated matrix games to stochastic game models under appropriate recurrence conditions a decision maker p1 who wishes to maximize his long term average reward is facing a markovian environment which may also be affected by arbitrary actions of other agents the latter are collectively modeled as a second player p2 whose strategy is arbitrary both states and actions are fully observed by both players while p1 may obviously secure the min max value of the game he may wish to improve on that when the opponent is not playing a worst case strategy for repeated matrix games an achievable goal is presented by the bayes envelope that traces p1 s best response payoff against the observable frequencies of p2 s actions we propose a generalization to the stochastic game framework under recurrence conditions that amount to fixed state reachability the empirical bayes envelope ebe is defined as p1 s best response payoff against the stationary strategies of p2 that agree with the observed state action frequencies because the ebe may not be attainable in general we consider its lower convex hull the convex bayes envelope cbe which is proved to be achievable by p1 the analysis relies on blackwell s approachability theory the cbe is lower bounded by the value of the game and for irreducible games turns out to be strictly above the value whenever p2 s frequencies deviate from a worst case strategy in the special case of single controller games where p2 alone affects the state transitions the ebe itself is shown to be attainable