an axiomatic approach to the measurement of the amount of interaction among criteria or players in the framework of multicriteria decision making based on fuzzy measures or cooperative game theory an axiomatization of the concept of amount of interaction index is given it is based on four axioms a natural property called absence of interaction a boundary condition for which the index reduces to the shapley or banzhaf interaction index a pseudo additivity axiom and a recursivity property the resulting index which can be interpreted as a measure of the average degree of independence among criteria or players takes a particularly simple and natural form for supermodular or submodular measures the approach to the notion of interaction adopted in this paper can be seen as complementary to that recently adopted by grabisch and roubens

another approach to the existence of value functions of stochastic differential games abstract the existence of value functions for general two player zero sum stochastic differential games has been obtained by fleming and souganidis in this paper we present a new approach to this problem we prove optimality inequalities of dynamic programming for viscosity sub and supersolutions of the associated bellman isaacs equations these inequalities are well known for deterministic differential games but are new for stochastic differential games it then easily follows that value functions are the unique viscosity solutions of the bellman isaacs equations and satisfy the principle of dynamic programming the results presented here are not the same as those of fleming and souganidis because we work with different reference spaces and the independence of value functions of the choice of reference spaces is not clear to us

signals on graphs uncertainty principle and sampling in many applications the observations can be represented as a signal defined over the vertices of a graph the analysis of such signals requires the extension of standard signal processing tools in this paper first we provide a class of graph signals that are maximally concentrated on the graph domain and on its dual then building on this framework we derive an uncertainty principle for graph signals and illustrate the conditions for the recovery of band limited signals from a subset of samples we show an interesting link between uncertainty principle and sampling and propose alternative signal recovery algorithms including a generalization to frame based reconstruction methods after showing that the performance of signal recovery algorithms is significantly affected by the location of samples we suggest and compare a few alternative sampling strategies finally we provide the conditions for perfect recovery of a useful signal corrupted by sparse noise showing that this problem is also intrinsically related to vertex frequency localization properties

optimal investment with high watermark performance fee we consider the problem of optimal investment and consumption when the investment opportunity is represented by a hedge fund charging proportional fees on profit the value of the fund evolves as a geometric brownian motion and the performance of the investment and consumption strategy is measured using discounted power utility from consumption on infinite horizon the resulting stochastic control problem is solved using dynamic programming arguments we show by analytical methods that the associated hamilton jacobi bellman equation has a smooth solution and then obtain the existence and representation of the optimal control in feedback form using verification arguments

an axiomatic approach to the definition of the entropy of a discrete choquet capacity to extend the classical shannon entropy to nonadditive measures marichal recently introduced the concept of generalized entropy for discrete choquet capacities we provide a first axiomatization of this new concept on the basis of three axioms the symmetry property a boundary condition for which the entropy reduces to the shannon entropy and a generalized version of the well known recursivity property we also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy like measure lastly we provide an interpretation of it in the framework of aggregation by the discrete choquet integral

on the dispersion measure of owa operators we describe some basic features of the owa operator we turn to the problem of determining the weights associated with this operator and particularly the maximal dispersion entropy approach we consider the possibility of using minimization of dispersion after discussing concerns with both maximization and minimization of dispersion we investigate the possibility of finding an optimal solution intermediate to these extremes we next consider alternative measures of dispersion we introduce a fundamental requirement for a measure of dispersion called the preference for equal division a number of general classes of dispersion measures are provided notable among these are those based on t norm and t conorm operators

two person zero sum game in weak formulation and path dependent bellman isaacs equation in this paper we study a two person zero sum stochastic differential game in weak formulation unlike the standard literature which uses strategy type controls the weak formulation allows us to consider the game with control against control we shall prove the existence of game value under natural conditions another main feature of the paper is that we allow for non markovian structure and thus the game value is a random process we characterize the value process as the unique viscosity solution of the corresponding path dependent bellman isaacs equation a notion recently introduced by ekren et al ann probab 42 2014 pp 204 236 and ekren touzi and zhang stochastic process to appear preprint arxiv 1210 0006v2 preprint arxiv 1210 0007v2

graph sampling for matrix completion using recurrent gershgorin disc shift matrix completion algorithms fill missing entries in a large matrix given a subset of observed samples the problem of how to pre select a subset of entries for sampling to maximize the reconstructed matrix fidelity is largely unaddressed in this paper we propose two sampling algorithms to tackle this problem i a fast base sampling algorithm on general single graphs and ii an extended sampling algorithm from the base algorithm for active matrix completion assuming the double graph smoothness prior where rows columns of the matrix signal are smooth with respect to row column factor graphs we first show that a matrix signal can be recovered from partial observations by solving a system of linear equations mathbf q mathbf x mathbf y where the sought matrix is vectorized as mathbf x and interpreted as a signal residing on a single graph on this single graph to promote small reconstruction error and stability of the linear system we maximize the smallest eigenvalue lambda min of mathbf q by greedily selecting samples corresponding to the largest magnitude entries in the first eigenvectors of mathbf q based on a corollary of the gershgorin circle theorem extending to the active matrix completion case we alternately choose entries in the rows and columns corresponding to the largest magnitude entries in the first eigenvectors of the row column coefficient matrices mathbf q r and mathbf q c our algorithm benefits computationally from warm start as the first eigenvectors are computed recurrently for more samples extensive experiments on both synthetic and real world datasets show that our extended graph sampling algorithm outperforms existing sampling schemes for matrix completion when combined with a range of modern matrix completion algorithms

determination of weights of interacting criteria from a reference set abstract in this paper we present a model allowing to determine the weights related to interacting criteria this is done on the basis of the knowledge of a partial ranking over a reference set of alternatives prototypes a partial ranking over the set of criteria and a partial ranking over the set of interactions between pairs of criteria

asymptotic perron s method and simple markov strategies in stochastic games and control we introduce a modification of perron s method where semisolutions are considered in a carefully defined asymptotic sense with this definition we can show in a rather elementary way that in a zero sum game or a control problem with or without model uncertainty the value function over all strategies coincides with the value function over markov strategies discretized in time therefore there are always discretized markov varepsilon optimal strategies uniform with respect to the bounded initial condition with a minor modification the method produces a value and approximate saddle points for an asymmetric game of feedback strategies versus counterstrategies

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

efficient sampling set selection for bandlimited graph signals using graph spectral proxies we study the problem of selecting the best sampling set for bandlimited reconstruction of signals on graphs a frequency domain representation for graph signals can be defined using the eigenvectors and eigenvalues of variation operators that take into account the underlying graph connectivity smoothly varying signals defined on the nodes are of particular interest in various applications and tend to be approximately bandlimited in the frequency basis sampling theory for graph signals deals with the problem of choosing the best subset of nodes for reconstructing a bandlimited signal from its samples most approaches to this problem require a computation of the frequency basis i e the eigenvectors of the variation operator followed by a search procedure using the basis elements this can be impractical in terms of storage and time complexity for real datasets involving very large graphs we circumvent this issue in our formulation by introducing quantities called graph spectral proxies defined using the powers of the variation operator in order to approximate the spectral content of graph signals this allows us to formulate a direct sampling set selection approach that does not require the computation and storage of the basis elements we show that our approach also provides stable reconstruction when the samples are noisy or when the original signal is only approximately bandlimited furthermore the proposed approach is valid for any choice of the variation operator thereby covering a wide range of graphs and applications we demonstrate its effectiveness through various numerical experiments

stochastic perron s method for the probability of lifetime ruin problem under transaction costs we apply the stochastic perron s method to a singular control problem where an individual targets at a given consumption rate invests in a risky financial market in which trading is subject to proportional transaction costs and seeks to minimize her probability of lifetime ruin without relying on the dynamic programming principle we characterize the value function as the unique viscosity solution of an associated hamilton jacobi bellman variational inequality we also provide a complete proof of the comparison principle which is the main assumption of the stochastic perron s method

mysteries around the graph laplacian eigenvalue 4 we describe our current understanding on the phase transition phenomenon of the graph laplacian eigenvectors constructed on a certain type of unweighted trees which we previously observed through our numerical experiments the eigenvalue distribution for such a tree is a smooth bell shaped curve starting from the eigenvalue 0 up to 4 then at the eigenvalue 4 there is a sudden jump interestingly the eigenvectors corresponding to the eigenvalues below 4 are semi global oscillations like fourier modes over the entire tree or one of the branches on the other hand those corresponding to the eigenvalues above 4 are much more localized and concentrated like wavelets around junctions branching vertices for a special class of trees called starlike trees we obtain a complete understanding of such phase transi tion phenomenon for a general graph we prove the number of the eigenvalues larger than 4 is bounded from above by the number of vertices whose degrees is strictly larger than 2 moreover we also prove that if a graph contains a branching path then the magnitudes of the components of any eigenvector corresponding to the eigenvalue greater than 4 decay exponentially from the branching vertex toward the leaf of that branch we have also identia a a ed a unique class of trees that can have an eigenvalue exactly equal to 4

assessing non additive utility for multicriteria decision aid abstract in the framework of multi attribute utility theory maut several methods have been proposed to build a decision maker s dm utility function representing his her preferences among such methods the uta method infers an additive utility function from a set of exemplary decisions using linear programming however the uta method does not guarantee to find a utility function which is coherent with the available information this drawback is due to the underlying utility model of uta viz the additive one which does not allow to include additional information such as an interaction among criteria in this paper we present a methodology for building a non additive utility function in the framework of the so called fuzzy integrals which permits to model preference structures with interaction between criteria like in the uta method we aim at searching a utility function representing the dm s preferences but unlike uta the functional form is a specific fuzzy integral choquet integral as a result we obtain weights which can be interpreted as the importance of coalitions of criteria exploiting the potential interaction between criteria as already proposed by other authors however within the same framework we obtain also the marginal utility functions relative to each one of the considered criteria that are evaluated on a common scale as a consequence of the implemented methodology finally we illustrate our approach with an example

sensor placement by maximal projection on minimum eigenspace for linear inverse problems this paper presents two new greedy sensor placement algorithms named minimum nonzero eigenvalue pursuit mnep and maximal projection on minimum eigenspace mpme for linear inverse problems with greater emphasis on the mpme algorithm for performance comparison with existing approaches in both mnep and mpme we select the sensing locations one by one in this way the least number of required sensor nodes can be determined by checking whether the estimation accuracy is satisfied after each sensing location is determined for the mpme algorithm the minimum eigenspace is defined as the eigenspace associated with the minimum eigenvalue of the dual observation matrix for each sensing location the projection of its observation vector onto the minimum eigenspace is shown to be monotonically decreasing w r t the worst case error variance wcev of the estimated parameters we select the sensing location whose observation vector has the maximum projection onto the minimum eigenspace of the current dual observation matrix the proposed mpme is shown to be one of the most computationally efficient algorithms our monte carlo simulations showed that mpme outperforms the convex relaxation method the sparsense method and the framesense method in terms of wcev and the mean square error mse of the estimated parameters especially when the number of available sensor nodes is very limited

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

stochastic differential games and viscosity solutions of hamilton jacobi bellman isaacs equations in this paper we study zero sum two player stochastic differential games with the help of the theory of backward stochastic differential equations bsdes more precisely we generalize the results of the pioneering work of fleming and souganidis indiana univ math j 38 1989 pp 293 314 by considering cost functionals defined by controlled bsdes and by allowing the admissible control processes to depend on events occurring before the beginning of the game this extension of the class of admissible control processes has the consequence that the cost functionals become random variables however by making use of a girsanov transformation argument which is new in this context we prove that the upper and the lower value functions of the game remain deterministic apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information its combination with bsde methods in particular that of the notion of stochastic backward semigroups introduced by peng bsde and stochastic optimizations in topics in stochastic analysis science press beijing 1997 allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way the upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower hamilton jacobi bellman isaacs equations respectively for this peng s bsde method is extended from the framework of stochastic control theory into that of stochastic differential games

zero sum stochastic differential games and backward equations abstract in this paper we are concerned with the applications to stochastic zero sum differential games of the results on backward stochastic differential equations using these techniques the existence of a saddle point in the bounded case is obtained if the isaacs condition holds furthermore this technique is also a very simple approach for the existence of an optimal strategy in the case of controlled diffusions

completing any low rank matrix provably matrix completion i e the exact and provable recovery of a low rank matrix from a small subset of its elements is currently only known to be possible if the matrix satisfies a restrictive structural constraint known as incoherence on its row and column spaces in these cases the subset of elements is assumed to be sampled uniformly at random in this paper we show that any rank r n by n matrix can be exactly recovered from as few as o nr log2 n randomly chosen elements provided this random choice is made according to a specific biased distribution suitably dependent on the coherence structure of the matrix the probability of any element being sampled should be at least a constant times the sum of the leverage scores of the corresponding row and column moreover we prove that this specific form of sampling is nearly necessary in a natural precise sense this implies that many other perhaps more intuitive sampling schemes fail we further establish three ways to use the above result for the setting when leverage scores are not known a priori a we describe a provably correct sampling strategy for the case when only the column space is incoherent and no assumption or knowledge of the row space is required b we propose a two phase sampling procedure for general matrices that first samples to estimate leverage scores followed by sampling for exact recovery these two approaches assume control over the sampling procedure c by using our main theorem in a reverse direction we provide an analysis showing the advantages of the empirically successful weighted nuclear trace norm minimization approach over the vanilla un weighted formulation given non uniformly distributed observed elements this approach does not require controlled sampling or knowledge of the leverage scores

minimum variance capacity identification abstract in the framework of multi criteria decision making whose aggregation process is based on the choquet integral we present a maximum entropy like method enabling to determine if it exists the least specific capacity compatible with the initial preferences of the decision maker the proposed approach consists in solving a strictly convex quadratic program whose objective function is equivalently either the opposite of a generalized entropy measure or the variance of the capacity the application of the proposed approach is illustrated on two examples

differential games optimal control and directional derivatives of viscosity solutions of bellman s and isaacs equations recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two person zero sum differential games problems and the new notion of viscosity solutions of hamilton jacobi pde s introduced by m g crandall and p l lions in particular it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated hamilton jacobi bellman and isaacs equations in the present work it is shown that viscosity super and subsolutions of these equations must satisfy some inequalities called super and subdynamic programming principle respectively this is then used to prove the equivalence between the notion of viscosity solutions and the conditions introduced by a subbotin concerning the sign of certain generalized directional derivatives

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

discrete signal processing on graphs sampling theory we propose a sampling theory for signals that are supported on either directed or undirected graphs the theory follows the same paradigm as classical sampling theory we show that perfect recovery is possible for graph signals bandlimited under the graph fourier transform the sampled signal coefficients form a new graph signal whose corresponding graph structure preserves the first order difference of the original graph signal for general graphs an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise for graphs whose graph fourier transforms are frames with maximal robustness to erasures as well as for erd s renyi graphs random sampling leads to perfect recovery with high probability we further establish the connection to the sampling theory of finite discrete time signal processing and previous work on signal recovery on graphs to handle full band graph signals we propose a graph filter bank based on sampling theory on graphs finally we apply the proposed sampling theory to semi supervised classification of online blogs and digit images where we achieve similar or better performance with fewer labeled samples compared to previous work

tolerant or intolerant character of interacting criteria in aggregation by the choquet integral abstract in many multi criteria decision making problems the decision criteria present some interaction whose nature may vary from one situation to another for example some criteria may be statistically correlated thus making them somewhat redundant or opposed some others may be somewhat substitutive or complementary depending on the behavior of the decision maker some others may be decisive in the sense that the global score of any alternative obtained by aggregation is bounded by the partial score along one of them in this paper we analyze this latter form of interaction when a criterion bounds the global score from above it is called a blocker or a veto due to its rather intolerant character when it bounds the global score from below it is then called a pusher or a favor we thus investigate the tolerance of criteria or equivalently the tolerance of the weighted aggregation operator here the choquet integral that is used to aggregate criteria more specifically we propose axiomatically indices to appraise the extent to which each criterion behaves like a veto or a favor in the aggregation by the choquet integral previous to this we also propose global tolerance degrees measuring the extent to which the choquet integral is conjunctive or disjunctive

training with noise is equivalent to tikhonov regularization it is well known that the addition of noise to the input data of a neural network during training can in some circumstances lead to significant improvements in generalization performance previous work has shown that such training with noise is equivalent to a form of regularization in which an extra term is added to the error function however the regularization term which involves second derivatives of the error function is not bounded below and so can lead to difficulties if used directly in a learning algorithm based on error minimization in this paper we show that for the purposes of network training the regularization term can be reduced to a positive semi definite form that involves only first derivatives of the network mapping for a sum of squares error function the regularization term belongs to the class of generalized tikhonov regularizers direct minimization of the regularized error function provides a practical alternative to training with noise

learning k maxitive fuzzy measures from data by mixed integer programming abstract fuzzy measures model interactions between the inputs in aggregation problems their complexity grows exponentially with the dimensionality of the problem and elicitation of fuzzy measure coefficients either from domain experts or from empirical data poses a significant challenge the notions of k additivity and k maxitivity simplify the fuzzy measures by limiting interactions to subsets of up to k elements learning fuzzy measures from data is an important elicitation technique which relies on solving an optimisation problem a heuristic learning algorithm to identify k maxitive fuzzy measures from the data on the basis of hlms heuristic least mean squares was recently presented in murillo et al 2017 11 we present an alternative formulation of the fitting problem which delivers a globally optimal solution through the solution of a mixed integer programming mip problem to deal with high computational cost of mip in moderate to large dimensions we also propose a simple mip relaxation technique which involves solving two related linear programming problems we also provide a linear programming formulation for fitting k tolerant fuzzy measures we discuss implementations of the fitting methods and present the results of numerical experiments

entropy of discrete choquet capacities we introduce a measure of entropy for any discrete choquet capacity and we interpret it in the setting of aggregation by the choquet integral

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

preference learning using the choquet integral the case of multipartite ranking we propose a novel method for preference learning or more specifically learning to rank where the task is to learn a ranking model that takes a subset of alternatives as input and produces a ranking of these alternatives as output just like in the case of conventional classifier learning training information is provided in the form of a set of labeled instances with labels or say preference degrees taken from an ordered categorical scale this setting is known as multipartite ranking in the literature our approach is based on the idea of using the discrete choquet integral as an underlying model for representing ranking functions being an established aggregation function in fields such as multiple criteria decision making and information fusion the choquet integral offers a number of interesting properties that make it attractive from a machine learning perspective too the learning problem itself comes down to properly specifying the fuzzy measure on which the choquet integral is defined this problem is formalized as a margin maximization problem and solved by means of a cutting plane algorithm the performance of our method is tested on a number of benchmark datasets

stochastic perron s method for hamilton jacobi bellman equations we show that the value function of a stochastic control problem is the unique solution of the associated hamilton jacobi bellman equation completely avoiding the proof of the so called dynamic programming principle dpp using the stochastic perron s method we construct a supersolution lying below the value function and a subsolution dominating it a comparison argument easily closes the proof the program has the precise meaning of verification for viscosity solutions obtaining the dpp as a conclusion it also immediately follows that the weak and strong formulations of the stochastic control problem have the same value using this method we also capture the possible face lifting phenomenon in a straightforward manner

graph laplacian regularization for image denoising analysis in the continuous domain inverse imaging problems are inherently underdetermined and hence it is important to employ appropriate image priors for regularization one recent popular prior the graph laplacian regularizer assumes that the target pixel patch is smooth with respect to an appropriately chosen graph however the mechanisms and implications of imposing the graph laplacian regularizer on the original inverse problem are not well understood to address this problem in this paper we interpret neighborhood graphs of pixel patches as discrete counterparts of riemannian manifolds and perform analysis in the continuous domain providing insights into several fundamental aspects of graph laplacian regularization for image denoising specifically we first show the convergence of the graph laplacian regularizer to a continuous domain functional integrating a norm measured in a locally adaptive metric space focusing on image denoising we derive an optimal metric space assuming non local self similarity of pixel patches leading to an optimal graph laplacian regularizer for denoising in the discrete domain we then interpret graph laplacian regularization as an anisotropic diffusion scheme to explain its behavior during iterations e g its tendency to promote piecewise smooth signals under certain settings to verify our analysis an iterative image denoising algorithm is developed experimental results show that our algorithm performs competitively with state of the art denoising methods such as bm3d for natural images and outperforms them significantly for piecewise smooth images

fast graph sampling set selection using gershgorin disc alignment graph sampling set selection where a subset of nodes are chosen to collect samples to reconstruct a smooth graph signal is a fundamental problem in graph signal processing gsp previous works employ an unbiased least squares ls signal reconstruction scheme and select samples via expensive extreme eigenvector computation instead we assume a biased graph laplacian regularization glr based scheme that solves a system of linear equations for reconstruction we then choose samples to minimize the condition number of the coefficient matrix specifically maximize the smallest eigenvalue lambda min circumventing explicit eigenvalue computation we maximize instead the lower bound of lambda min designated by the smallest left end of all gershgorin discs of the matrix to achieve this efficiently we first convert the optimization to a dual problem where we minimize the number of samples needed to align all gershgorin disc left ends at a chosen lower bound target t algebraically the dual problem amounts to optimizing two disc operations i shifting of disc centers due to sampling and ii scaling of disc radii due to a similarity transformation of the matrix we further reinterpret the dual as an intuitive disc coverage problem bearing strong resemblance to the famous np hard set cover sc problem the reinterpretation enables us to derive a fast approximation scheme from a known sc error bounded approximation algorithm we find an appropriate target t efficiently via binary search extensive simulation experiments show that our disc based sampling algorithm runs substantially faster than existing sampling schemes and outperforms other eigen decomposition free sampling schemes in reconstruction error

point cloud denoising via feature graph laplacian regularization point cloud is a collection of 3d coordinates that are discrete geometric samples of an object s 2d surfaces imperfection in the acquisition process means that point clouds are often corrupted with noise building on recent advances in graph signal processing we design local algorithms for 3d point cloud denoising specifically we design a signal dependent feature graph laplacian regularizer sdfglr that assumes surface normals computed from point coordinates are piecewise smooth with respect to a signal dependent graph laplacian matrix using sdfglr as a signal prior we formulate an optimization problem with a general ell p norm fidelity term that can explicitly remove only two types of additive noise small but non sparse noise like gaussian using ell 2 fidelity term and large but sparser noise like laplacian using ell 1 fidelity term to establish a linear relationship between normals and 3d point coordinates we first perform bipartite graph approximation to divide the point cloud into two disjoint node sets red and blue we then optimize the red and blue nodes coordinates alternately for ell 2 norm fidelity term we iteratively solve an unconstrained quadratic programming qp problem efficiently computed using conjugate gradient with a bounded condition number to ensure numerical stability for ell 1 norm fidelity term we iteratively minimize an ell 1 ell 2 cost function using accelerated proximal gradient apg where a good step size is chosen via lipschitz continuity analysis finally we propose simple mean and median filters for flat patches of a given point cloud to estimate the noise variance given the noise type which in turn is used to compute a weight parameter trading off the fidelity term and signal prior in the problem formulation extensive experiments show state of the art denoising performance among local methods using our proposed algorithms

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

on the rate of convergence of finite difference approximations for bellmans equations with variable coefficients the estimates presented here for parabolic bellman s equations with variable coefficients extend the ones earlier obtained for constant coefficients

2 additive capacity identification methods from multicriteria correlation preference information the essential role of the particular families of capacities and the capacity identification methods is to help the decision maker to deal with the exponential complexity inherent in the construction process of the capacity the 2 additive capacities appear to be the most popular among the particular families of capacities since they permit to model interactions between criteria while preserving simplicity besides the preference with respect to the decision criteria most of the capacity identification methods also need to provide the desired overall evaluations of the decision alternatives in the learning set which is a time consuming task for the decision maker in this paper we propose some models to identify 2 additive capacities only from a kind of refined preference information with respect to the decision criteria called the multicriteria correlation preference information mccpi the mccpi is a group of 2 d preference information which can be described and obtained by the refined diamond diagram the common principle of the proposed models is to minimize the different kinds of deviations between the mccpi and the most desired 2 additive capacity ies a multicriteria decision making example is presented to show the feasibility of the proposed methods and a 2 d scale of the mccpi is also introduced in the further discussion of the illustrative example

on martingale problems with continuous time mixing and values of zero sum games without the isaacs condition we consider a zero sum stochastic differential game over elementary mixed feedback strategies these are strategies based only on the knowledge of the past state randomized continuously in time from a sampling distribution which is kept constant in between some stopping rules once both players choose such strategies the state equation admits a unique solution in the sense of the martingale problem of stroock and varadhan we show that the game defined over martingale solutions has a value which is the unique continuous viscosity solution of the randomized isaacs equation

stochastic target games and dynamic programming via regularized viscosity solutions we study a class of stochastic target games where one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the opponent our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non linear partial differential equation because abstract measurable selection arguments cannot be used in this context the main obstacle is the construction of measurable almost optimal strategies we propose a novel approach where smooth supersolutions are used to define almost optimal strategies of markovian type similarly as in verification arguments for classical solutions of hamilton jacobi bellman equations the smooth supersolutions are constructed by an extension of krylov s method of shaken coefficients we apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending

value in mixed strategies for zero sum stochastic differential games without isaacs condition in the present work we consider 2 person zero sum stochastic differential games with a nonlinear pay off functional which is defined through a backward stochastic differential equation our main objective is to study for such a game the problem of the existence of a value without isaacs condition not surprising this requires a suitable concept of mixed strategies which to the authors best knowledge was not known in the context of stochastic differential games for this we consider nonanticipative strategies with a delay defined through a partition pi of the time interval 0 t the underlying stochastic controls for the both players are randomized along pi by a hazard which is independent of the governing brownian motion and knowing the information available at the left time point t j 1 of the subintervals generated by pi the controls of players 1 and 2 are conditionally independent over t j 1 t j it is shown that the associated lower and upper value functions w pi and u pi converge uniformly on compacts to a function v the so called value in mixed strategies as the mesh of pi tends to zero this function v is characterized as the unique viscosity solution of the associated hamilton jacobi bellman isaacs equation

a review of methods for capacity identification in choquet integral based multi attribute utility theory applications of the kappalab r package the application of multi attribute utility theory whose aggregation process is based on the choquet integral requires the prior identification of a capacity the main approaches to capacity identification proposed in the literature are reviewed and their advantages and inconveniences are discussed all the reviewed methods have been implemented within the kappalab r package their application is illustrated on a detailed example

k order additive discrete fuzzy measures and their representation abstract in order to face with the complexity of discrete fuzzy measures we propose the concept of k orderadditive fuzzy measure including usual additive measures and fuzzy measures every discrete fuzzy measure is a k order additive fuzzy measure for a unique k a related topic of the paper is to introduce an alternative representation of fuzzy measures called the interaction representation which sets and extends in a common framework the shapley value and the interaction index proposed by murofushi and soneda

learning choquet integral based metrics for semisupervised clustering we consider an application of fuzzy measures to the problem of metric learning in semisupervised clustering we investigate the necessary and sufficient conditions on the underlying fuzzy measure that make the discrete choquet integral suitable for defining a metric as a byproduct we can obtain the analogous conditions for the ordered weighted averaging owa operators which constitute a special case we then generalize these results for power based choquet and owa operators we show that this metric learning problem can be formulated as a linear programming problem and specify the required sets of linear constraints we present the results of numerical experiments on artificial and real world datasets which illustrate the potential usefulness and limitations of this construction

on the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the isaacs equations we prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time homogeneous diffusion processes considered up to the first exit time from a domain in contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well there is no assumption about solvability of the the isaacs equation in any sense classical or viscosity the zeroth order coefficient and the free term are only assumed to be measurable in the space variable we also prove that value functions are uniquely determined by the functions defining the corresponding isaacs equations and thus stochastic games with the same isaacs equation have the same value functions

non additive robust ordinal regression a multiple criteria decision model based on the choquet integral abstract within the multicriteria aggregation disaggregation framework ordinal regression aims at inducing the parameters of a decision model for example those of a utility function which have to represent some holistic preference comparisons of a decision maker dm usually among the many utility functions representing the dm s preference information only one utility function is selected since such a choice is arbitrary to some extent recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the dm s preference information until now robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence in this paper we propose a non additive robust ordinal regression on a set of alternatives a whose utility is evaluated in terms of the choquet integral which permits to represent the interaction among criteria modelled by the fuzzy measures parameterizing our approach in our methodology besides holistic pairwise preference comparisons of alternatives from a subset of reference alternatives a a the dm is also requested to express the intensity of preference on pairs of alternatives from a and to supply pairwise comparisons on the importance of criteria and the sign and intensity of interaction among pairs of criteria the output of our approach defines a set of fuzzy measures capacities such that the corresponding choquet integral is compatible with the dm s preference information moreover using linear programming our decision model establishes two preference relations for any a b a the necessary weak preference relation if for all compatible fuzzy measures the utility of a is not smaller than the utility of b and the possible weak preference relation if for at least one compatible fuzzy measure the utility of a is not smaller than the utility of b

the controller and stopper game for a linear diffusion consider a process x x t 0 t 1 with values in the interval i 0 1 absorption at the boundary points of i and dynamics dx t t dt t dw t x 0 x the values t t are selected by a controller from a subset of 0 1 that depends on the current position x t for every t 0 at any stopping rule of his choice a second player called stopper can halt the evolution of the process x upon which he receives from the controller the amount e u x here 2 0 1 is a discount factor and u 0 1 is a continuous reward function under appropriate conditions on this function and on the contoller s set of choices it is shown that the two players have a saddle point of optimal strategies these can be described fairly explicitly by reduction to a suitable problem of optimal stopping whose maximal expected reward v coincides with the value of the game v sup inf x e e u x inf x sup e e u x

stochastic perron s method and elementary strategies for zero sum differential games we develop here the stochastic perron method in the framework of two player zero sum differential games we consider the formulation of the game where both players play symmetrically feedback strategies as opposed to the elliott kalton formulation prevalent in the literature the class of feedback strategies we use is carefully chosen so that the state equation admits strong solutions and the technicalities involved in the stochastic perron method carry through in a rather simple way more precisely we define the game over elementary strategies which are well motivated by intuition within this framework the stochastic perron method produces a viscosity subsolution of the upper isaacs equation dominating the upper value of the game and a viscosity supersolution of the upper isaacs equation lying below the upper value of the game using a viscosity comparison result we obtain that the upper value is the unique and continuous viscosity solution of the upper isaacs equation an identical statement hol

sampling in paley wiener spaces on combinatorial graphs a notion of paley wiener spaces on combinatorial graphs is introduced it is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets such uniqueness sets are described in terms of poincare wirtinger type inequalities a reconstruction algorithm of paley wiener functions from uniqueness sets which uses the idea of frames in hilbert spaces is developed special consideration is given to the n dimensional lattice homogeneous trees and eigenvalue and eigenfunction problems on finite graphs

compromise principle based methods of identifying capacities in the framework of multicriteria decision analysis abstract the basic aim of the compromise principle employed in this paper is to seek the capacity by which each alternative has a relatively equal chance to reach as close as possible to its highest reachable overall evaluation according to the compromise principle three types of capacity identification methods the simple arithmetic average based compromise method the least squares based compromise method and the linear programming based compromise method are proposed the input information required for the compromise principle based identification methods consists of the preference information with respect to the decision criteria and the partial evaluations of the alternatives provided by the decision maker an illustrative example is given to show the processes of the proposed methods and a comparison analysis with the maximum entropy principle based identification method is also presented