the value of the stochastic solution in stochastic linear programs with fixed recourse stochastic linear programs have been rarely used in practical situations largely because of their complexity in evaluating these problems without finding the exact solution a common method has been to find bounds on the expected value of perfect information in this paper we consider a different method we present bounds on the value of the stochastic solution that is the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters these bounds are calculated by solving smaller programs related to the stochastic recourse problem

bounding the expectation of a saddle function with application to stochastic programming we previously obtained tight upper and lower bounds to the expectation of a saddle function of multivariate random variables using first and cross moments of the random variables without independence assumptions these bounds are applicable when domains of the random vectors are compact sets in the euclidean space in this paper we extend the results to the case of unbounded domains similar in spirit to the extensions by birge and wets in the pure convex case the relationship of these bounds to a certain generalized moment problem is also investigated finally for solving stochastic linear programs utilizing the above bounding procedures a computationally more appealing order cone decomposition scheme is proposed which behaves quadratically in the number of random variables moreover the resulting upper and lower approximations are amenable to efficient solution techniques

existence of primitive divisors of lucas and lehmer numbers we prove that for n 30 every n th lucas and lehmer number has a primitive divisor this allows us to list all lucas and lehmer numbers without a primitive divisor

multiplicity estimates on group varieties tions the field k will be the field q of algebraic numbers we remark at this point that the results remain true if we take instead of the field of complex numbers c its p adic analogue c for some fixed prime p and for k a corresponding subfield kp of cp we restrict ourselves to the complex case in order to avoid some minor complications appearing in the p adic domain these complications always arise when analytic functions come in our functions are defined globally in the case of complex numbers but only locally in the case when we deal with the p adic domain these difficulties can be avoided by a purely algebraic approach if we took this approach the only condition on the ground field would be that it should be algebraically closed and of characteristic zero but we prefer to avoid such an approach in order to keep the text understandable also for those who are mainly interested in the applications in transcendence

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

fast approximate k nn graph construction for high dimensional data via recursive lanczos bisection nearest neighbor graphs are widely used in data mining and machine learning a brute force method to compute the exact knn graph takes dn2 time for n data points in the d dimensional euclidean space we propose two divide and conquer methods for computing an approximate knn graph in dnt time for high dimensional data large d the exponent t 1 2 is an increasing function of an internal parameter which governs the size of the common region in the divide step experiments show that a high quality graph can usually be obtained with small overlaps that is for small values of t a few of the practical details of the algorithms are as follows first the divide step uses an inexpensive lanczos procedure to perform recursive spectral bisection after each conquer step an additional refinement step is performed to improve the accuracy of the graph finally a hash table is used to avoid repeating distance calculations during the divide and conquer process the combination of these techniques is shown to yield quite effective algorithms for building knn graphs

stochastic programming approach to optimization under uncertainty in this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems we argue that two stage say linear stochastic programming problems can be solved with a reasonable accuracy by monte carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages we discuss an extension of coherent risk measures to a multistage setting and in particular dynamic programming equations for such problems

aggregation and discretization in multistage stochastic programming multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events in order to ensure computational tractability continuous state stochastic programs are usually discretized and frequently the curse of dimensionality dictates that decision stages must be aggregated in this article we construct two discrete stage aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem the approximate problems involve finitely many decisions and constraints thus principally allowing for numerical solution

computing bounds for stochastic programming problems by means of a generalized moment problem bounds on the expected value of a convex function are obtained by means of an approximating generalized moment problem numerical implementation is discussed in the context of stochastic programming problems

on the abc conjecture ii let x y and z be coprime positive integers with x y z in this paper we give upper bounds for z in terms of the greatest square free factor of xyz

nested decomposition of multistage nonlinear programs with recourse nested decomposition is extended to the case of arborescent nonlinear programs duals of extensive forms of nonlinear multistage stochastic programs constitute a particular class of those problems the method is tested on a set of problems of that type

patch alignment manifold matting image matting is generally modeled as a space transform from the color space to the alpha space by estimating the alpha factor of the model the foreground of an image can be extracted however there is some dimensional information redundancy in the alpha space it usually leads to the misjudgments of some pixels near the boundary between the foreground and the background in this paper a manifold matting framework named patch alignment manifold matting is proposed for image matting in particular we first propose a part modeling of color space in the local image patch we then perform whole alignment optimization for approximating the alpha results using subspace reconstructing error furthermore we utilize nesterov s algorithm to solve the optimization problem finally we apply some manifold learning methods in the framework and obtain several image matting methods such as named isomap matting and its derived cascade isomap matting the experimental results reveal that the manifold matting framework and its two examples are effective when compared with several representative matting methods

stochastic optimization problems with incomplete information on distribution functions the main purpose of this paper is to discuss numerical optimization procedures based on duality theory for stochastic extremal problems in which the distribution function is only partially known we formulate such problems as minimax problems in which the inner problem involves optimization with respect to probability measures the latter problem is solved using generalized linear programming techniques then we state the dual problem to the initial stochastic optimization problem numerical procedures that avoid the difficulties associated with solving the inner problem are proposed

on divisors of lucas and lehmer numbers let un be the nth term of a lucas sequence or a lehmer sequence in this article we shall establish an estimate from below for the greatest prime factor of un which is of the form n exp log n 104 log log n in doing so we are able to resolve a question of schinzel from 1962 and a conjecture of erd s from 1965 in addition we are able to give the first general improvement on results of bang from 1886 and carmichael from 1912

epi convergent discretizations of multistage stochastic programs in many dynamic stochastic optimization problems in practice the uncertain factors are best modeled as random variables with an infinite support this results in infinite dimensional optimization problems that can rarely be solved directly therefore the random variables stochastic processes are often approximated by finitely supported ones scenario trees which result in finite dimensional optimization problems that are more likely to be solvable by available optimization tools this paper presents conditions under which such finite dimensional optimization problems can be shown to epi converge to the original infinite dimensional problem epi convergence implies the convergence of optimal values and solutions as the discretizations are made finer our convergence result applies to a general class of convex problems where neither linearity nor complete recourse are assumed

primitive divisors of lucas and lehmer sequences stewart reduced the problem of determining all lucas and lehmer sequences whose nth element does not have a primitive divisor to solving certain thue equations using the method of tzanakis and de weger for solving thue equations we determine such sequences for n 30 further computations lead us to conjecture that for n 30 the nth element of such sequences always has a primitive divisor

tight bounds for stochastic convex programs variable and row aggregation as a technique of simplifying a mathematical program is utilized to develop bounds for two stage stochastic convex programs with random right hand sides if one is able to utilize the problem structure along with only first moment information a tighter bound than the usual mean model bound based on jensen s inequality may be obtained moreover it is possible to construct examples for which the mean model bound will be arbitrarily poor consequently one can tighten jensen s bound for stochastic programs when the distribution has a compact support this bound may be improved further by partitioning the support using conditional first moments with regard to first moment upper bounds the gassmann ziemba inequality is used for the stochastic convex program to seek a model which can be solved using standard convex programming techniques moreover it allows one to easily construct upper bounds using the solution of the lower bounding problem finally the results are extended to multistage stochastic convex programming problems

second order lower bounds on the expectation of a convex function we develop a class of lower bounds on the expectation of a convex function the bounds utilize the first two moments of the underlying random variable whose support is contained in a bounded interval or hyperrectangle our bounds have applications to stochastic programs whose random parameters are known only through limited moment information computational results are presented for two stage stochastic linear programs

applying the minimax criterion in stochastic recourse programs we consider an optimization problem in which some uncertain parameters are replaced by random variables the minimax approach to stochastic programming concerns the problem of minimizing the worst expected value of the objective function with respect to the set of probability measures that are consistent with the available information on the random data only very few practicable solution procedures have been proposed for this problem and the existing ones rely on simplifying assumptions in this paper we establish a number of stability results for the minimax stochastic program justifying in particular the approach of restricting attention to probability measures with support in some known finite set following this approach we elaborate solution procedures for the minimax problem in the setting of two stage stochastic recourse models considering the linear recourse case as well as the integer recourse case since the solution procedures are modifications of well known algorithms their efficacy is immediate from the computational testing of these procedures and we do not report results of any computational experiments

network unfolding map by vertex edge dynamics modeling the emergence of collective dynamics in neural networks is a mechanism of the animal and human brain for information processing in this paper we develop a computational technique using distributed processing elements in a complex network which are called particles to solve semisupervised learning problems three actions govern the particles dynamics generation walking and absorption labeled vertices generate new particles that compete against rival particles for edge domination active particles randomly walk in the network until they are absorbed by either a rival vertex or an edge currently dominated by rival particles the result from the model evolution consists of sets of edges arranged by the label dominance each set tends to form a connected subnetwork to represent a data class although the intrinsic dynamics of the model is a stochastic one we prove that there exists a deterministic version with largely reduced computational complexity specifically with linear growth furthermore the edge domination process corresponds to an unfolding map in such way that edges stretch and shrink according to the vertex edge dynamics consequently the unfolding effect summarizes the relevant relationships between vertices and the uncovered data classes the proposed model captures important details of connectivity patterns over the vertex edge dynamics evolution in contrast to the previous approaches which focused on only vertex or only edge dynamics computer simulations reveal that the new model can identify nonlinear features in both real and artificial data including boundaries between distinct classes and overlapping structures of data

learning with ell 1 graph for image analysis the graph construction procedure essentially determines the potentials of those graph oriented learning algorithms for image analysis in this paper we propose a process to build the so called directed 1 graph in which the vertices involve all the samples and the ingoing edge weights to each vertex describe its 1 norm driven reconstruction from the remaining samples and the noise then a series of new algorithms for various machine learning tasks e g data clustering subspace learning and semi supervised learning are derived upon the 1 graphs compared with the conventional k nearest neighbor graph and ball graph the 1 graph possesses the advantages 1 greater robustness to data noise 2 automatic sparsity and 3 adaptive neighborhood for individual datum extensive experiments on three real world datasets show the consistent superiority of 1 graph over those classic graphs in data clustering subspace learning and semi supervised learning tasks

bounds for two stage stochastic programs with fixed recourse this paper develops upper and lower bounds on two stage stochastic linear programs using limited moment information the case considered is when both the right hand side as well as the objective coefficients of the second stage problem are random random variables are allowed to have arbitrary multivariate probability distributions with bounded support first upper and lower bounds are obtained using first and cross moments from which we develop bounds using only first moments the bounds are shown to solve the respective general moment problems

the scenario generation algorithm for multistage stochastic linear programming a multistage stochastic linear program mslp is a model of sequential stochastic optimization where the objective and constraints are linear when any of the random variables used in the mslp are continuous the problem is infinite dimensional to numerically tackle such a problem we usually replace it with a finite dimensional approximation even when all the random variables have finite support the problem is often computationally intractable and must be approximated by a problem of smaller dimension one of the primary challenges in the field of stochastic programming deals with discovering effective ways to evaluate the importance of scenarios and to use that information to trim the scenario tree in such a way that the solution to the smaller optimization problem is not much different from the problem stated with the original tree the scenario generation sg algorithm proposed in this paper is a finite element method that addresses this problem for the class of mslp with random right hand sides

scenario tree modeling for multistage stochastic programs an important issue for solving multistage stochastic programs consists in the approximate representation of the multivariate stochastic input process in the form of a scenario tree in this paper we develop stability theory based heuristics for generating scenario trees out of an initial set of scenarios they are based on forward or backward algorithms for tree generation consisting of recursive scenario reduction and bundling steps conditions are established implying closeness of optimal values of the original process and its tree approximation respectively by relying on a recent stability result in heitsch romisch and strugarek siam j optim 17 511 525 2006 for multistage stochastic programs numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management

epi convergent discretizations of multistage stochastic programs via integration quadratures this paper presents procedures for constructing numerically solvable discretizations of multistage stochastic programs that epi converge to the original problem as the discretizations are made finer epi convergence implies in particular that the cluster points of the first stage solutions of the discretized problems are optimal first stage solutions of the original problem the discretization procedures apply to a general class of nonlinear stochastic programs where the uncertain factors are driven by time series models using existing routines for numerical integration allows for an easy and efficient implementation of the procedures

mtc a fast and robust graph based transductive learning method despite the great success of graph based transductive learning methods most of them have serious problems in scalability and robustness in this paper we propose an efficient and robust graph based transductive classification method called minimum tree cut mtc which is suitable for large scale data motivated from the sparse representation of graph we approximate a graph by a spanning tree exploiting the simple structure we develop a linear time algorithm to label the tree such that the cut size of the tree is minimized this significantly improves graph based methods which typically have a polynomial time complexity moreover we theoretically and empirically show that the performance of mtc is robust to the graph construction overcoming another big problem of traditional graph based methods extensive experiments on public data sets and applications on web spam detection and interactive image segmentation demonstrate our method s advantages in aspect of accuracy speed and robustness

p adic logarithmic forms and a problem of erdős for any natural number m 1 let p m denote the greatest prime divisor of m by the problem of erd s in the title of the present paper we mean the general version of his problem that is his conjecture from 1965 that frac p 2 n 1 n to infty quad rm as n to infty see erd s 10 and its far reaching generalization to lucas and lehmer numbers in the present paper the author delivers three refinements upon yu 40 required by c l stewart for solving completely the problem of erd s see stewart 25 the author gives also some remarks on the solution of this problem aiming to be more streamlined with respect to the p adic theory of logarithmic forms

sample average approximation method for compound stochastic optimization problems the paper studies stochastic optimization programming problems with compound functions containing expectations and extreme values of other random functions as arguments compound functions arise in various applications a typical example is a variance function of nonlinear outcomes other examples include stochastic minimax problems econometric models with latent variables and multilevel and multicriteria stochastic optimization problems as a solution technique a sample average approximation saa method also known as statistical or empirical sample mean method is used the method consists in approximation of all expectation functions by their empirical means and solving the resulting approximate deterministic optimization problems in stochastic optimization this method is widely used for optimization of standard expectation functions under constraints in this paper saa method is extended to general compound stochastic optimization problems the conditions for convergence in mean almost sure

interactive image segmentation via adaptive weighted distances an interactive algorithm for soft segmentation of natural images is presented in this paper the user first roughly scribbles different regions of interest and from them the whole image is automatically segmented this soft segmentation is obtained via fast linear complexity computation of weighted distances to the user provided scribbles the adaptive weights are obtained from a series of gabor filters and are automatically computed according to the ability of each single filter to discriminate between the selected regions of interest we present the underlying framework and examples showing the capability of the algorithm to segment diverse images

asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems we study the asymptotic behavior of the statistical estimators that maximize a not necessarily dieren tiable criterion function possibly subject to side constraints equalities and inequalities the consistency results generalize those of wald and huber conditions are also given under which one is still able to obtain asymptotic normality the analysis brings to the fore the relationship between the problem of nding statistical estimators and that of nding the optimal solutions of stochastic optimization problems with partial information the last section is devoted to the properties of the saddle points of the associated lagrangians

extreme points of sets of randomized strategies in constrained optimization and control problems this paper concerns the existence and characterization of optimal randomized strategies for some constrained optimization and control problems we first present a characterization of the extreme points of a set of randomized strategies that satisfy n moment like constraints conditions are given under which those extreme points are randomizations of at most n 1 deterministic strategies this result is then applied to obtain the existence and characterization of optimal strategies for a class of deterministic allocation like optimization problems and their young relaxations similar results are obtained for constrained markov control processes in borel spaces

process flexibility a distribution free bound on the performance of k chain process flexibility has been widely applied in many industries as a competitive strategy to improve responsiveness to demand uncertainty an important flexibility concept is the long chain proposed by jordan and graves 1995 jordan wc graves sc 1995 principles on the benefits of manufacturing process flexibility management sci 41 4 577 594 the effectiveness of the long chain has been investigated via numerical as well as theoretical analysis for specific probability distributions of the random demand in this paper we obtain in closed form a distribution free bound on the ratio of the expected sales of the long chain relative to that of full flexibility our bound depends only on the mean and standard deviation of the random demand but compares very well with the ratio that uses complete information of the demand distribution this suggests the robustness of the performance of the long chain under different distributions we also prove a similar result for k chain a more general flexibility

barycentric scenario trees in convex multistage stochastic programming this work deals with the approximation of convex stochastic multistage programs allowing prices and demand to be stochastic with compact support based on earlier results sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem error bounds for the optimal policies of the approximate problem and duality analysis with respect to the stochastic data determine the scenarios which improve the approximation convergence of the approximate solutions is proven under the stated assumptions preliminary computational results are outlined

parallel decomposition of multistage stochastic programming problems a new decomposition method for multistage stochastic linear programming problems is proposed a multistage stochastic problem is represented in a tree like form and with each node of the decision tree a certain linear or quadratic subproblem is associated the subproblems generate proposals for their successors and some backward information for their predecessors the subproblems can be solved in parallel and exchange information in an asynchronous way through special buffers after a finite time the method either finds an optimal solution to the problem or discovers its inconsistency an analytical illustrative example shows that parallelization can speed up computation over every sequential method computational experiments indicate that for large problems we can obtain substantial gains in efficiency with moderate numbers of processors

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

solving slp recourse problems with arbitrary multivariate distributions the dependent case bounding the expectation of a convex function of a multivariate random variable is of great importance for solving stochastic linear programming problems slp with recourse the classic bounds are those of jensen and edmundson madansky whereby jensen s inequality is available in the independent as well as in the dependent case in contrast with the edmundson madansky e m inequality madansky a 1959 bounds on the expectation of a convex function of multivariate random variable ann math statist 30 743 746 which is valid only in the independent case this paper presents a straightforward extension of the e m inequality for the dependent case yielding a discrete distribution that is extremal with respect to a partial ordering of a set of distribution functions further we prove that the bounds for the expectation behave monotonically by applying the obtained inequality to smaller and smaller subintervals as this requires the evaluation of conditional probabilities and expectations we replace the given distribution by a discrete one resulting from sampling and determine the sampling size that keeps the statistical error negligible finally we conclude with the application to slp recourse problems and state some computational results illustrating the effort for solving recourse problems

on a class of minimax stochastic programs for a particular class of minimax stochastic programming models we show that the problem can be equivalently reformulated into a standard stochastic programming problem this permits the direct use of standard decomposition and sampling methods developed for stochastic programming we also show that this class of minimax stochastic programs is closely related to a large family of mean risk stochastic programs where risk is measured in terms of deviations from a quantile

version independence and nested distributions in multistage stochastic optimization the amount of stagewise available information is crucial in multistage stochastic optimization but unlike data which directly enter the profit loss functions of a decision problem information is invariant w r t bijective transformations the usual concept to deal with information in multistage stochastic programming is by introducing filtrations i e increasing sequences of sigma algebras to which the decisions must be adapted for the definition of filtrations one has to fix a certain probability space while random variables are typically given by their distributions only and all realizations of this distribution on some probability space are equivalent we introduce here the new concept of nested distributions to describe the information structure as well as the scenario process of a stochastic optimization program in a way which is independent of specific versions of probability spaces and random variables the setting is totally in distribution two stochastic programs with identical objective function and constraints are equivalent if and only if the scenario processes have the same nested distribution as a byproduct we analyze the question of whether introducing extra randomness by defining randomized decisions would lead to improvement in the objective value in the language of information this would mean that enlarging the filtration based on available information by conditionally independent additional random variables would have a positive effect we show that in general the answer is yes while for compound convex objectives the answer is no finally we define a distance between nested distributions which generalizes the well known kantorovich distance of probability distributions and demonstrates that this distance may be used in quantifying the quality of approximation between a continuous stochastic program and a tree discretization or between two tree discretizations

the perfect order subset conjecture for simple groups abstract this paper verifies the perfect order subset conjecture for simple groups for all but one family of finite simple groups specifically for each nonabelian finite simple group g there is some n such that the cardinality of the nonempty subset of all elements of order n in g does not divide the order of g unless g is an orthogonal group of plus type in dimension 4n for some n 2

a compromise solution for the multiobjective stochastic linear programming under partial uncertainty abstract this paper solves the multiobjective stochastic linear program with partially known probability we address the case where the probability distribution is defined by crisp inequalities we propose a chance constrained approach and a compromise programming approach to transform the multiobjective stochastic linear program with linear partial information on probability distribution into its equivalent uniobjective problem the resulting program is then solved using the modified l shaped method we illustrate our results by an example

bounds and approximations for multistage stochastic programs consider typically large multistage stochastic programs which are defined on scenario trees as the basic data structure it is well known that the computational complexity of the solution depends on the size of the tree which itself increases typically exponentially fast with its height i e the number of decision stages for this reason approximations which replace the problem by a simpler one and allow bounding the optimal value are of great importance in this paper we study several methods to obtain lower and upper bounds for multistage stochastic programs and we demonstrate their use in a multistage inventory problem

guaranteed bounds for general nondiscrete multistage risk averse stochastic optimization programs in general multistage stochastic optimization problems are formulated on the basis of continuous distributions describing the uncertainty such infinite problems are practically impossible to solve as they are formulated and finite tree approximations of the underlying stochastic processes are used as proxies in this paper we demonstrate how one can find guaranteed bounds i e finite tree models for which the optimal values give upper and lower bounds for the optimal value of the original infinite problem typically there is a gap between the two bounds however this gap can be made arbitrarily small by making the approximating trees bushier we consider approximations in the first order stochastic sense in the convex order sense and based on subgradient approximations their use is shown in a multistage risk averse production problem

a stability result for linear markovian stochastic optimization problems in this paper we propose a semi metric for markov processes that allows to bound optimal values of linear markovian stochastic optimization problems similar to existing notions of distance for general stochastic processes our distance is based on transportation metrics as opposed to the extant literature the proposed distance is problem specific i e dependent on the data of the problem whose objective value we want to bound as a result we are able to consider problems with randomness in the constraints as well as in the objective function and therefore relax an assumption in the extant literature we derive several properties of the proposed semi metric and demonstrate its use in a stylized numerical example

toward the optimization of normalized graph laplacian normalized graph laplacian has been widely used in many practical machine learning algorithms e g spectral clustering and semisupervised learning however all of them use the euclidean distance to construct the graph laplacian which does not necessarily reflect the inherent distribution of the data in this brief we propose a method to directly optimize the normalized graph laplacian by using pairwise constraints the learned graph is consistent with equivalence and nonequivalence pairwise relationships and thus it can better represent similarity between samples meanwhile our approach unlike metric learning automatically determines the scale factor during the optimization the learned normalized laplacian matrix can be directly applied in spectral clustering and semisupervised learning algorithms comprehensive experiments demonstrate the effectiveness of the proposed approach

a measure approximation for distributionally robust pde constrained optimization problems in numerous applications scientists and engineers acquire varied forms of data that partially characterize the inputs to an underlying physical system this data is then used to inform decisions such as controls and designs consequently it is critical that the resulting control or design is robust to the inherent uncertainties associated with the unknown probabilistic characterization of the model inputs in this paper we consider optimal control and design problems constrained by partial differential equations with uncertain inputs we do not assume a known probabilistic model for the inputs but rather we formulate the problem as a distributionally robust optimization problem where the outer minimization problem determines the control or design while the inner maximization problem determines the worst case probability measure that matches desired characteristics of the data we analyze the inner maximization problem in the space of measures and introduce a novel measure approximation technique bas