compact location problems we consider the problem of placing a specified number p of facilities on the nodes of a network so as to minimize some measure of the distances between facilities this formulation models a number of problems arising in facility location statistical clustering pattern recognition and also a processor allocation problem in multiprocessor systems

better bounds on online unit clustering unit clustering is the problem of dividing a set of points from a metric space into a minimal number of subsets such that the points in each subset are enclosable by a unit ball we continue the work recently initiated by chan and zarrabi zadeh on determining the competitive ratio of the online version of this problem for the one dimensional case we develop a deterministic algorithm improving the best known upper bound of 7 4 by epstein and van stee to 5 3 this narrows the gap to the best known lower bound of 8 5 to only 1 15 our algorithm automatically leads to improvements in all higher dimensions as well finally we strengthen the deterministic lower bound in two dimensions and higher from 2 to 13 6

segmentation problems we study a novel genre of optimization problems which we call segmentation problems motivated in part by certain aspects of clustering and data mining for any classical optimization problem the corresponding segmentation problem seeks to partition a set of cost vectors into several segments so that the overall cost is optimized we focus on two natural and interesting but maxsnp complete problems in this class the hypercube segmentation problem and the catalog segmentation problem and present approximation algorithms for them we also present a general greedy scheme which can be specialized to approximate any segmentation problem

computing maximally separated sets in the plane let s be a set of n points in reals 2 given an integer 1 le k le n we wish to find a maximally separated subset i subseteq s of size k this is a subset for which the minimum among the k choose 2 pairwise distances between its points is as large as possible the decision problem associated with this problem is to determine whether there exists i subseteq s i k so that all k choose 2 pairwise distances in i are at least 2 this problem can also be formulated in terms of disk intersection graphs let d be the set of unit disks centered at the points of s the disk intersection graph g of d has as edges all pairs of disks with nonempty intersection any set i with the above properties is then the set of centers of disks that form an independent set in the graph g this problem is known to be np complete if k is part of the input in this paper we first present a linear time eps approximation algorithm for any constant k next we give exact algorithms for the cases k 3 and k 4 that run in time o n 4 3 polylog n we also present a simpler n o sqrt k time exact algorithm as compared with the recent algorithm in j alber and j fiala j algorithms 52 2004 pp 134 151 for arbitrary values of k

covering moving points with anchored disks consider a set of mobile clients represented by n points in the plane moving at constant speed along n different straight lines we study the problem of covering all mobile clients using a set of k disks centered at k fixed centers each disk exists only at one instant and while it does covers any client within its coverage radius the task is to select an activation time and a radius for each disk such that every mobile client is covered by at least one disk in particular we study the optimization problem of minimizing the maximum coverage radius first we prove that although the static version of the problem is polynomial the kinetic version is np hard moreover we show that the problem is not approximable by a constant factor unless p np we then present a generic framework to solve it for fixed values of k which in turn allows us to solve more general optimization problems our algorithms are efficient for constant values of k

improved combinatorial algorithms for facility location problems we present improved combinatorial approximation algorithms for the uncapacitated facility location problem two central ideas in most of our results are cost scaling and greedy improvement we present a simple greedy local search algorithm which achieves an approximation ratio of 2 414 epsilon in tilde o n 2 epsilon time this also yields a bicriteria approximation tradeoff of 1 gamma 1 2 gamma for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to jain and vazirani we get an approximation ratio of 1 853 in tilde o n 3 time this is very close to the approximation guarantee of the best known algorithm which is linear programming lp based further combined with the best known lp based algorithm for facility location we get a very slight improvement in the approximation factor for facility location achieving 1 728 we also consider a variant of the capacitated facility location problem and present improved approximation algorithms for this

the online set cover problem let x 1 2 ldots n be a ground set of n elements and let cal s be a family of subsets of x cal s m with a positive cost c s associated with each s in cal s consider the following online version of the set cover problem described as a game between an algorithm and an adversary an adversary gives elements to the algorithm from x one by one once a new element is given the algorithm has to cover it by some set of cal s containing it we assume that the elements of x and the members of cal s are known in advance to the algorithm however the set x subseteq x of elements given by the adversary is not known in advance to the algorithm in general x may be a strict subset of x the objective is to minimize the total cost of the sets chosen by the algorithm let cal c denote the family of sets in cal s that the algorithm chooses at the end of the game the adversary also produces offline a family of sets cal c opt that covers x the performance of the algorithm is the ratio between the cost of cal c and the cost of cal c opt the maximum ratio taken over all input sequences is the competitive ratio of the algorithm we present an o log m log n competitive deterministic algorithm for the problem and establish a nearly matching omega bigl frac log n log m log log m log log n bigr lower bound for all interesting values of m and n the techniques used are motivated by similar techniques developed in computational learning theory for online prediction e g the winnow algorithm together with a novel way of converting a fractional solution into a deterministic online algorithm

ectropy of diversity measures for populations in euclidean space measures to evaluate the diversity of a set of points population in euclidean space play an important role in a variety of areas of science and engineering well known measures are often used without a clear insight into their quality and many of them do not appropriately penalize populations with a few distant groups of collocated or closely located points to the best of our knowledge there is a lack of rigorous criteria to compare diversity measures and help select an appropriate one in this work we define a mathematical notion of ectropy for classifying diversity measures in terms of the extent to which they tend to penalize point collocation we investigate the advantages and disadvantages of several known measures and we propose some novel ones that exhibit a good ectropic behavior in particular we introduce a quasi entropy measure based on a geometric covering problem three measures based on discrepancy from uniform distribution and one based on euclidean minimum spanning trees all considered measures are tested and compared on a large set of random and structured populations special attention is also devoted to the complexity of computing the measures most of the novel measures compare favorably with the classical ones in terms of ectropy the measure based on euclidean minimum spanning trees turns out to be the most promising one in terms of the tradeoff between the ectropic behavior and the computational complexity

on min max r gatherings we consider a min max version of the previously studied r gathering problem with unit demands the problem we consider is a metric facility location problem in which each open facility must serve at least r customers and the maximum of all the facility and connection costs should be minimized rather than their sum this problem is motivated by scenarios in which r customers are required for a facility to be worth opening and the costs represent the time until the facility connection will be available i e we want to have the complete solution ready as soon as possible we present a 3 approximation algorithm for this problem and prove that it cannot be approximated better assuming p np next we consider this problem with the additional natural requirement that each customer will be assigned to a nearest open facility and present a 9 approximation algorithm we further consider previously introduced special cases and variants and obtain improved algorithmic and hardness results

fcm based model selection algorithms for determining the number of clusters clustering is an important research topic that has practical applications in many fields it has been demonstrated that fuzzy clustering using algorithms such as the fuzzy c means fcm has clear advantages over crisp and probabilistic clustering methods like most clustering algorithms however fcm and its derivatives need the number of clusters in the given data set as one of their initializing parameters the main goal of this paper is to develop an effective fuzzy algorithm for automatically determining the number of clusters after a brief review of the relevant literature we present a new algorithm for determining the number of clusters in a given data set and a new validity index for measuring the goodness of clustering experimental results and comparisons are given to illustrate the performance of the new algorithm

online unit covering in euclidean space abstract we revisit the online unit covering problem in higher dimensions given a set of n points in r d that arrive one by one cover the points by balls of unit radius so as to minimize the number of balls used in this paper we work in r d using the euclidean distance i we give an online deterministic algorithm with competitive ratio o 1 321 d thereby improving on the previous record o 2 d d log d due to charikar et al 2004 by an exponential factor in particular the competitive ratios are 5 in the plane and 12 in 3 space the previous ratios were 7 and 21 respectively for d 3 the ratio of our online algorithm matches the ratio of the current best offline algorithm for the same problem due to biniaz et al 2017 which is remarkable and rather unusual ii we show that the competitive ratio of every deterministic online algorithm for unit covering in r d under the l 2 norm is at least d 1 for every d 1 this greatly improves upon the previous best lower bound log d log log log d due to charikar et al 2004 iii we generalize the above result to unit covering in r d under the l c norm where c is a centrally symmetric convex body via the illumination number iv we obtain lower bounds of 4 and 5 for the competitive ratio of any deterministic algorithm for online unit covering in r 2 and r 3 respectively the previous best lower bounds were 3 for both cases v when the input points are from the square or hexagonal lattice in r 2 we give deterministic online algorithms for unit covering with an optimal competitive ratio of 3 for the cubic lattice in r 3 we give a deterministic online algorithm with a competitive ratio of 5

parameterized approximation algorithms for some location problems in graphs we develop efficient parameterized with additive error approximation algorithms for the connected r domination problem and the connected p center problem for unweighted and undirected graphs given a graph g we show how to construct a connected big r mathcal o mu big dominating set d with d le d efficiently here d is a minimum connected r dominating set of g and mu is our graph parameter which is the tree breadth or the cluster diameter in a layering partition of g additionally we show that a mathcal o mu approximation for the connected p center problem on g can be computed in polynomial time our interest in these parameters stems from the fact that in many real world networks including internet application networks web networks collaboration networks social networks biological networks and others and in many structured classes of graphs these parameters are small constants

kinetic clustering of points on the line the problem of clustering a set of points moving on the line consists of the following given positive integers n and k the initial position and the velocity of n points find an optimal k clustering of the points we consider two classical quality measures for the clustering minimizing the sum of the clusters diameters and minimizing the maximum diameter of a cluster for the former we present polynomial time algorithms under some assumptions and for the latter a 2 71 e approximation

k dimensional invariant cones of random dynamical systems in rn with applications abstract we investigated a linear random dynamical system which strongly preserves a cone c of dimension k abbr k cone in r n under some general assumptions it is shown that such system admits a measurable family of k dimensional subspaces and a measurable family of n k dimensional subspaces which are complementary to each other and form into a tempered invariant splitting of r n we further apply the measurable bundles so obtained to study the linear random monotone cyclic feedback systems as well as the linear competitive cooperative tridiagonal systems this generalizes the floquet theory for these deterministic non autonomous or time periodic systems to the random systems

on random linear dynamical systems in a banach space i multiplicative ergodic theorem and krein rutman type theorems abstract for linear random dynamical systems in a separable banach space x we derived a series of krein rutman type theorems with respect to co invariant cone family with rank k which present a quasi equivalence relation between the measurably co invariant cone family and the measurably dominated splitting of x moreover such quasi equivalence relation turns out to be an equivalence relation whenever i k 1 or ii in the frame of the multiplicative ergodic theorem with certain lyapunov exponent being greater than the negative infinity for the second case we thoroughly investigated the relations between the lyapunov exponents the co invariant cone family and the measurably dominated splitting for linear random dynamical systems in x

approximation schemes for covering and packing problems in image processing and vlsi a unified and powerful approach is presented for devising polynomial approximation schemes for many strongly np complete problems such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e o with running time that is polynomial when e is fixed though the polynomiality of these algorithms depends on the degree of approximation e being fixed they cannot be improved owing to a negative result stating that there are no fully polynomial approximation schemes for strongly np complete problems unless np p the unified technique that is introduced here referred to as the shifting strategy is applicable to numerous geometric covering and packing problems the method of using the technique and how it varies with problem parameters are illustrated a similar technique independently devised by b s baker was shown to be applicable for covering and packing problems on planar graphs

random walks for image segmentation a novel method is proposed for performing multilabel interactive image segmentation given a small number of pixels with user defined or predefined labels one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels by assigning each pixel to the label for which the greatest probability is calculated a high quality image segmentation may be obtained theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits this algorithm is formulated in discrete space i e on a graph using combinatorial analogues of standard operators and principles from continuous potential theory allowing it to be applied in arbitrary dimension on arbitrary graphs

nonlinear dimensionality reduction by locally linear inlaying high dimensional data is involved in many fields of information processing however sometimes the intrinsic structures of these data can be described by a few degrees of freedom to discover these degrees of freedom or the low dimensional nonlinear manifold underlying a high dimensional space many manifold learning algorithms have been proposed here we describe a novel algorithm locally linear inlaying lli which combines simple geometric intuitions and rigorously established optimality to compute the global embedding of a nonlinear manifold using a divide and conquer strategy lli gains some advantages in itself first its time complexity is linear in the number of data points and hence lli can be implemented efficiently second lli overcomes problems caused by the nonuniform sample distribution third unlike existing algorithms such as isometric feature mapping isomap local tangent space alignment ltsa and locally linear coordination llc lli is robust to noise in addition to evaluate the embedding results quantitatively two criteria based on information theory and kolmogorov complexity theory respectively are proposed furthermore we demonstrated the efficiency and effectiveness of our proposal by synthetic and real world data sets

extremal properties of rate distortion functions an attempt is made to determine whether it is true that for a fixed distortion level delta the rate distortion function r p delta has in the distribution p no local maxima with values different from the global maximum it is shown that in general the answer is negative however the answer is positive for hamming distortion measures moreover their r is schur concave

a nearly linear time approximation scheme for the euclidean k median problem this paper provides a randomized approximation scheme for the k median problem when the input points lie in the d dimensional euclidean space the worst case running time is o 2 o log 1 epsilon varepsilon d 1 n log d 6 n which is nearly linear for any fixed varepsilon and d moreover our method provides the first polynomial time approximation scheme for and uncapacitated facility location instances in d dimensional euclidean space for any fixed d 2 our work extends techniques introduced originally by arora for the euclidean traveling salesman problem tsp to obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions the theorem is based on an adaptive decomposition scheme which guesses at every level of the hierarchy the structure of the optimal solution and accordingly modifies the parameters of the decomposition we believe that our methodology is of independent interest and may find applications to further geometric problems

a probabilistic analysis of em for mixtures of separated spherical gaussians we show that given data from a mixture of k well separated spherical gaussians in d a simple two round variant of em will with high probability learn the parameters of the gaussians to near optimal precision if the dimension is high d ln k we relate this to previous theoretical and empirical work on the em algorithm

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

spectral generalized multi dimensional scaling multidimensional scaling mds is a family of methods that embed a given set of points into a simple usually flat domain the points are assumed to be sampled from some metric space and the mapping attempts to preserve the distances between each pair of points in the set distances in the target space can be computed analytically in this setting generalized mds is an extension that allows mapping one metric space into another that is mds into target spaces in which distances are evaluated numerically rather than analytically here we propose an efficient approach for computing such mappings between surfaces based on their natural spectral decomposition where the surfaces are treated as sampled metric spaces the resulting spectral gmds procedure enables efficient embedding by incorporating smoothness of the metric structure into the problem thereby substantially reducing the complexity involved in its solution while practically overcoming its non convex nature the method is compared to existing techniques that compute dense correspondence between shapes numerical experiments of the proposed method demonstrate its efficiency and accuracy compared to state of the art approaches especially when isometry invariance is a dominant property

an indication of unification for different clustering approaches the question of finding generic concepts and properties common to the different clustering approaches is a current problem this inquire is addressed most thoroughly in kleinberg s paper on the impossibility theorem see 1 kleinberg introduced the notion of clustering function a function that takes a dissimilarity measure defined on a data set s and returns a partition of s and a set of simple properties for the study of such functions scale invariance richness and consistency the main result of 1 is the impossibility theorem there is no clustering method satisfying all these properties this study has been accepted as a rigorous proof of the difficulty in finding a unified framework for different clustering approaches our goal in this paper is to provide primary concepts and results for the formal study of the various clustering approaches to accomplish this we discuss and expand on the ideas introduced by kleinberg our guiding philosophy is to incorporate a crucial fact overlooked in the study conducted in 1 clustering methods not only depend on the dissimilarity measure but also on other parameters such as dissimilarity thresholds centroids stop criteria among others this paper gives a formal definition of clustering method and reformulates the afore mentioned properties even it introduces some new contrary to the result obtained in 1 many of the methods discussed here satisfy all of our properties with all these grounds in hand we glimpse a clue of unification among the different clustering approaches

topology invariant similarity of nonrigid shapes this paper explores the problem of similarity criteria between nonrigid shapes broadly speaking such criteria are divided into intrinsic and extrinsic the first referring to the metric structure of the object and the latter to how it is laid out in the euclidean space both criteria have their advantages and disadvantages extrinsic similarity is sensitive to nonrigid deformations while intrinsic similarity is sensitive to topological noise in this paper we approach the problem from the perspective of metric geometry we show that by unifying the extrinsic and intrinsic similarity criteria it is possible to obtain a stronger topology invariant similarity suitable for comparing deformed shapes with different topology we construct this new joint criterion as a tradeoff between the extrinsic and intrinsic similarity and use it as a set valued distance numerical results demonstrate the efficiency of our approach in cases where using either extrinsic or intrinsic criteria alone would fail

dynamics of the periodic type k competitive kolmogorov systems abstract for time periodic dissipative and irreducible type k competitive kolmogorov systems it is proved that there is a canonically defined countable family f of unordered disjoint invariant sets with the property that for every persistent trajectory whose limit set is not a cycle there exists a unique trajectory in some element of f such that these two trajectories are asymptotic and the corresponding points in these two trajectories are k related

a bad instance for k means k means is a seeding technique for the k means method with an expected approximation ratio of o logk where k denotes the number of clusters examples are known on which the expected approximation ratio of k means is w logk showing that the upper bound is asymptotically tight however it remained open whether k means yields a constant approximation with probability 1 poly k or even with constant probability we settle this question and present instances on which k means achieves an approximation ratio no better than 2 email protected logk with probability exponentially close to 1

a mapping approach to rate distortion computation and analysis in rate distortion theory results are often derived and stated in terms of the optimizing density over the reproduction space in this paper the problem is reformulated in terms of the optimal mapping from the unit interval with lebesgue measure that would induce the desired reproduction probability density this results in optimality conditions that are random relatives of the known lloyd 1982 optimality conditions for deterministic quantizers the validity of the mapping approach is assured by fundamental isomorphism theorems for measure spaces we show that for the squared error distortion the optimal reproduction random variable is purely discrete at supercritical distortion where the shannon 1948 lower bound is not tight the gaussian source is thus the only source that produces continuous reproduction variables for the entire range of positive rate to analyze the evolution of the optimal reproduction distribution we use the mapping formulation and establish an analogy to statistical mechanics the solutions are given by the distribution at isothermal statistical equilibrium and are parameterized by the temperature in direct correspondence to the parametric solution of the variational equations in rate distortion theory the analysis of an annealing process shows how the number of symbols grows as the system undergoes phase transitions thus an algorithm based on the mapping approach often needs but a few variables to find the exact solution while the blahut 1972 algorithm would only approach it at the limit of infinite resolution finally a quick deterministic annealing algorithm to generate the rate distortion curve is suggested the resulting curve is exact as long as continuous phase transitions in the process are accurately followed

nonlinear dimensionality reduction by topologically constrained isometric embedding many manifold learning procedures try to embed a given feature data into a flat space of low dimensionality while preserving as much as possible the metric in the natural feature space the embedding process usually relies on distances between neighboring features mainly since distances between features that are far apart from each other often provide an unreliable estimation of the true distance on the feature manifold due to its non convexity distortions resulting from using long geodesics indiscriminately lead to a known limitation of the isomap algorithm when used to map non convex manifolds presented is a framework for nonlinear dimensionality reduction that uses both local and global distances in order to learn the intrinsic geometry of flat manifolds with boundaries the resulting algorithm filters out potentially problematic distances between distant feature points based on the properties of the geodesics connecting those points and their relative distance to the boundary of the feature manifold thus avoiding an inherent limitation of the isomap algorithm since the proposed algorithm matches non local structures it is robust to strong noise we show experimental results demonstrating the advantages of the proposed approach over conventional dimensionality reduction techniques both global and local in nature

on the performance of data compression algorithms based upon string matching lossless and lossy data compression algorithms based on string matching are considered in the lossless case a result of wyner and ziv 1989 is extended in the lossy case a data compression algorithm based on approximate string matching is analyzed in the following two frameworks 1 the database and the source together form a markov chain of finite order 2 the database and the source are independent with the database coming from a markov model and the source from a general stationary ergodic model in either framework it is shown that the resulting compression rate converges with probability one to a quantity computable as the infimum of an information theoretic functional over a set of auxiliary random variables the quantity is strictly greater than the rate distortion function of the source except in some symmetric cases in particular this result implies that the lossy algorithm proposed by steinberg and gutman 1993 is not optimal even for memoryless or markov sources

ecmdd evidential c medoids clustering with multiple prototypes in this work a new prototype based clustering method named evidential c medoids ecmdd which belongs to the family of medoid based clustering for proximity data is proposed as an extension of fuzzy c medoids fcmdd on the theoretical framework of belief functions in the application of fcmdd and original ecmdd a single medoid prototype which is supposed to belong to the object set is utilized to represent one class for the sake of clarity this kind of ecmdd using a single medoid is denoted by secmdd in real clustering applications using only one pattern to capture or interpret a class may not adequately model different types of group structure and hence limits the clustering performance in order to address this problem a variation of ecmdd using multiple weighted medoids denoted by wecmdd is presented unlike secmdd in wecmdd objects in each cluster carry various weights describing their degree of representativeness for that class this mechanism enables each class to be represented by more than one object experimental results in synthetic and real data sets clearly demonstrate the superiority of secmdd and wecmdd moreover the clustering results by wecmdd can provide richer information for the inner structure of the detected classes with the help of prototype weights

approximating uniform triangular meshes in polygons we consider the problem of triangulating a convex polygon using n steiner points under the following optimality criteria 1 minimizing the overall edge length ratio 2 minimizing the maximum edge length and 3 minimizing the maximum triangle perimeter we establish a relation of these problems to a certain extreme packing problem based on this relationship we develop a heuristic producing constant approximations for all the optimality criteria above provided n is chosen sufficiently large that is the produced triangular mesh is uniform in these respects the method is easy to implement and runs in o n2log n time and o n space the observed runtime is much less moreover for criterion 1 the method works within the same complexity and approximation bounds for arbitrary polygons with possible holes and for criteria 2 and 3 it does so for a large subclass

active clustering of biological sequences given a point set s and an unknown metric d on s we study the problem of efficiently partitioning s into k clusters while querying few distances between the points in our model we assume that we have access to one versus all queries that given a point s s return the distances between s and all other points we show that given a natural assumption about the structure of the instance we can efficiently find an accurate clustering using only o k distance queries our algorithm uses an active selection strategy to choose a small set of points that we call landmarks and considers only the distances between landmarks and other points to produce a clustering we use our procedure to cluster proteins by sequence similarity this setting nicely fits our model because we can use a fast sequence database search program to query a sequence against an entire data set we conduct an empirical study that shows that even though we query a small fraction of the distances between the points we produce clusterings that are close to a desired clustering given by manual classification

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems ii finite dimensional systems abstract this is the second part in a series of papers concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part focuses on applications of general theory developed in the authors paper j mierczy ski w shen principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory trans amer math soc in press http dx doi org 10 1090 s0002 9947 2013 05814 x preprint available at http arxiv org abs 1209 3475 to positive random dynamical systems on finite dimensional ordered banach spaces it is shown under some quite general assumptions that measurable linear skew product semidynamical systems generated by measurable families of positive matrices and by strongly cooperative or type k strongly monotone systems of linear ordinary differential equations admit measurable families of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations

clustering to minimize the maximum intercluster distance the problem of clustering a set of points so as to minimize the maximum intercluster distance is studied an o kn approximation algorithm where n is the number of points and k is the number of clusters that guarantees solutions with an objective function value within two times the optimal solution value is presented this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality we also show that our approximation algorithm is best possible with respect to the approximation bound if pz np

constrained clustering by constraint programming cluster analysis is an important task in data mining with hundreds of different approaches in the literature since the last decade the cluster analysis has been extended to constrained clustering also called semi supervised clustering so as to integrate previous knowledge on data to clustering algorithms in this dissertation we explore constraint programming cp for solving the task of constrained clustering the main principles in cp are 1 users specify declaratively the problem in a constraint satisfaction problem 2 solvers search for solutions by constraint propagation and search relying on cp has two main advantages the declarativity which enables to easily add new constraints and the ability to find an optimal solution satisfying all the constraints when there exists one we propose two models based on cp to address constrained clustering tasks the models are flexible and general and supports instance level constraints and different cluster level constraints it also allows the users to choose among different optimization criteria in order to improve the efficiency different aspects have been studied in the dissertation experiments on various classical datasets show that our models are competitive with other exact approaches we show that our models can easily be embedded in a more general process and we illustrate this on the problem of finding the pareto front of a bi criterion optimization process

a context quantization approach to universal denoising we revisit the problem of denoising a discrete time continuous amplitude signal corrupted by a known memoryless channel by modifying our earlier approach to the problem we obtain a scheme that is much more tractable than the original one and at the same time retains the universal optimality properties the universality refers to the fact that the proposed denoiser asymptotically with increasing block length of the data achieves the performance of an optimum denoiser that has full knowledge of the distribution of a source generating the underlying clean sequence the only restriction being that the distribution is stationary the optimality in a sense we will make precise of the denoiser also holds in the case where the underlying clean sequence is unknown and deterministic and the only source of randomness is in the noise the schemes involve a simple preprocessing step of quantizing the noisy symbols to generate quantized contexts the quantized context value corresponding to each sequence component is then used to partition the unquantized symbols into subsequences a universal symbol by symbol denoiser for unquantized sequences is then separately employed on each of the subsequences we identify a rate at which the context length and quantization resolution should be increased so that the resulting scheme is universal the proposed family of schemes is computationally attractive with an upper bound on complexity which is independent of the context length and the quantization resolution initial experimentation seems to indicate that these schemes are not only superior from a computational viewpoint but also achieve better denoising in practice

asymmetric k center is log n hard to approximate in the asymmetric k center problem the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality the goal is to choose a set of k points to serve as centers and to assign all the points to the centers so that the maximum distance of any point from its center is as small as possible we show that the asymmetric k center problem is hard to approximate up to a factor of log n o 1 unless np dtime nlog log n since an o log n approximation algorithm is known for this problem this resolves the asymptotic approximability of asymmetric k center this is the first natural problem whose approximability threshold does not polynomially relate to the known approximation classes we also resolve the approximability threshold of the metric symmetric k center problem with costs

estimation of the rate distortion function motivated by questions in lossy data compression and by theoretical considerations the problem of estimating the rate distortion function of an unknown not necessarily discrete valued source from empirical data is examined the focus is the behavior of the so called ldquoplug inrdquo estimator which is simply the rate distortion function of the empirical distribution of the observed data sufficient conditions are given for its consistency and examples are provided demonstrating that in certain cases it fails to converge to the true rate distortion function the analysis of its performance is complicated by the fact that the rate distortion function is not continuous in the source distribution the underlying mathematical problem is closely related to the classical problem of establishing the consistency of maximum likelihood estimators mles general consistency results are given for the plug in estimator applied to a broad class of sources including all stationary and ergodic ones a more general class of estimation problems is also considered arising in the context of lossy data compression when the allowed class of coding distributions is restricted analogous results are developed for the plug in estimator in that case finally consistency theorems are formulated for modified e g penalized versions of the plug in and for estimating the optimal reproduction distribution

incremental clustering and dynamic information retrieval motivated by applications such as document and image classification in information retrieval we consider the problem of clustering dynamic point sets in a metric space we propose a model called incremental clustering which is based on a careful analysis of the requirements of the information retrieval application and which should also be useful in other applications the goal is to efficiently maintain clusters of small diameter as new points are inserted we analyze several natural greedy algorithms and demonstrate that they perform poorly we propose new deterministic and randomized incremental clustering algorithms which have a provably good performance and which we believe should also perform well in practice we complement our positive results with lower bounds on the performance of incremental algorithms finally we consider the dual clustering problem where the clusters are of fixed diameter and the goal is to minimize the number of clusters

spectral representation and reduced order modeling of the dynamics of stochastic reaction networks via adaptive data partitioning dynamical analysis tools are well established for deterministic models however for many biochemical phenomena in cells the molecule count is low leading to stochastic behavior that causes deterministic macroscale reaction models to fail the main mathematical framework representing these phenomena is based on jump markov processes that model the underlying stochastic reaction network conventional dynamical analysis tools do not readily generalize to the stochastic setting due to nondifferentiability and absence of explicit state evolution equations we developed a reduced order methodology for dynamical analysis that relies on the karhunen loeve decomposition and polynomial chaos expansions the methodology relies on adaptive data partitioning to obtain an accurate representation of the stochastic process especially in the case of multimodal behavior as a result a mixture model is obtained that represents the reduced order dynamics of the system the schlogl model is used as a prototype bistable process that exhibits time scale separation and leads to multimodality in the reduced order model

large group decision making incorporating decision risk and risk attitude a statistical approach abstract in this study we propose a statistical method incorporating decision risk and risk attitude into large group decision making the decision making groups are divided into subgroups based on their attitudes to risk and the evaluation information for the decision makers within the same subgroup is combined to form the sample dataset next the internal decision risk levels of all subgroups are measured using sample standard deviations and reduced through a feedback mechanism significance testing is used to determine the criterion weights and to measure the external decision risk levels of subgroups the internal and external decision risk levels are then combined to yield the subgroup weights confidence interval is used to transform the sample data into interval numbers which are then aggregated and analyzed to yield the decision making results meanwhile risk attitudes are taken into account throughout the decision making process by various means a case study and comparison analyses along with sensitivity analyses are used to illustrate the feasibility and rationality of the proposed method our experiments suggest that decision risk and risk attitude matter in large group decision making

generalized p center problems complexity results and approximation algorithms abstract in an earlier paper two alternative p center problems where the centers serving costumers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected were shown to be np complete this paper considers a generalized version of these problems in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range we refer to these problems as the set p center problems we establish that the triangle inequality inequality versions of these problems in which the edge weights are assumed to satisfy the triangle inequality are also np complete we also provide a polynomial time approximation algorithm for the two inequality set p center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee unless p np for the special case alternative p center problems which we refer to as the pair p center problems we extend the previous results in several ways for example the results mentioned above for the set p center problems also apply to the pair p center problems furthermore we establish and exploit a correspondence between satisfiability and the dominating set type of problems that naturally arise when considering the decision versions of the pair p center problems

fault tolerant k center problems the basic k center problem is a fundamental facility location problem where we are asked to locate k facilities in a graph and to assign vertices to facilities so as to minimize the maximum distance from a vertex to the facility to which it is assigned this problem is known to be np hard and several optimal approximation algorithms that achieve an approximation factor of 2 have been developed for it we focus our attention on a generalization of this problem where each vertex is required to have a set of k centers close to it in particular we study two different versions of this problem in the first version each vertex is required to have at least centers close to it in the second version each vertex that does not have a center placed on it is required to have at least centers close to it for both these versions we are able to provide polynomial time approximation algorithms that achieve constant approximation factors for any for the first version we give an algorithm that achieves an approximation factor of 3 for any and achieves an approximation factor of 2 for 4 for the second version we provide algorithms with approximation factors of 2 for any the best possible approximation factor for even the basic k center problem is 2 assuming p np in addition we give a polynomial time approximation algorithm for a generalization of the k supplier problem where a subset of at most k supplier nodes must be selected as centers so that every demand node has at least centers close to it for this version our approximation factor is 3 the best possible approximation factor for even the basic k supplier problem is 3 assuming p np

duality of i projections and maximum likelihood estimation for log linear models under cone constraints abstract important order restrictions and general log linear models for multinomial experiments can often be expressed by requiring that the vector composed of the logs of the probabilities fall within a closed convex cone here we exhibit a duality relationship by way of a fenchel duality theorem between these types of problems and projections in i divergence geometry we show that many cone constrained maximum likelihood estimation problems are exactly equivalent to an i projection onto a translation of the negative polar cone this duality relationship permits the concise characterization of cone restricted maximum likelihood estimates and the use of iterative algorithms proposed by csiszar 1975 and dykstra 1985b which are analogous to the iterative proportional fitting procedure computational aspects of maximum likelihood estimates are discussed and a relationship with least squares projections onto isotonic cones is presented we discuss examples in detail to exhibit the usefulness and power

nonparametric entropy estimation for stationary processes and random fields with applications to english text we discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a doeblin type mixing condition the estimators are cesaro averages of longest match lengths and their consistency follows from a generalized ergodic theorem due to maker 1940 we provide examples of their performance on english text and we generalize our results to countable alphabet processes and to random fields

the generalized asymptotic equipartition property necessary and sufficient conditions suppose a string x1 n x1 x2 xn generated by a memoryless source xn n ges 1 with distribution p is to be compressed with distortion no greater than d ges 0 using a memoryless random codebook with distribution q the compression performance is determined by the generalized asymptotic equipartition property aep which states that the probability of finding a d close match between x1 n and any given codeword y1 n is approximately 2 nr p q d where the rate function r p q d can be expressed as an infimum of relative entropies the main purpose here is to remove various restrictive assumptions on the validity of this result that have appeared in the recent literature necessary and sufficient conditions for the generalized aep are provided in the general setting of abstract alphabets and unbounded distortion measures all possible distortion levels d ges 0 are considered the source xn n ges 1 can be stationary and ergodic and the codebook distribution can have memory moreover the behavior of the matching probability is precisely characterized even when the generalized aep is not valid natural characterizations of the rate function r p q d are established under equally general conditions

universal entropy estimation via block sorting in this correspondence we present a new universal entropy estimator for stationary ergodic sources prove almost sure convergence and establish an upper bound on the convergence rate for finite alphabet finite memory sources the algorithm is motivated by data compression using the burrows wheeler block sorting transform bwt by exploiting the property that the bwt output sequence is close to a piecewise stationary memoryless source we can segment the output sequence and estimate probabilities in each segment experimental results show that our algorithm outperforms lempel ziv lz string matching based algorithms

on finite block length quantization distortion we investigate the upper and lower bounds on the quantization distortions for independent and identically distributed continuous sources in the finite block length regime we derive a lower bound on the quantization distortion as well as an upper bound on the quantization distortion based on random quantization codebooks moreover we apply the obtained bounds to the continuous gaussian source for the gaussian source we propose a computationally tractable method to numerically compute the upper and lower bounds for both bounded and unbounded quantization codebooks numerical results show that the gap between the upper and lower bounds is small for the block length of several hundreds and the upper and lower bounds are tighter than those proposed in the existing works

hybrid manifold embedding in this brief we present a novel supervised manifold learning framework dubbed hybrid manifold embedding hyme unlike most of the existing supervised manifold learning algorithms that give linear explicit mapping functions the hyme aims to provide a more general nonlinear explicit mapping function by performing a two layer learning procedure in the first layer a new clustering strategy called geodesic clustering is proposed to divide the original data set into several subsets with minimum nonlinearity in the second layer a supervised dimensionality reduction scheme called locally conjugate discriminant projection is performed on each subset for maximizing the discriminant information and minimizing the dimension redundancy simultaneously in the reduced low dimensional space by integrating these two layers in a unified mapping function a supervised manifold embedding framework is established to describe both global and local manifold structure as well as to preserve the discriminative ability in the learned subspace experiments on various data sets validate the effectiveness of the proposed method

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

a general framework of hierarchical clustering and its applications hierarchical clustering problem is a traditional topic in computer science which aims to discover a consistent hierarchy of clusters with different granularities one of the most important open questions on hierarchical clustering is the identification of the meaningful clustering levels in the hierarchical structure in this paper we answer this question from algorithmic point of view in particular we derive a quantitative analysis on the impact of the low level clustering costs on high level clusters when agglomerative algorithms are run to construct the hierarchy this analysis enables us to find meaningful clustering levels which are independent of the clusters hierarchically beneath it we thus propose a general agglomerative hierarchical clustering framework which automatically constructs meaningful clustering levels this framework is proven to be generally applicable to any k clustering problem in any relaxed metric space in which strict triangle inequality is relaxed within some constant factor to fully utilize the hierarchical clustering framework we conduct some case studies on k median and k means clustering problems in both of which our framework achieves better approximation factor than the state of the art methods we also extend our framework to handle the data stream clustering problem which allows only one scan on the whole data set by incorporating our framework into guha s data stream clustering algorithm the clustering quality is greatly enhanced with only small extra computation cost incurred the extensive experiments show that our proposal is superior to the distance based agglomerative hierarchical clustering and data stream clustering algorithms on a variety of data sets

spreading speeds and linear determinacy of time dependent diffusive cooperative competitive systems abstract the current paper is devoted to the study of spreading speeds and linear determinacy of diffusive cooperative competitive systems of the form u t t x a u t x f t u t x x h where h r n or z n a is a random or nonlocal dispersal operator in the cases h r n and is a discrete dispersal operator in the case h z n f t u is cooperative or competitive in u and is recurrent in t which includes f t u is periodic and almost periodic in t as special cases first the notion of spreading speed intervals for diffusive cooperative systems is introduced via the natural features of spreading speeds and some basic spreading properties are established next some principal lyapunov exponent and principal floquent bundle theory for linear cooperative systems of ordinary differential equations is developed which plays an important role in the study of spreading speeds of diffusive cooperative competitive systems with time dependence and is also of independent interest in terms of the principal lyapunov exponent and principal floquent bundle theory some upper and lower bounds for the spreading speed intervals for diffusive cooperative systems are then established under certain conditions it is proved that a diffusive cooperative system has a single spreading speed and is linearly determinant finally by transforming a diffusive competitive system to a diffusive cooperative system and applying established results for diffusive cooperative systems upper and lower bounds for the spreading speed intervals for diffusive competitive systems are obtained and linearly determinacy for such systems is discussed

optimization on lie manifolds and pattern recognition several pattern recognition problems can be reduced in a natural way to the problem of optimizing a nonlinear function over a lie manifold however optimization on lie manifolds involves in general a large number of nonlinear equality constraints and is hence one of the hardest optimization problems we show that exploiting the special structure of lie manifolds allows one to devise a method for optimizing on lie manifolds in a computationally efficient manner the new method relies on the differential geometry of lie manifolds and the underlying connections between lie groups and their associated lie algebras we describe an application of the new lie group method to the problem of diagnosing malignancy in the cytological extracts of breast tumors the diagnosis method that we present has a mean sensitivity of 98 086 and a predictive index of 0 0602 making it the most accurate and reliable diagnostic method reported thus far

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

online primal dual algorithms for covering and packing we study a wide range of online covering and packing optimization problems in an online covering problem a linear cost function is known in advance but the linear constraints that define the feasible solution space are given one by one in rounds in an online packing problem the profit function as well as the packing constraints are not known in advance in each round additional information i e a new variable about the profit function and the constraints is revealed an online algorithm needs to maintain a feasible solution in each round in addition the solutions generated over the different rounds need to satisfy a monotonicity property we provide general deterministic primal dual algorithms for online fractional covering and packing problems we also provide deterministic algorithms for several integral online covering and packing problems our algorithms are designed via a novel online primal dual technique and are evaluated via competitive analysis

the correntropy mace filter the minimum average correlation energy mace filter is well known for object recognition this paper proposes a nonlinear extension to the mace filter using the recently introduced correntropy function correntropy is a positive definite function that generalizes the concept of correlation by utilizing second and higher order moments of the signal statistics because of its positive definite nature correntropy induces a new reproducing kernel hilbert space rkhs taking advantage of the linear structure of the rkhs it is possible to formulate the mace filter equations in the rkhs induced by correntropy and obtained an approximate solution due to the nonlinear relation between the feature space and the input space the correntropy mace cmace can potentially improve upon the mace performance while preserving the shift invariant property additional computation for all shifts will be required in the cmace to alleviate the computation complexity of the solution this paper also presents the fast cmace using the fast gauss transform fgt we apply the cmace filter to the mstar public release synthetic aperture radar sar data set as well as pie database of human faces and show that the proposed method exhibits better distortion tolerance and outperforms the linear mace in both generalization and rejection abilities

spectral multidimensional scaling an important tool in information analysis is dimensionality reduction there are various approaches for large data simplification by scaling its dimensions down that play a significant role in recognition and classification tasks the efficiency of dimension reduction tools is measured in terms of memory and computational complexity which are usually a function of the number of the given data points sparse local operators that involve substantially less than quadratic complexity at one end and faithful multiscale models with quadratic cost at the other end make the design of dimension reduction procedure a delicate balance between modeling accuracy and efficiency here we combine the benefits of both and propose a low dimensional multiscale modeling of the data at a modest computational cost the idea is to project the classical multidimensional scaling problem into the data spectral domain extracted from its laplace beltrami operator there embedding into a small dimensional euclidean space is accomplished while optimizing for a small number of coefficients we provide a theoretical support and demonstrate that working in the natural eigenspace of the data one could reduce the process complexity while maintaining the model fidelity as examples we efficiently canonize nonrigid shapes by embedding their intrinsic metric into a method often used for matching and classifying almost isometric articulated objects finally we demonstrate the method by exposing the style in which handwritten digits appear in a large collection of images we also visualize clustering of digits by treating images as feature points that we map to a plane

a nonparametric statistical approach to clustering via mode identification a new clustering approach based on mode identification is developed by applying new optimization techniques to a nonparametric density estimator a cluster is formed by those sample points that ascend to the same local maximum mode of the density function the path from a point to its associated mode is efficiently solved by an em style algorithm namely the modal em mem this method is then extended for hierarchical clustering by recursively locating modes of kernel density estimators with increasing bandwidths without model fitting the mode based clustering yields a density description for every cluster a major advantage of mixture model based clustering moreover it ensures that every cluster corresponds to a bump of the density the issue of diagnosing clustering results is also investigated specifically a pairwise separability measure for clusters is defined using the ridgeline between the density bumps of two clusters the ridgeline is solved for by the ridgeline em rem algorithm an extension of mem based upon this new measure a cluster merging procedure is created to enforce strong separation experiments on simulated and real data demonstrate that the mode based clustering approach tends to combine the strengths of linkage and mixture model based clustering in addition the approach is robust in high dimensions and when clusters deviate substantially from gaussian distributions both of these cases pose difficulty for parametric mixture modeling a c package on the new algorithms is developed for public access at http www stat psu edu jiali hmac

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data

on kissing numbers and spherical codes in high dimensions abstract we prove a lower bound of d 3 2 2 3 d on the kissing number in dimension d this improves the classical lower bound of chabauty shannon and wyner by a linear factor in the dimension we obtain a similar linear factor improvement to the best known lower bound on the maximal size of a spherical code of acute angle in high dimensions

a framework for evaluating approximation methods for gaussian process regression gaussian process gp predictors are an important component of many bayesian approaches to machine learning however even a straightforward implementation of gaussian process regression gpr requires o n2 space and o n3 time for a data set of n examples several approximation methods have been proposed but there is a lack of understanding of the relative merits of the different approximations and in what situations they are most useful we recommend assessing the quality of the predictions obtained as a function of the compute time taken and comparing to standard baselines e g subset of data and fitc we empirically investigate four different approximation algorithms on four different prediction problems and make our code available to encourage future comparisons