on the stable set polytope of a series parallel graph we give a short proof of chvatal s conjecture that the nontrivial facets of the stable set polytope of a series parallel graph all come from edges and odd holes

the steiner tree polytope and related polyhedra we consider the vertex weighted version of the undirected steiner tree problem in this problem a cost is incurred both for the vertices and the edges present in the steiner tree we completely describe the associated polytope by linear inequalities when the underlying graph is series parallel for general graphs this formulation can be interpreted as a partial extended formulation for the steiner tree problem by projecting this formulation we obtain some very large classes of facet defining valid inequalities for the steiner tree polytope

facets with fixed defect of the stable set polytope the stable set polytope of a graph is the convex hull of the 0 1 vectors corresponding to stable sets of vertices to any nontrivial facet v v a v x v b of this polytope we associate an integer called the defect of the facet by v v a v 2b we show that for any fixed there is a finite collection of graphs called basis such that any graph with a facet of defect contains a subgraph which can be obtained from one of the graphs in the basis by a simple subdivision operation

gear composition of stable set polytopes and g perfection graphs obtained by applying the gear composition to a given graph h are called geared graphs we show how a linear description of the stable set polytope stab g of a geared graph g can be obtained by extending the linear inequalities defining stab h and stab he where he is the graph obtained from h by subdividing the edge e we also introduce the class of g perfect graphs i e graphs whose stable set polytope is described by nonnegativity inequalities rank inequalities lifted 5 wheel inequalities and some special inequalities called geared inequalities and g lifted inequalities we prove that graphs obtained by repeated applications of the gear composition to a given graph h are g perfect provided that any graph obtained from h by subdividing a subset of its simplicial edges is g perfect in particular we show that a large subclass of claw free graphs is g perfect

feature discovery in non metric pairwise data pairwise proximity data given as similarity or dissimilarity matrix can violate metricity this occurs either due to noise fallible estimates or due to intrinsic non metric features such as they arise from human judgments so far the problem of non metric pairwise data has been tackled by essentially omitting the negative eigenvalues or shifting the spectrum of the associated pseudo covariance matrix for a subsequent embedding however little attention has been paid to the negative part of the spectrum itself in particular no answer was given to whether the directions associated to the negative eigenvalues would at all code variance other than noise related we show by a simple exploratory analysis that the negative eigenvalues can code for relevant structure in the data thus leading to the discovery of new features which were lost by conventional data analysis techniques the information hidden in the negative eigenvalue part of the spectrum is illustrated and discussed for three data sets namely usps handwritten digits text mining and data from cognitive psychology

stable set polytopes for a class of circulant graphs we study the stable set polytope p gn for the graph gn with n nodes and edges i j with j in i 1 i 2 i 1 ldots n and where nodes n 1 and 1 resp n 2 and 2 are identified this graph coincides with the antiweb bar w n 3 a minimal linear system defining p gn is determined the system consists of certain rank inequalities with some number theoretic flavor a characterization of the vertices of a natural relaxation of p gn is also given

on claw free t perfect graphs a graph is called t perfect if its stable set polytope is defined by non negativity edge and odd cycle inequalities we characterise the class of all claw free t perfect graphs by forbidden t minors and show that they are 3 colourable moreover we determine the chromatic number of claw free h perfect graphs and give a polynomial time algorithm to compute an optimal colouring

shintani cocycles and the order of vanishing of p adic hecke l series at s 0 let chi be a hecke character of finite order of a totally real number field f by using hill s shintani cocycle we provide a cohomological construction of the p adic l series l p chi s associated to chi this is used to show that l p chi s has a trivial zero at s 0 of order at least equal to the number of places of f above p where the local component of chi is trivial

classification with nonmetric distances image retrieval and class representation a key problem in appearance based vision is understanding how to use a set of labeled images to classify new images systems that model human performance or that use robust image matching methods often use nonmetric similarity judgments but when the triangle inequality is not obeyed most pattern recognition techniques are not applicable exemplar based nearest neighbor methods can be applied to a wide class of nonmetric similarity functions the key issue however is to find methods for choosing good representatives of a class that accurately characterize it we show that existing condensing techniques are ill suited to deal with nonmetric dataspaces we develop techniques for solving this problem emphasizing two points first we show that the distance between images is not a good measure of how well one image can represent another in nonmetric spaces instead we use the vector correlation between the distances from each image to other previously seen images second we show that in nonmetric spaces boundary points are less significant for capturing the structure of a class than in euclidean spaces we suggest that atypical points may be more important in describing classes we demonstrate the importance of these ideas to learning that generalizes from experience by improving performance we also suggest ways of applying parametric techniques to supervised learning problems that involve a specific nonmetric distance functions showing how to generalize the idea of linear discriminant functions in a way that may be more useful in nonmetric spaces

kernel based nonlinear blind source separation we propose ktdsep a kernel based algorithm for nonlinear blind source separation bss it combines complementary research fields kernel feature spaces and bss using temporal information this yields an efficient algorithm for nonlinear bss with invertible nonlinearity key assumptions are that the kernel feature space is chosen rich enough to approximate the nonlinearity and that signals of interest contain temporal information both assumptions are fulfilled for a wide set of real world applications the algorithm works as follows first the data are implicitly mapped to a high possibly infinite dimensional kernel feature space in practice however the data form a smaller submanifold in feature space even smaller than the number of training data points a fact that has already been used by for example reduced set techniques for support vector machines we propose to adapt to this effective dimension as a preprocessing step and to construct an orthonormal basis of this submanifold the latter dimension reduction step is essential for making the subsequent application of bss methods computationally and numerically tractable in the reduced space we use a bss algorithm that is based on second order temporal decorrelation finally we propose a selection procedure to obtain the original sources from the extracted nonlinear components automatically experiments demonstrate the excellent performance and efficiency of our ktdsep algorithm for several problems of nonlinear bss and for more than two sources

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

svms for histogram based image classification traditional classification approaches generalize poorly on image classification tasks because of the high dimensionality of the feature space this paper shows that support vector machines svm can generalize well on difficult image classification problems where the only features are high dimensional histograms heavy tailed rbf kernels of the form k x y e p i x i y i with a 1 and b 2 are evaluated on the classification of images extracted from the corel stock photo collection and shown to far outperform traditional polynomial or gaussian rbf kernels moreover we observed that a simple remapping of the input xi x a i improves the performance of linear svms to such an extend that it makes them for this problem a valid alternative to rbf kernels keywords support vector machines radial basis functions image histogram image classification corel

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

feature space interpretation of svms with indefinite kernels kernel methods are becoming increasingly popular for various kinds of machine learning tasks the most famous being the support vector machine svm for classification the svm is well understood when using conditionally positive definite cpd kernel functions however in practice non cpd kernels arise and demand application in svm the procedure of plugging these indefinite kernels in svm often yields good empirical classification results however they are hard to interpret due to missing geometrical and theoretical understanding in this paper we provide a step toward the comprehension of svm classifiers in these situations we give a geometric interpretation of svm with indefinite kernel functions we show that such svm are optimal hyperplane classifiers not by margin maximization but by minimization of distances between convex hulls in pseudo euclidean spaces by this we obtain a sound framework and motivation for indefinite svm this interpretation is the basis for further theoretical analysis e g investigating uniqueness and for the derivation of practical guidelines like characterizing the suitability of indefinite svm

suffix arrays a new method for on line string searches a new and conceptually simple data structure called a suffix array for on line string searches is introduced in this paper constructing and querying suffix arrays is reduced to a sort and search paradigm that employs novel algorithms the main advantage of suffix arrays over suffix trees is that in practice they use three to five times less space from a complexity standpoint suffix arrays permit on line string searches of the type is w a substring of a to be answered in time o p log n where p is the length of w and n is the length of a which is competitive with and in some cases slightly better than suffix trees the only drawback is that in those instances where the underlying alphabet is finite and small suffix trees can be constructed in o n time in the worst case versus o n log n time for suffix arrays however an augmented algorithm is given that regardless of the alphabet size constructs suffix arrays in o n expected time albeit with lesser space efficiency it is

optimal cluster preserving embedding of nonmetric proximity data for several major applications of data analysis objects are often not represented as feature vectors in a vector space but rather by a matrix gathering pairwise proximities such pairwise data often violates metricity and therefore cannot be naturally embedded in a vector space concerning the problem of unsupervised structure detection or clustering in this paper a new embedding method for pairwise data into euclidean vector spaces is introduced we show that all clustering methods which are invariant under additive shifts of the pairwise proximities can be reformulated as grouping problems in euclidian spaces the most prominent property of this constant shift embedding framework is the complete preservation of the cluster structure in the embedding space restating pairwise clustering problems in vector spaces has several important consequences such as the statistical description of the clusters by way of cluster prototypes the generic extension of the grouping procedure to a discriminative prediction rule and the applicability of standard preprocessing methods like denoising or dimensionality reduction

wheel inequalities for stable set polytopes we introduce new classes of valid inequalities called wheel inequalities for the stable set polytopepg of a graphg each wheel configuration gives rise to two such inequalities the simplest wheel configuration is an odd subdivisionw of a wheel and for these we give necessary and sufficient conditions for the wheel inequality to be facet inducing forpw generalizations arise by allowing subdivision paths to intersect and by replacing the hub of the wheel by a clique the separation problem for these inequalities can be solved in polynomial time

on the information and representation of non euclidean pairwise data two common data representations are mostly used in intelligent data analysis namely the vectorial and the pairwise representation pairwise data which satisfy the restrictive conditions of euclidean spaces can be faithfully translated into a euclidean vectorial representation by embedding non metric pairwise data with violations of symmetry reflexivity or triangle inequality pose a substantial conceptual problem for pattern recognition since the amount of predictive structural information beyond what can be measured by embeddings is unclear we show by systematic modeling of non euclidean pairwise data that there exists metric violations which can carry valuable problem specific information furthermore euclidean and non metric data can be unified on the level of structural information contained in the data stable component analysis selects linear subspaces which are particularly insensitive to data fluctuations experimental results from different domains support our pattern recognition strategy

support vector machines for histogram based image classification traditional classification approaches generalize poorly on image classification tasks because of the high dimensionality of the feature space this paper shows that support vector machines svm can generalize well on difficult image classification problems where the only features are high dimensional histograms heavy tailed rbf kernels of the form k x y e sup spl rho spl sigma sub i sup xia yia b with a spl les 1 and b spl les 2 are evaluated on the classification of images extracted from the corel stock photo collection and shown to far outperform traditional polynomial or gaussian radial basis function rbf kernels moreover we observed that a simple remapping of the input x sub i spl rarr x sub i sup a improves the performance of linear svm to such an extend that it makes them for this problem a valid alternative to rbf kernels

fractional and integral colourings letg v e be an undirected graph andc any vector in v g denote by gc resp gc the chromatic number resp fractional chromatic number ofg with respect toc we study graphs for which gc gc 1 we show that for the class of graphs satisfying gc gc a class generalizing perfect graphs an analogue of the duplication lemma does not hold we also describe a 2 vertex cut decomposition procedure related to the integer decomposition property we use this procedure to show that gc gc for series parallel graphs and gc gc 1 for graphs that do not have the 4 wheel as a minor

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

model complexity control for regression using vc generalization bounds it is well known that for a given sample size there exists a model of optimal complexity corresponding to the smallest prediction generalization error hence any method for learning from finite samples needs to have some provisions for complexity control existing implementations of complexity control include penalization or regularization weight decay in neural networks and various greedy procedures aka constructive growing or pruning methods there are numerous proposals for determining optimal model complexity aka model selection based on various asymptotic analytic estimates of the prediction risk and on resampling approaches nonasymptotic bounds on the prediction risk based on vapnik chervonenkis vc theory have been proposed by vapnik this paper describes application of vc bounds to regression problems with the usual squared loss an empirical study is performed for settings where the vc bounds can be rigorously applied i e linear models and penalized linear models where the vc dimension can be accurately estimated and the empirical risk can be reliably minimized empirical comparisons between model selection using vc bounds and classical methods are performed for various noise levels sample size target functions and types of approximating functions our results demonstrate the advantages of vc based complexity control with finite samples

linear time computation of similarity measures for sequential data efficient and expressive comparison of sequences is an essential procedure for learning with sequential data in this article we propose a generic framework for computation of similarity measures for sequences covering various kernel distance and non metric similarity functions the basis for comparison is embedding of sequences using a formal language such as a set of natural words k grams or all contiguous subsequences as realizations of the framework we provide linear time algorithms of different complexity and capabilities using sorted arrays tries and suffix trees as underlying data structures experiments on data sets from bioinformatics text processing and computer security illustrate the efficiency of the proposed algorithms enabling peak performances of up to 106 pairwise comparisons per second the utility of distances and non metric similarity measures for sequences as alternatives to string kernels is demonstrated in applications of text categorization network intrusion detection and transcription site recognition in dna