approximate labeling via graph cuts based on linear programming a new framework is presented for both understanding and developing graph cut based combinatorial algorithms suitable for the approximate optimization of a very wide class of markov random fields mrfs that are frequently encountered in computer vision the proposed framework utilizes tools from the duality theory of linear programming in order to provide an alternative and more general view of state of the art techniques like the alpha expansion algorithm which is included merely as a special case moreover contrary to alpha expansion the derived algorithms generate solutions with guaranteed optimality properties for a much wider class of problems for example even for mrfs with nonmetric potentials in addition they are capable of providing per instance suboptimality bounds in all occasions including discrete mrfs with an arbitrary potential function these bounds prove to be very tight in practice that is very close to 1 which means that the resulting solutions are almost optimal our algorithms effectiveness is demonstrated by presenting experimental results on a variety of low level vision tasks such as stereo matching image restoration image completion and optical flow estimation as well as on synthetic problems

motivic bivariant characteristic classes let k0 v x k0 v x be the relative grothendieck group of varieties over x obj v x obj v with v vk qp resp v vcan the category of quasi projective algebraic resp compact complex analytic varieties over a base field k then we constructed the motivic hirzebruch class transformation ty k0 v x h x q y ty k0 v x h x q y in the algebraic context for k of characteristic zero with h x ch x h x ch x resp in the complex algebraic or analytic context with h x h2 bm x it unifies the well known three characteristic class transformations of singular varieties macpherson s chern class baum fulton macpherson s todd class and the l class of goresky macpherson and cappell shaneson in this paper we construct a bivariant relative grothendieck group k0 v x y k0 v x y for v vk qp resp vcan so that k0 v x pt k0 v x k0 v x pt k0 v x in the algebraic context with k of characteristic zero resp complex analytic context we also construct in the algebraic context in any characteristic two grothendieck transformations mcy ymot k0 vqp x y kalg x y z y and ty k0 vqp x y h x y q y ty k0 vqp x y h x y q y with kalg f kalg f the bivariant algebraic k theory of f perfect complexes and hh the bivariant operational chow groups or the even degree bivariant homology in the case k ck c evaluating at y 0y 0 we get a motivic lift t0t0 of fulton macpherson s bivariant riemann roch transformation kalg h q kalg h q the covariant transformations mcy k0 vqp x pt g0 x z y mcy k0 vqp x pt g0 x z y and ty k0 vqp x pt h x q y ty k0 vqp x pt h x q y agree for k of characteristic zero with our motivic chern and hirzebruch class transformations defined on k0 vqp x k0 vqp x finally evaluating at y 1y 1 for k of characteristic zero we get a motivic lift t 1t 1 of ernstrom yokura s bivariant chern class transformation f ch

expander flows geometric embeddings and graph partitioning we give a o slog n approximation algorithm for the sparsest cut edge expansion balanced separator and graph conductance problems this improves the o log n approximation of leighton and rao 1988 we use a well known semidefinite relaxation with triangle inequality constraints central to our analysis is a geometric theorem about projections of point sets in rd whose proof makes essential use of a phenomenon called measure concentration we also describe an interesting and natural approximate certificate for a graph s expansion which involves embedding an n node expander in it with appropriate dilation and congestion we call this an expander flow

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

multiple subjects connectivity based parcellation using hierarchical dirichlet process mixture models we propose a hierarchical infinite mixture model approach to address two issues in connectivity based parcellations i choosing the number of clusters and ii combining data from different subjects in a bayesian setting we model voxel wise anatomical connectivity profiles as an infinite mixture of multivariate gaussian distributions with a dirichlet process prior on the cluster parameters this type of prior allows us to conveniently model the number of clusters and estimate its posterior distribution directly from the data an important benefit of using bayesian modelling is the extension to multiple subjects clustering via a hierarchical mixture of dirichlet processes data from different subjects are used to infer on class parameters and the number of classes at individual and group level such a method accounts for inter subject variability while still benefiting from combining different subjects data to yield more robust estimates of the individual clusterings

estimating the gap of finite metric spaces of strict p negative type abstract let x d be a finite metric space this paper first discusses the spectrum of the p distance matrix of a finite metric space of p negative type and then gives upper and lower bounds for the so called gap x p of a finite metric space of strict p negative type furthermore estimations for x p under a certain gluing construction for finite metric spaces are given and finally be applied to finite ultrametric spaces

non simple semistable vector bundles over a curve let x be a non singular algebraic curve over c of genus g 2 let s 3 0 be the set of indecomposable semistable vector bundles over x of rank 3 and degree 0 in 21 we prove that non simple vector bundles in s 3 0 must have algebra of endomorphisms isomorphic to c t t2 c t t 3 o r c r s r s 2 in this paper we prove that if we fix the algebra of endomorphisms and make a further subdivision in the case c t t2 then there exists a moduli space mi for such vector bundles in s 3 0 for some i m i is a fine moduli space i e there exists a universal family parametrized by mg see theorems 6 and 9 mainly we use the universal extensions constructed in 3 and we prove that under a suitable equivalence relation these extensions define the required moduli spaces the equivalence relations are defined by considering the c c x c c x c and the gl 2 c action in sect 1 we recall from 2 the conditions for a vector bundle to have a particular algebra of endomorphisms in the following sections we construct the moduli spaces my

silhouettes a graphical aid to the interpretation and validation of cluster analysis a new graphical display is proposed for partitioning techniques each cluster is represented by a so called silhouette which is based on the comparison of its tightness and separation this silhouette shows which objects lie well within their cluster and which ones are merely somewhere in between clusters the entire clustering is displayed by combining the silhouettes into a single plot allowing an appreciation of the relative quality of the clusters and an overview of the data configuration the average silhouette width provides an evaluation of clustering validity and might be used to select an appropriate number of clusters

trees and ultrametric spaces a categorical equivalence abstract there is a well known correspondence between infinite trees and ultrametric spaces that comes from considering the end space of the tree the correspondence is interpreted here as an equivalence between two categories one of which encodes the geometry of trees at infinity and the other encodes the micro geometry of complete ultrametric spaces

residue formulae vector partition functions and lattice points in rational polytopes we obtain residue formulae for certain functions of several vari ables as an application we obtain closed formulae for vector partition func tions and for their continuous analogs they imply an euler maclaurin sum mation formula for vector partition functions and for rational convex poly topes as well we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope institut fourier b p 74 38402 saint martin d heres cedex france e mail address mbrion fourier ujf grenoble fr ecole normale superieure 45 rue d ulm 75005 paris cedex 05 france e mail address vergne dmi ens fr

stability based validation of clustering solutions data clustering describes a set of frequently employed techniques in exploratory data analysis to extract natural group structure in data such groupings need to be validated to separate the signal in the data from spurious structure in this context finding an appropriate number of clusters is a particularly important model selection question we introduce a measure of cluster stability to assess the validity of a cluster model this stability measure quantifies the reproducibility of clustering solutions on a second sample and it can be interpreted as a classification risk with regard to class labels produced by a clustering algorithm the preferred number of clusters is determined by minimizing this classification risk as a function of the number of clusters convincing results are achieved on simulated as well as gene expression data sets comparisons to other methods demonstrate the competitive performance of our method and its suitability as a general validation tool for clustering solutions in real world problems

enhanced negative type for finite metric trees abstract a finite metric tree is a finite connected graph that has no cycles endowed with an edge weighted path metric finite metric trees are known to have strict 1 negative type in this paper we introduce a new family of inequalities 1 that encode the best possible quantification of the strictness of the non trivial 1 negative type inequalities for finite metric trees these inequalities are sufficiently strong to imply that any given finite metric tree t d must have strict p negative type for all p in an open interval 1 1 where 0 may be chosen so as to depend only upon the unordered distribution of edge weights that determine the path metric d on t in particular if the edges of the tree are not weighted then it follows that depends only upon the number of vertices in the tree we also give an example of an infinite metric tree that has strict 1 negative type but does not have p negative type for any p 1 this shows that the maximal p negative type of a metric space can be strict

on the gap of finite metric spaces of p negative type let x d be a metric space of p negative type recently i doust and a weston introduced a quantification of the p negative type property the so called gap of x this paper gives some formulas for the gap of a finite metric space of strict p negative type and applies them to evaluate for some concrete finite metric spaces

polygonal equalities and virtual degeneracy in lp spaces abstract suppose 0 p 2 and that is a measure space for which l p is at least two dimensional the central results of this paper provide a complete description of the subsets of l p that have strict p negative type in order to do this we study non trivial p polygonal equalities in l p these are equalities that can after appropriate rearrangement and simplification be expressed in the form j i 1 n j i z j z i p p 0 where z 1 z n is a subset of l p and 1 n are non zero real numbers that sum to zero we provide a complete classification of the non trivial p polygonal equalities in l p the cases p 2 and p 2 are substantially different and are treated separately the case p 1 generalizes an elegant result of elsner han koltracht neumann and zippin another reason for studying non trivial p polygonal equalities in l p is due to the fact that they preclude the existence of certain types of isometry for example our techniques show that if x d is a metric space that has strict q negative type for some q p then 1 x d is not isometric to any linear subspace w of l p that contains a pair of disjointly supported non zero vectors and 2 x d is not isometric to any subset of l p that has non empty interior furthermore in the case p 2 it also follows that x d is not isometric to any affinely dependent subset of l 2 more generally we show that if y is a metric space whose generalized roundness is finite and if x d is a metric space that has strict q negative type for some q then x d is not isometric to any metric subspace of y that admits a non trivial p 1 polygonal equality for some p 1 q it is notable in all of these statements that the metric space x d can for instance be any ultrametric space as a result we obtain new insights into sophisticated embedding theorems of lemin and shkarin we conclude the paper by constructing some pathological infinite dimensional linear subspaces of l p that do not have strict p negative type

functional density and edge maps characterizing functional architecture in individuals and improving cross subject registration population level inferences and individual level analyses are two important aspects in functional magnetic resonance imaging fmri studies extracting reliable and informative features from fmri data that capture biologically meaningful inter subject variation is critical for aligning and comparing functional networks across subjects and connecting the properties of functional brain organization with variations in behavior cognition and genetics in this study we derive two new measures which we term functional density map and edge map and demonstrate their usefulness in characterizing the function of individual brains specifically using data from the human connectome project hcp we show that 1 both functional maps capture intrinsic properties of the functional connectivity pattern in individuals while exhibiting large variation across subjects 2 functional maps derived from either resting state or task evoked fmri can be used to accurately identify subjects from a population and 3 cross subject alignment using these functional maps considerably reduces functional variation and improves functional correspondence across subjects over state of the art multimodal registration algorithms our results suggest that the proposed functional density and edge maps are promising features in characterizing the functional architecture in individuals and provide an alternative way to explore the functional variation across subjects

characterization stability and convergence of hierarchical clustering methods we study hierarchical clustering schemes under an axiomatic view we show that within this framework one can prove a theorem analogous to one of kleinberg 2002 in which one obtains an existence and uniqueness theorem instead of a non existence result we explore further properties of this unique scheme stability and convergence are established we represent dendrograms as ultrametric spaces and use tools from metric geometry namely the gromov hausdorff distance to quantify the degree to which perturbations in the input metric space affect the result of hierarchical methods

numerical characterization of the kahler cone of a compact kahler manifold the goal of this work is to give a precise numerical description of the kahler cone of a compact kahler manifold our main result states that the kahler cone depends only on the intersection form of the cohomology ring the hodge structure and the homology classes of analytic cycles if x is a compact kahler manifold the kahler cone k of x is one of the connected components of the set p of real 1 1 cohomology classes which are numerically positive on analytic cycles i e y p 0 for every irreducible analytic set y in x p dim y this result is new even in the case of projective manifolds where it can be seen as a generalization of the well known nakai moishezon criterion and it also extends previous results by campana peternell and eyssidieux the principal technical step is to show that every nef class which has positive highest self intersection numberx n 0 contains a kahler current this is done by using the calabi yau theorem and a mass concentration technique for monge ampere equations the main result admits a number of variants and corollaries including a description of the cone of numerically effective 1 1 classes and their dual cone another important consequence is the fact that for an arbitrary deformation x s of compact kahler manifolds the kahler cone of a very general fibre xt is independent of t i e invariant by parallel transport under the 1 1 component of the gauss manin connection

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

on the generalized roundness of finite metric spaces in this paper we show that the generalized roundness of a finite metric space can be bounded from below by a strictly positive constant that only depends upon the number of points in the space

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

asymptotic negative type properties of finite ultrametric spaces abstract negative type inequalities arise in the study of embedding properties of metric spaces but they often reduce to intractable combinatorial problems in this paper we study more quantitative versions of these inequalities involving the so called p negative type gap in particular we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining let x d be a given finite ultrametric space with minimum non zero distance then the p negative type gap x p of x d is positive for all p 0 in this paper we compute the value of the limit x lim p x p p it turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula this formula allows us to characterize when the ratio x p p is a constant independent of p the determination of x also leads to new asymptotically sharp families of enhanced p negative type inequalities for x d indeed suppose that g 0 x then for all sufficiently large p the inequality g p 2 k 1 n k 2 j i 1 n d z j z i p j i 0 holds for each finite subset z 1 z n x and each scalar n tuple 1 n r n that satisfies 1 n 0 notably these results do not extend to general finite metric spaces

estimates for hilbertian koszul homology abstract the objective of this paper is to give new kind of estimates for hilbertian koszul homology inspired by commutative algebra in multivariable fredholm theory

analytic cycles bott chern forms and singular sets for the yang mills flow on kahler manifolds abstract it is shown that the singular set for the yang mills flow on unstable holomorphic vector bundles over compact kahler manifolds is completely determined by the harder narasimhan seshadri filtration of the initial holomorphic bundle we assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves we derive a singular bott chern formula relating the second chern form of a smooth metric on the bundle to the chern current of an admissible metric on the associated graded sheaf this is used to show that the multiplicities of the top dimensional bubbling locus defined via the yang mills density agree with the corresponding multiplicities for the harder narasimhan seshadri filtration the set theoretic equality of singular sets is a consequence

the fredholm index of a pair of commuting operators ii this paper concerns fredholm theory in several variables and its applications to hilbert spaces of analytic functions one feature is the introduction of ideas from commutative algebra to operator theory specifically we introduce a method to calculate the fredholm index of a pair of commuting operators to achieve this we define and study the hilbert space analogs of samuel multiplicities in commutative algebra then the theory is applied to the symmetric fock space in particular our results imply a satisfactory answer to arveson s program on developing a fredholm theory for pure d contractions when d 2 including both the fredholmness problem and the calculation of indices we also show that arveson s curvature invariant is in fact always equal to the samuel multiplicity for an arbitrary pure d contraction with finite defect rank it follows that the curvature is a similarity invariant

polar and coisotropic actions on kahler manifolds the main result of the paper is that a polar action on a compact irreducible homogeneous kahler manifold is coisotropic this is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics

hyperpolygon spaces and moduli spaces of parabolic higgs bundles abstract given an n tuple of positive real numbers we consider the hyperpolygon space x the hyperkahler quotient analogue to the kahler moduli space of polygons in r 3 we prove the existence of an isomorphism between hyperpolygon spaces and moduli spaces of stable rank 2 holomorphically trivial parabolic higgs bundles over c p 1 with fixed determinant and trace free higgs field this isomorphism allows us to prove that hyperpolygon spaces x undergo an elementary transformation in the sense of mukai as crosses a wall in the space of its admissible values we describe the changes in the core of x as a result of this transformation as well as the changes in the nilpotent cone of the corresponding moduli spaces of parabolic higgs bundles moreover we study the intersection rings of the core components of x in particular we find generators of these rings prove a recursion relation in n for their intersection numbers and use it to obtain explicit formulas for the computation of these numbers using our isomorphism we obtain similar formulas for each connected component of the nilpotent cone of the corresponding moduli spaces of parabolic higgs bundles thus determining their intersection rings as a final application of our isomorphism we describe the cohomology ring structure of these moduli spaces of parabolic higgs bundles and of the components of their nilpotent cone

grasp geodesic graph based segmentation with shape priors for the functional parcellation of the cortex abstract resting state functional mri is a powerful technique for mapping the functional organization of the human brain however for many types of connectivity analysis high resolution voxelwise analyses are computationally infeasible and dimensionality reduction is typically used to limit the number of network nodes most commonly network nodes are defined using standard anatomic atlases that do not align well with functional neuroanatomy or regions of interest covering a small portion of the cortex data driven parcellation methods seek to overcome such limitations but existing approaches are highly dependent on initialization procedures and produce spatially fragmented parcels or overly isotropic parcels that are unlikely to be biologically grounded in this paper we propose a novel graph based parcellation method that relies on a discrete markov random field framework the spatial connectedness of the parcels is explicitly enforced by shape priors the shape of the parcels is adapted to underlying data through the use of functional geodesic distances our method is initialization free and rapidly segments the cortex in a single optimization the performance of the method was assessed using a large developmental cohort of more than 850 subjects compared to two prevalent parcellation methods our approach provides superior reproducibility for a similar data fit furthermore compared to other methods it avoids incoherent parcels finally the method s utility is demonstrated through its ability to detect strong brain developmental effects that are only weakly observed using other methods

hodge genera of algebraic varieties i the aim of this paper is to study the behavior of hodge theoretic intersection homology genera and their associated characteristic classes under proper morphisms of complex algebraic varieties we obtain formulae that relate parametrized families of global invariants of a complex algebraic variety x to such invariants of singularities of proper algebraic maps defined on x such formulae severely constrain both topologically and analytically the singularities of complex maps even between smooth varieties similar results were announced by the first and third author in 13 32 2007 wiley periodicals inc

maximal degree subsheaves of torsion free sheaves on singular projective curves fix integers r k g with r k 0 and g 2 let x be an integral projective curve with g pa x and e a rank r torsion free sheaf on x which is a flat limit of a family of locally free sheaves on x here we prove the existence of a rank k subsheaf a of e such that r deg a k deg e k r k g we show that for every g 9 there is an integral projective curve x x not gorenstein and a rank 2 torsion free sheaf e on x with no rank 1 subsheaf a with 2 deg a deg e g we show the existence of torsion free sheaves on non gorenstein projective curves with other pathological properties

finite metric spaces of strictly negative type abstract we prove that if a finite metric space is of strictly negative type then its transfinite diameter is uniquely realized by the infinite extender load vector finite metric spaces that have this property include all spaces on two three or four points all trees and all finite subspaces of euclidean spaces we prove that if the distance matrix is both hypermetric and regular then it is of strictly negative type we show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points in connection with an open problem raised by kelly we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular and hence of strictly negative type

universal families of extensions if x denotes a projective variety over a field k and jr and y coherent sheaves on x the vector space exti sr y parametrizes the extensions of f by v on x and there is a universal extension of p t by p y on xx exti y 5 such that for every k rational point u of exti ir y its restriction to x x u is just the extension represented by u moreover p p extk st v parametrizes the classes of nonsplitting extensions of 9 by y on x module k and there is a universal family of extensions of pt f by pf5 p 1 on x x p over p such that for every k rational point p of p its restriction to x x p represents just the class of extensions given by p it is the purpose of this paper not only to prove these two statements but to generalize them to the relative case of a flat projective morphism f x y of noetherian schemes they turn out to be a consequence of grothendieck s universal properties of vector bundles and projective bundles cp 2 section i 91 special cases of these results appeared several times in the literature to the best of my knowledge first in 5 lemma 3 11 and second in 6 lemma 2 41 the technical tool is the base change theory for relative ext sheaves which is outlined in section 1 in section 2 some relations between relative ext sheaves and families of extensions are given section 3 contains the construction of the universal family of extensions and section 4 the construction of the universal family of classes of nonsplitting extensions concerning the notation f x y means a flat projective morphism of noetherian schemes and r and ce coherent ox modules flat over y the proofs of sections 2 to 4 are more generally correct if only there is a base change theory for relative ext sheaves for example if one uses the base change results of 1 valid for arbitrary proper flat morphisms of complex spaces one gets the corresponding results in this case too 101 oozl 8693183 3 00