metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

metric differentiation monotonicity and maps to l1 this is one of a series of papers on lipschitz maps from metric spaces to l1 here we present the details of results which were announced in cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 sect 1 8 a new approach to the infinitesimal structure of lipschitz maps into l1 and as a first application an alternative proof of the main theorem of cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 that the heisenberg group does not admit a bi lipschitz embedding in l1 the proof uses the metric differentiation theorem of pauls commun anal geom 9 5 951 982 2001 and the cut metric description in cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 to reduce the nonembedding argument to a classification of monotone subsets of the heisenberg group a quantitative version of this classification argument is used in our forthcoming joint paper with assaf naor cheeger et al in arxiv 0910 2026 2009

the unique games conjecture integrality gap for cut problems and embeddability of negative type metrics into l 1 in this article we disprove a conjecture of goemans and linial namely that every negative type metric embeds into e1 with constant distortion we show that for an arbitrarily small constant 0 for all large enough n there is an n point negative type metric which requires distortion at least log log n 1 6 to embed into e1 surprisingly our construction is inspired by the unique games conjecture ugc establishing a previously unsuspected connection between probabilistically checkable proof systems pcps and the theory of metric embeddings we first prove that the ugc implies a super constant hardness result for the nonuniform s parsest c ut problem though this hardness result relies on the ugc we demonstrate nevertheless that the corresponding pcp reduction can be used to construct an integrality gap instance for s parsest c ut towards this we first construct an integrality gap instance for a natural sdp relaxation of u nique g ames then we simulate the pcp reduction and translate the integrality gap instance of u nique g ames to an integrality gap instance of s parsest c ut this enables us to prove a log log n 1 6 integrality gap for s parsest c ut which is known to be equivalent to the metric embedding lower bound

semi supervised classification via local spline regression this paper presents local spline regression for semi supervised classification the core idea in our approach is to introduce splines developed in sobolev space to map the data points directly to be class labels the spline is composed of polynomials and green s functions it is smooth nonlinear and able to interpolate the scattered data points with high accuracy specifically in each neighborhood an optimal spline is estimated via regularized least squares regression with this spline each of the neighboring data points is mapped to be a class label then the regularized loss is evaluated and further formulated in terms of class label vector finally all of the losses evaluated in local neighborhoods are accumulated together to measure the global consistency on the labeled and unlabeled data to achieve the goal of semi supervised classification an objective function is constructed by combining together the global loss of the local spline regressions and the squared errors of the class labels of the labeled data in this way a transductive classification algorithm is developed in which a globally optimal classification can be finally obtained in the semi supervised learning setting the proposed algorithm is analyzed and addressed into the laplacian regularization framework comparative classification experiments on many public data sets and applications to interactive image segmentation and image matting illustrate the validity of our method

isomorphisms between spaces of lipschitz functions abstract we develop tools for proving isomorphisms of normed spaces of lipschitz functions over various doubling metric spaces and banach spaces in particular we show that lip 0 z d lip 0 r d for all d n more generally we e g show that lip 0 lip 0 g where is from a large class of finitely generated nilpotent groups and g is its mal cev closure or that lip 0 l p lip 0 l p for all 1 p we leave a large area for further possible research

compression bounds for lipschitz maps from the heisenberg group to l 1 we prove a quantitative bi lipschitz non embedding theorem for the heisenberg group with its carnot caratheodory metric and apply it to give a lower bound on the integrality gap of the goemans linial semidefinite relaxation of the sparsest cut problem

formal series of generalized functions and their application to deformation quantization foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed

scalable similarity search with topology preserving hashing hashing based similarity search techniques is becoming increasingly popular in large data sets to capture meaningful neighbors the topology of a data set which represents the neighborhood relationships between its subregions and the relative proximities between the neighbors of each subregion e g the relative neighborhood ranking of each subregion should be exploited however most existing hashing methods are developed to preserve neighborhood relationships while ignoring the relative neighborhood proximities moreover most hashing methods lack in providing a good result ranking since there are often lots of results sharing the same hamming distance to a query in this paper we propose a novel hashing method to solve these two issues jointly the proposed method is referred to as topology preserving hashing tph tph is distinct from prior works by also preserving the neighborhood ranking based on this framework we present three different tph methods including linear unsupervised tph semisupervised tph and kernelized tph particularly our unsupervised tph is capable of mining semantic relationship between unlabeled data without supervised information extensive experiments on four large data sets demonstrate the superior performances of the proposed methods over several state of the art unsupervised and semisupervised hashing techniques

image clustering using local discriminant models and global integration in this paper we propose a new image clustering algorithm referred to as clustering using local discriminant models and global integration ldmgi to deal with the data points sampled from a nonlinear manifold for each data point we construct a local clique comprising this data point and its neighboring data points inspired by the fisher criterion we use a local discriminant model for each local clique to evaluate the clustering performance of samples within the local clique to obtain the clustering result we further propose a unified objective function to globally integrate the local models of all the local cliques with the unified objective function spectral relaxation and spectral rotation are used to obtain the binary cluster indicator matrix for all the samples we show that ldmgi shares a similar objective function with the spectral clustering sc algorithms e g normalized cut ncut in contrast to ncut in which the laplacian matrix is directly calculated based upon a gaussian function a new laplacian matrix is learnt in ldmgi by exploiting both manifold structure and local discriminant information we also prove that k means and discriminative k means diskmeans are both special cases of ldmgi extensive experiments on several benchmark image datasets demonstrate the effectiveness of ldmgi we observe in the experiments that ldmgi is more robust to algorithmic parameter when compared with ncut thus ldmgi is more appealing for the real image clustering applications in which the ground truth is generally not available for tuning algorithmic parameters

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

equivalent definitions of bv space and of total variation on metric measure spaces abstract in this paper we introduce a new definition of bv based on measure upper gradients and prove the equivalence of this definition and the coincidence of the corresponding notions of total variation with the definitions based on relaxation of l 1 norm of the slope of lipschitz functions or upper gradients as in the previous work by the first author with gigli and savare in the sobolev case the proof requires neither local compactness nor doubling and poincare

differentiating maps into l1 and the geometry of bv functions this is one of a series of papers examining the interplay between differentiation theory for lipschitz maps x v and bi lipschitz nonembeddability where x is a metric measure space and v is a banach space here we consider the case v l 1 where differentiability fails we establish another kind of differentiability for certain x including n and h the heisenberg group with its carnot caratheodory metric it follows that does not bi lipschitz embed into l 1 as conjectured by j lee and a naor when combined with their work this provides a natural counterexample to the goemans linial conjecture in theoretical computer science the first such counterexample was found by khot vishnoi kv05 a key ingredient in the proof of our main theorem is a new connection between lipschitz maps to l 1 and functions of bounded variation which permits us to exploit results on the structure of bv functions on the heisenberg group fssc01

spline regression hashing for fast image search techniques for fast image retrieval over large databases have attracted considerable attention due to the rapid growth of web images one promising way to accelerate image search is to use hashing technologies which represent images by compact binary codewords in this way the similarity between images can be efficiently measured in terms of the hamming distance between their corresponding binary codes although plenty of methods on generating hash codes have been proposed in recent years there are still two key points that needed to be improved 1 how to precisely preserve the similarity structure of the original data and 2 how to obtain the hash codes of the previously unseen data in this paper we propose our spline regression hashing method in which both the local and global data similarity structures are exploited to better capture the local manifold structure we introduce splines developed in sobolev space to find the local data mapping function furthermore our framework simultaneously learns the hash codes of the training data and the hash function for the unseen data which solves the out of sample problem extensive experiments conducted on real image datasets consisting of over one million images show that our proposed method outperforms the state of the art techniques

product quantization for nearest neighbor search this paper introduces a product quantization based approach for approximate nearest neighbor search the idea is to decompose the space into a cartesian product of low dimensional subspaces and to quantize each subspace separately a vector is represented by a short code composed of its subspace quantization indices the euclidean distance between two vectors can be efficiently estimated from their codes an asymmetric version increases precision as it computes the approximate distance between a vector and a code experimental results show that our approach searches for nearest neighbors efficiently in particular in combination with an inverted file system results for sift and gist image descriptors show excellent search accuracy outperforming three state of the art approaches the scalability of our approach is validated on a data set of two billion vectors

full space local topology extraction for cross modal retrieval with the ever increasing availability of various kinds of multimedia data cross modal retrieval which enables information retrieval from various types of data given various types of query has become a research hotspot hashing based techniques have been developed to solve this problem however most previous works cannot capture the shared underlying structure of real world multimodal data which degrades their retrieval performances in this paper we propose a novel hashing method based on the extraction of the common manifold structure shared among different feature spaces to faithfully represent the common structure two kinds of local topology information are exploited in our method local angles are incorporated within the extraction of local topology of each feature space which is then used to learn a common intermediate subspace after heterogeneous features being embedded into this subspace local similarities are exploited to extract the local topology between different feature spaces and learn compact hamming embeddings to facilitate cross modal retrieval the proposed method is referred to as full space local topology extraction for hashing extensive comparisons with other state of the art methods on three benchmark multimedia data sets demonstrate the superiority of our proposed method in terms of retrieval recall and search accuracy

from local to global in quasiconformal structures abstract we exhibit a large class of metric spaces whose infinitesimal quasiconformal structure is strong enough to capture the global quasiconformal structure a sufficient condition for this to happen is described in terms of a poincare type inequality

coarse differentiation and quantitative nonembeddability for carnot groups abstract we give lower bound estimates for the macroscopic scale of coarse differentiability of lipschitz maps from a carnot group with the carnot caratheodory metric g d c c to a few different classes of metric spaces using this result we derive lower bound estimates for quantitative nonembeddability of lipschitz embeddings of g into a metric space x d x if x is either an alexandrov space with nonpositive or nonnegative curvature a superreflexive banach space or another carnot group that does not admit a bilipschitz homomorphic embedding of g for the same targets we can further give lower bound estimates for the bilipschitz distortion of every embedding f b n x where b n is the ball of radius n of a finitely generated nonabelian torsion free nilpotent group g we also prove an analogue of bourgain s discretization theorem for carnot groups and show that carnot groups have nontrivial markov convexity these give the first examples of metric spaces that have nontrivial markov convexity but cannot bilipschitzly embed into banach spaces of nontrivial markov convexity

neighborhood discriminant hashing for large scale image retrieval with the proliferation of large scale community contributed images hashing based approximate nearest neighbor search in huge databases has aroused considerable interest from the fields of computer vision and multimedia in recent years because of its computational and memory efficiency in this paper we propose a novel hashing method named neighborhood discriminant hashing ndh for short to implement approximate similarity search different from the previous work we propose to learn a discriminant hashing function by exploiting local discriminative information i e the labels of a sample can be inherited from the neighbor samples it selects the hashing function is expected to be orthogonal to avoid redundancy in the learned hashing bits as much as possible while an information theoretic regularization is jointly exploited using maximum entropy principle as a consequence the learned hashing function is compact and nonredundant among bits while each bit is highly informative extensive experiments are carried out on four publicly available data sets and the comparison results demonstrate the outperforming performance of the proposed ndh method over state of the art hashing techniques

new characterizations of sobolev spaces on the heisenberg group abstract the main aim of this paper is to present some new characterizations of sobolev spaces on the heisenberg group h first among several results theorem 1 1 theorem 1 2 we prove that if f l p h p 1 then f w 1 p h if and only if sup 0 1 f u f v p u 1 v q p d u d v this characterizations is in the spirit of the work by bourgain brezis and mironescu 5 in particular the work by hoai minh nguyen 29 in the euclidean spaces our work extends that of nguyen to sobolev spaces w 1 p h for p 1 in the setting of heisenberg group second corresponding to the case p 1 we give a characterizations of bv functions on the heisenberg group theorem 4 1 theorem 4 2 third we give some more generalized characterizations of sobolev spaces on the heisenberg groups theorem 5 1 theorem 5 2 it is worth to note that the underlying geometry of the euclidean spaces such as that any two points in r n can be connected by a line segment plays an important role in the proof of the main theorems in 29 thus one of the main techniques in 29 is to use the uniformity in every directions of the unit sphere in the euclidean spaces more precisely to deal with the general case s n 1 it is often assumed that e n 0 0 1 and hence one just needs to work on the one dimensional case this can be done by using the rotation in the euclidean spaces due to the non commutative nature of the heisenberg group the absence of this uniformity on the heisenberg group creates extra difficulties for us to handle hence we need to find a different approach to establish this characterization

hashing on nonlinear manifolds learning based hashing methods have attracted considerable attention due to their ability to greatly increase the scale at which existing algorithms may operate most of these methods are designed to generate binary codes preserving the euclidean similarity in the original space manifold learning techniques in contrast are better able to model the intrinsic structure embedded in the original high dimensional data the complexities of these models and the problems with out of sample data have previously rendered them unsuitable for application to large scale embedding however in this paper how to learn compact binary embeddings on their intrinsic manifolds is considered in order to address the above mentioned difficulties an efficient inductive solution to the out of sample data problem and a process by which nonparametric manifold learning may be used as the basis of a hashing method are proposed the proposed approach thus allows the development of a range of new hashing techniques exploiting the flexibility of the wide variety of manifold learning approaches available it is particularly shown that hashing on the basis of t distributed stochastic neighbor embedding outperforms state of the art hashing methods on large scale benchmark data sets and is very effective for image classification with very short code lengths it is shown that the proposed framework can be further improved for example by minimizing the quantization error with learned orthogonal rotations without much computation overhead in addition a supervised inductive manifold hashing framework is developed by incorporating the label information which is shown to greatly advance the semantic retrieval performance

principal manifolds and nonlinear dimensionality reduction via tangent space alignment we present a new algorithm for manifold learning and nonlinear dimensionality reduction based on a set of unorganized data points sampled with noise from a parameterized manifold the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold we also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases we illustrate our algorithm using curves and surfaces both in two dimensional three dimensional 2d 3d euclidean spaces and in higher dimensional euclidean spaces we also address several theoretical and algorithmic issues for further research and improvements

lp compression traveling salesmen and stable walks we show that if h is a group of polynomial growth whose growth rate is at least quadratic then the lp compression of the wreath product z oh equals max 1 p 1 2 we also show that the lp compression of z oz equals max p 2p 1 2 3 and the lp compression of z oz 0 the zero section of z oz equipped with the metric induced from z o z equals max p 1 2p 3 4 the fact that the hilbert compression exponent of z o z equals 3 while the hilbert compression exponent of z o z 0 equals 3 4 is used to show that there exists a lipschitz function f z o z 0 l2 which cannot be extended to a lipschitz function defined on all of z o z