randomized clustering forests for image classification some of the most effective recent methods for content based image classification work by quantizing image descriptors and accumulating histograms of the resulting visual word codes large numbers of descriptors and large codebooks are required for good results and this becomes slow using k means we introduce extremely randomized clustering forests ensembles of randomly created clustering trees and show that they provide more accurate results much faster training and testing and good resistance to background clutter second an efficient image classification method is proposed it combines erc forests and saliency maps very closely with the extraction of image information for a given image a classifier builds a saliency map online and uses it to classify the image we show in several state of the art image classification tasks that this method can speed up the classification process enormously finally we show that the proposed erc forests can also be used very successfully for learning distance between images the distance computation algorithm consists of learning the characteristic differences between local descriptors sampled from pairs of same or different objects these differences are vector quantized by erc forests and the similarity measure is computed from this quantization the similarity measure has been evaluated on four very different datasets and always outperforms the state of the art competitive approaches

boosting the margin a new explanation for the effectiveness of voting methods one of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large and often is observed to decrease even after the training error reaches zero in this paper we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label we show that techniques used in the analysis of vapnik s support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error we also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples finally we compare our explanation to those based on the bias variance decomposition

the ring structure on the cohomology of coordinate subspace arrangements every simplicial complex delta subset2 n on the vertex set n 1 ldots n defines a real resp complex arrangement of coordinate subspaces in mathbb r n resp mathbb c n via the correspondence delta ni sigma mapsto rm span e i i in sigma the linear structure of the cohomology of the complement of such an arrangement is explicitly given in terms of the combinatorics of delta and its links by the goresky macpherson formula here we derive by combinatorial means the ring structure on the integral cohomology in terms of data of delta we provide a non trivial example of different cohomology rings in the real and complex case furthermore we give an example of a coordinate arrangement that yields non trivial multiplication of torsion elements

stratified feature sampling method for ensemble clustering of high dimensional data high dimensional data with thousands of features present a big challenge to current clustering algorithms sparsity noise and correlation of features are common characteristics of such data another common phenomenon is that clusters in such high dimensional data often exist in different subspaces ensemble clustering is emerging as a prominent technique for improving robustness stability and accuracy of high dimensional data clustering in this paper we propose a stratified sampling method for generating subspace component data sets in ensemble clustering of high dimensional data instead of randomly sampling a subset of features for each component data set in this method we first cluster the features of high dimensional data into a few feature groups called feature strata using stratified sampling we randomly sample some features from each feature stratum and merge the sampled features from different feature strata to generate a component data set in this way the component data sets have better representations of the clustering structure in the original data set comparing with random sampling and random projection methods in synthetic data analysis the component clustering by stratified sampling has demonstrated that the average clustering accuracy was increased without sacrificing clustering diversity we carried out a series of experiments on eight real world data sets from microarray text and image domains to evaluate ensemble clustering methods using three subspace component data generation methods and four consensus functions the experimental results consistently showed that the stratified sampling method produced the best ensemble clustering results in all data sets the ensemble clustering with stratified sampling also outperformed three other ensemble clustering methods which generate component clusters from the entire space of the original data highlightsa new component generation approach is proposed to produce ensemble components stratified sampling is used to generate subspace component data sets the component data can well represent the characteristics of the original data the proposed method can achieve a consistent performance in ensemble clustering the proposed method is easy to implement

subspace arrangements over finite fields cohomological and enumerative aspects abstract the enumeration of points on or off the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function we discuss the basic relations between these two points of view counting points is also related to the l adic cohomology of the arrangement as a variety we describe the eigenvalues of the frobenius map acting on this cohomology which corresponds to a finer decomposition of the zeta function the l adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data finally it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases

rigid body constrained noisy point pattern matching noisy pattern matching problems arise in many areas e g computational vision robotics guidance and control stereophotogrammetry astronomy genetics and high energy physics least squares pattern matching over the euclidean space e sup n for unordered sets of cardinalities p and q is commonly formulated as a combinatorial optimization problem having complexity p p 1 spl middot spl middot spl middot p q 1 q spl les p since p and q may be 10 sup 3 or larger in typical applications less than satisfactory suboptimal methods are usually employed a hybrid approach is described for solving the pattern matching problem under rigid motion constraints which often apply the method reduces the complexity to l sub 21 spl middot n sup 4 l sub 12 spl middot p sup 3 where l sub 12 and l sub 21 are the number of iterations required by steepest ascent and singular value decomposition svd based procedures respectively

bi level path following for cross validated solution of kernel quantile regression we show how to follow the path of cross validated solutions to families of regularized optimization problems defined by a combination of a parameterized loss function and a regularization term a primary example is kernel quantile regression where the parameter of the loss function is the quantile being estimated even though the bi level optimization problem we encounter for every quantile is non convex the manner in which the optimal cross validated solution evolves with the parameter of the loss function allows tracking of this solution we prove this property construct the resulting algorithm and demonstrate it on real and artificial data this algorithm allows us to efficiently solve the whole family of bi level problems we show how it can be extended to cover other modeling problems like support vector regression and alternative in sample model selection approaches

use of the zero norm with linear models and kernel methods we explore the use of the so called zero norm of the parameters of linear models in learning minimization of such a quantity has many uses in a machine learning context for variable or feature selection minimizing training error and ensuring sparsity in solutions we derive a simple but practical method for achieving these goals and discuss its relationship to existing techniques of minimizing the zero norm the method boils down to implementing a simple modification of vanilla svm namely via an iterative multiplicative rescaling of the training data applications we investigate which aid our discussion include variable and feature selection on biological microarray data and multicategory classification

random forests with ensemble of feature spaces random forests receive much attention from researchers because of their excellent performance as breiman suggested the performance of random forests depends on the strength of the weak learners in the forests and the diversity among them however in the literature many researchers only considered pre processing of the data or post processing of the random forests models in this paper we propose a new method to increase the diversity of each tree in the forests and thereby improve the overall accuracy during the training process of each individual tree in the forest different rotation spaces are concatenated into a higher space at the root node then the best split is exhaustively searched within this higher space the location where the best split lies decides which rotation method to be used for all subsequent nodes the performance of the proposed method here is evaluated on 42 benchmark data sets from various research fields and compared with the standard random forests the results show that the proposed method improves the performance of the random forests in most cases

the polyhedral product functor a method of decomposition for moment angle complexes arrangements and related spaces this article gives a natural decomposition of the suspension of generalized moment angle complexes or partial product spaces which arise as polyhedral product functors described below the geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of goresky macpherson concerning complements of certain subspace arrangements as well as davis januszkiewicz and buchstaber panov concerning moment angle complexes since the stable decompositions here are geometric they provide corresponding homological decompositions for generalized moment angle complexes for any homology theory

a statistical approach to texture classification from single images we investigate texture classification from single images obtained under unknown viewpoint and illumination a statistical approach is developed where textures are modelled by the joint probability distribution of filter responses this distribution is represented by the frequency histogram of filter response cluster centres textons recognition proceeds from single uncalibrated images and the novelty here is that rotationally invariant filters are used and the filter response space is low dimensional classification performance is compared with the filter banks and methods of leung and malik ijcv 2001 schmid cvpr 2001 and cula and dana ijcv 2004 and it is demonstrated that superior performance is achieved here classification results are presented for all 61 materials in the columbia utrecht texture database we also discuss the effects of various parameters on our classification algorithm such as the choice of filter bank and rotational invariance the size of the texton dictionary as well as the number of training images used finally we present a method of reliably measuring relative orientation co occurrence statistics in a rotationally invariant manner and discuss whether incorporating such information can enhance the classifier s performance

shape quantization and recognition with randomized trees we explore a new approach to shape recognition based on a virtually infinite family of binary features queries of the image data designed to accommodate prior information about shape invariance and regularity each query corresponds to a spatial arrangement of several local topographic codes or tags which are in themselves too primitive and common to be informative about shape all the discriminating power derives from relative angles and distances among the tags the important attributes of the queries are a natural partial ordering corresponding to increasing structure and complexity semi invariance meaning that most shapes of a given class will answer the same way to two queries that are successive in the ordering and stability since the queries are not based on distinguished points and substructures no classifier based on the full feature set can be evaluated and it is impossible to determine a priori which arrangements are informative our approach is to select informative features and build tree classifiers at the same time by inductive learning in effect each tree provides an approximation to the full posterior where the features chosen depend on the branch that is traversed due to the number and nature of the queries standard decision tree construction based on a fixed length feature vector is not feasible instead we entertain only a small random sample of queries at each node constrain their complexity to increase with tree depth and grow multiple trees the terminal nodes are labeled by estimates of the corresponding posterior distribution over shape classes an image is classified by sending it down every tree and aggregating the resulting distributions the method is applied to classifying handwritten digits and synthetic linear and nonlinear deformations of three hundred l at e x symbols stateof the art error rates are achieved on the national institute of standards and technology database of digits the principal goal of the experiments on l at e x symbols is to analyze invariance generalization error and related issues and a comparison with artificial neural networks methods is presented in this context

on the difference between real and complex arrangements if b is an arrangement of linear complex hyperplanes in c d then the following can be constructed from knowledge of its intersection lattice a the cohomology groups of the complement br b the cohomology algebra of the complement os c the fundamental group of the complement if d 2 d the singularity link up to homeomorphism if d 3 e the singularity link up to homotopy type zz if bis more generally a 2 arrangement in ir 2d an arrangement of real subspaces of codimension 2 with even dimensional intersections then the intersection lattice still determines a the cohomology groups of the complement gm and e the homotopy type of the singularity link zz we show however that for 2 arrangements the data b c and d are not deter mined by the intersection lattice they require the knowledge of extra information on sign patterns which can be computed as determinants of linear relations or equiva lently as linking coefficients in the sense of knot theory

contrast trees and distribution boosting a method for decision tree induction is presented given a set of predictor variables formula see text and two outcome variables y and z associated with each x the goal is to identify those values of x for which the respective distributions of formula see text and formula see text or selected properties of those distributions such as means or quantiles are most different contrast trees provide a lack of fit measure for statistical models of such statistics or for the complete conditional distribution formula see text as a function of x they are easily interpreted and can be used as diagnostic tools to reveal and then understand the inaccuracies of models produced by any learning method a corresponding contrast boosting strategy is described for remedying any uncovered errors thereby producing potentially more accurate predictions this leads to a distribution boosting strategy for directly estimating the full conditional distribution of y at each x under no assumptions concerning its shape form or parametric representation

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

disjunctive normal random forests we develop a novel supervised learning classification method called disjunctive normal random forest dnrf a dnrf is an ensemble of randomly trained disjunctive normal decision trees dndt to construct a dndt we formulate each decision tree in the random forest as a disjunction of rules which are conjunctions of boolean functions we then approximate this disjunction of conjunctions with a differentiable function and approach the learning process as a risk minimization problem that incorporates the classification error into a single global objective function the minimization problem is solved using gradient descent dnrfs are able to learn complex decision boundaries and achieve low generalization error we present experimental results demonstrating the improved performance of dndts and dnrfs over conventional decision trees and random forests we also show the superior performance of dnrfs over state of the art classification methods on benchmark datasets

progressive subspace ensemble learning there are not many classifier ensemble approaches which investigate the data sample space and the feature space at the same time and this multi pronged approach will be helpful for constructing more powerful learning models for example the adaboost approach only investigates the data sample space while the random subspace technique only focuses on the feature space to address this limitation we propose the progressive subspace ensemble learning approach psel which takes into account the data sample space and the feature space at the same time specifically psel first adopts the random subspace technique to generate a set of subspaces then a progressive selection process based on new cost functions that incorporate current and long term information to select the classifiers sequentially will be introduced finally a weighted voting scheme is used to summarize the predicted labels and obtain the final result we also adopt a number of non parametric tests to compare psel and its competitors over multiple datasets the results of the experiments show that psel works well on most of the real datasets and outperforms a number of state of the art classifier ensemble approaches highlightsprogressive subspace ensemble learning it takes into account the data sample space and the feature space at the same time a progressive selection process based on new cost functions that incorporate current and long term information to select the classifiers sequentially will be introduced

the automorphism group of a free group is not linear a linear group has ascending chain condition on centralizers malcev s p 511 finitely generated linear groups are residually finite malcev s p 453 a solvable linear group is nilpotent by abelian byfinite tits 5 pp 14551461 a linear group either contains a free group of rank two or is solvable by locally finite lubotzky 3 has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions these conditions appear difficult to check although they have been used to show linearity for certain groups however we do not use his theorem the main result of this paper theorem 5 is that for n 2 3 the automorphism group of a free group of rank n is not a linear group the proof uses the representation theory of algebraic groups to show that a kind of diophantine equation between the irreducible representations of

learning subspace based rbfnn using coevolutionary algorithm for complex classification tasks many real world classification problems are characterized by samples of a complex distribution in the input space the classification accuracy is determined by intrinsic properties of all samples in subspaces of features this paper proposes a novel algorithm for the construction of radial basis function neural network rbfnn classifier based on subspace learning in this paper feature subspaces are obtained for every hidden node of the rbfnn during the learning process the connection weights between the input layer and the hidden layer are adjusted to produce various subspaces with dominative features for different hidden nodes the network structure and dominative features are encoded in two subpopulations that are cooperatively coevolved using the coevolutionary algorithm to achieve a better global optimality for the estimated rbfnn experimental results illustrate that the proposed algorithm is able to obtain rbfnn models with both better classification accuracy and simpler network structure when compared with other learning algorithms thus the proposed model provides a more flexible and efficient approach to complex classification tasks by employing the local characteristics of samples in subspaces

small rational model of subspace complement this paper concerns the rational cohomology ring of the complement m of a complex subspace arrangement we start with the de concini procesi differential graded algebra that is a rational model for m inside it we find a much smaller subalgebra d quasi isomorphic to the whole algebra d is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice l whence connecting the de concini procesi model with the goresky macpherson formula for the additive structure of h m the algebra d has a natural integral version that is a good candidate for an integral model of m if the rational local homology of l can be computed explicitly we obtain an explicit presentation of the ring h m q for example this is done for the cases where l is a geometric lattice and where m is a k equal manifold

hyperplane arrangement cohomology and monomials in the exterior algebra we show that if x is the complement of a complex hyperplane arrangement then the homology of x has linear free resolution as a module over the exterior algebra on the first cohomology of x we study invariants of x that can be deduced from this resolution a key ingredient is a result of aramova avramov and herzog 2000 on resolutions of monomial ideals in the exterior algebra we give a new conceptual proof of this result

maximal matching of 3 d points for multiple object motion estimation abstract determining the relative motion position between an observer and its environment is an important problem in computer vision a major task is to find corresponding object features in this paper an efficient algorithm for matching 3 d points of multiple rigid objects is presented the point matching algorithm determines the correspondence by initiating a pairing of a triplet of noncollinear sensed points with a triplet of reference points and searching for new pairs of corresponding points one at a time using local distance and angular constraints the pairing of each subsequent sensed point with a reference point is determined if the tetrahedron formed by the sensed point and the initial triplet is congruent to that formed by the corresponding reference points only simple computations are required in the algorithm global consistency of the pairings found by the algorithm is ensured without using model tests the algorithm can be easily extended to incorporate other geometrical or non geometrical object attributes to further prune the matching results of running the algorithm on synthetic and real data are given

on cohomology algebras of complex subspace arrangements the integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement we give a combinatorial presentation of the cohomology algebra in the spirit of the orlik solomon result on the cohomology algebras of complex hyperplane arrangements our methods are elementary we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space we describe simplicial cochains that generate the cohomology among them we distinguish a linear basis study cup product multiplication and derive an algebra presentation in terms of generators and relations

group actions on arrangements of linear subspaces and applications to configuration spaces for an arrangement of linear subspaces in ri that is invariant under a finite subgroup of the general linear group gin ir we develop a formula for the g module structure of the cohomology of the complement m a our formula specializes to the well known goresky macpherson theorem in case g 1 but for g 1 the formula shows that the g module structure of the complement is not a combinatorial invariant as an application we are able to describe the free part of the cohomology of the quotient space ma4ig our motivating examples are arrangements in cn that are invariant under the action of sn by permuting coordinates a particular case is the k equal arrangement first studied by bj6rner lovasz and yao motivated by questions in complexity theory in these cases ma4 and ma sn are spaces of ordered and unordered point configurations in cn many of whose properties are reduced by our formulas to combinatorial questions in partition lattices more generally we treat point configurations in rd and provide explicit results for the k equal and the k divisible cases

topology and geometry of cohomology jump loci we elucidate the key role played by formality in the theory of characteristic and resonance varieties we define relative characteristic and resonance varieties v k and r k related to twisted group cohomology with coefficients of arbitrary rank we show that the germs at the origin of v k and r k are analytically isomorphic if the group is 1 formal in particular the tangent cone to v k at 1 equals r k these new obstructions to 1 formality lead to a striking rationality property of the usual resonance varieties a detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth complex quasi projective algebraic variety this sheds new light on a classical problem of j p serre applications to arrangements configuration spaces coproducts of groups and artin groups are given

matching point features under small nonrigid motion this paper describes a method for matching point features between images of objects that have undergone small nonrigid motion feature points are assumed to be available and given a properly extracted set of feature points a robust matching is established under the condition that the local nonrigid motion of each point is restricted to a circle of radius where is not too large this is in contrast to other techniques for point matching which assume either rigid motion or nonrigid motion of a known kind the point matching problem is viewed in terms of weighted bipartite graph matching in order to account for the possibility that the feature selector can be imprecise we incorporate a greedy matching strategy with the weighted graph matching algorithm our algorithm is robust and insensitive to noise and missing features the resulting matching can be used with image warping or other techniques for nonrigid motion analysis image subtraction etc we present our experimental results on sequences of mammograms images of a deformable clay object and satellite cloud images in the first two cases we provide quantitative comparison with known ground truth

commuting elements and spaces of homomorphisms this article records basic topological as well as homological properties of the space of homomorphisms hom g where is a finitely generated discrete group and g is a lie group possibly non compact if is a free abelian group of rank equal to n then hom g is the space of ordered n tuples of commuting elements in g if g su 2 a complete calculation of the cohomology of these spaces is given for n 2 3 an explicit stable splitting of these spaces is also obtained as a special case of a more general splitting

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

the minimal model of the complement of an arrangement of hyperplanes in this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory let a be a finite collection of hyperplanes in cl and let m cl uhea h we say a is a rational k w 1 arrangement if the rational completion of m is aspherical for these arrangements an identity the lcs formula is established relating the lower central series of 7r1 m to the cohomology of m this identity was established by group theoretic means for the class of fiber type arrangements in previous work we reproduce this result by showing that the class of rational k w 1 arrangements contains all fiber type arrangements this class includes the reflection arrangements of types al and bl there is much interest in arrangements for which m is a k w 1 space the methods developed here do not apply directly because m is rarely a nilpotent space we give examples of k w 1 arrangements which are not rational k w 1 for which the lcs formula fails and k w 1 arrangements which are not rational k w 1 where the lcs formula holds it remains an open question whether rational k w 1 arrangements are necessarily k w 1 l introduction an arrangement of hyperplanes is a finite collection of clinear subspaces of dimension 1 l in c1 to such an arrangement a is associated an open 21 manifold the complement m cl u hih e a the connections between the topology of m and the combinatorial geometry of a are the source of much current research in this area the most successful investigations concern the cohomology of m 13 15 whereas the most difficult unsolved problems involve the homotopy groups of m 5 in this paper we study the link between cohomology and homotopy provided by sullivan s theory of minimal models 14 in 6 a numerical relationship was established between 1 m and h m for the class of fiber type arrangements this lcs formula reads as follows ri 1 tn y7n m pm t n l where the n m are the ranks of successive quotients in the lower central series of 1 m and pm t is the poincare polynomial of h m because the sequence n m is related to the l minimal model y of m we conjecture in 5 that the lds formula holds precisely when r determines h m with the methods developed in this paper we can resolve this conjecture specifically we show corollary 3 8 that the lcs formula holds when h 5 is isomorphic to h m arrangements satisfying the latter condition are called rational k 7r l corollary 3 8 may be received by the editors june 1 1987 1980 mathematicts subject clatstsification 1985 revitsion primary 55p62 57m05 secondary 06c10 r 1988 american mathematical society 0002 9947 88 1 00 25 per page

on the algebra associated with a geometric lattice abstract let l be a geometric lattice following p orlik and l solomon combinatorics and topology of complements of hyperplanes invent math 56 1980 167 189 we associate with l a graded commutative algebra a l in this paper we introduce a new invariant of the algebra a l which suffices to distinguish algebras for which all other known invariants coincide this result is applied to the study of arrangements of complex hyperplanes with l being the intersection lattice in this case a l is isomorphic to the cohomology algebra of the associated hyperplane complement the goal is to find examples of arrangements with non isomorphic lattices but homotopy equivalent complements the invariant introduced here effectively narrows the list of candidates nevertheless we exhibit combinatorially inequivalent arrangements for which all known invariants including coincide

topological representations of matroids one of the foundations of oriented matroid theory is the topological representation theorem of folkman and lawrence 8 it says that an oriented simple matroid can be realized uniquely as an arrangement of pseudospheres that there is no similar interpretation for the class of all matroids has been taken for granted for instance a non coordinatizable matroid of abstract origin may be thought of as a geometric object only in a purely formal way whereas an oriented matroid may always be thought of as a geometric topological configuration on the d sphere or in projective space 3 p 19 our main theorem is that the class of geometric lattices which is cryptomorphic to the category of simple matroids is the same as the class of intersection lattices of arrangements of homotopy spheres the interpretation of a geometric lattice as an arrangement of homotopy spheres is a natural generalization of the folkman lawrence theorem an oriented matroid realizable over r has a representation with geodesic spheres allowing pseudospheres i e those which are homeomorphic but possibly not isometric to the unit sphere leads to the category of simple oriented matroids if we further relax the conditions on the spheres to only homotopy equivalence to the standard sphere then we are led to the category of all simple matroids some of the theory of oriented matroids which only depends on the underlying matroid can be extended to homotopy sphere arrangements zaslavsky s enumerative theory for pseudosphere arrangements can be extended to the homotopy setting as in the oriented matroid representation theorem the arrangement can be forced to be antipodal so a realization as homotopy projective spaces is also possible the minimal cellular resolutions of orientable matroid ideals developed in 10 can be extended to arbitrary matroids by using arrangements of homotopy spheres our point of view is primarily through the lens of oriented matroids hence the homotopy spheres which represent the atoms of the geometric lattice have codimension one in the future we hope to examine the point of view of complex hyperplane arrangements and consider spheres of even codimension the matroid theory we require is in section 2 while section 3 presents matroid steiner complexes we review a very general theory of arrangements of topological

homotopy types of subspace arrangements via diagrams of spaces we prove combinatorial formulas for the homotopy type of the union of the subspaces in an affine compactified affine spherical or projective subspace arrangement from these formulas we derive results of goresky macpherson on the homology of the arrangement and the cohomology of its complement the union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset a closely related space is obtained by taking the homotopy direct limit of this diagram our method consists in constructing a combinatorial model diagram over the same poset whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces

random forests and adaptive nearest neighbors in this article we study random forests through their connection with a new framework of adaptive nearest neighbor methods we introduce a concept of potential nearest neighbors k pnns and show that random forests can be viewed as adaptively weighted k pnn methods various aspects of random forests can be studied from this perspective we study the effect of terminal node sizes on the prediction accuracy of random forests we further show that random forests with adaptive splitting schemes assign weights to k pnns in a desirable way for the estimation at a given target point these random forests assign voting weights to the k pnns of the target point according to the local importance of different input variables we propose a new simple splitting scheme that achieves desirable adaptivity in a straightforward fashion this simple scheme can be combined with existing algorithms the resulting algorithm is computationally faster and gives comparable results other possible aspects of random forests such

finding point correspondences using simulated annealing abstract identifying corresponding points between two recordings of a point set has always been an important problem in stereo vision applications we describe this matching problem in terms of cost minimization and present an algorithm to approach the minimal cost mapping using simulated annealing the algorithm calculates the costs to match all possible point pairs and tries to minimize the sum of the costs of all matched points starting from an initial mapping it uses a random rearrangement scheme to alter the mapping towards the optimal minimal cost mapping

on the homotopy type of the quillen complex of finite soluble groups abstract we prove that the homotopy type of the quillen complex of a finite soluble group at the prime p 2 is that of a wedge of spheres of possible different dimensions

k parabolic subspace arrangements in this paper we study k parabolic arrangements a generalization of the k equal arrangement for any finite real reflection group when k 2 these arrangements correspond to the well studied coxeter arrangements brieskorn 1971 showed that the fundamental group of the complement of the type w coxeter arrangement over mathbb c is isomorphic to the pure artin group of type w khovanov 1996 gave an algebraic description for the fundamental group of the complement of the 3 equal arrangement over mathbb r we generalize khovanov s result to obtain an algebraic description of the fundamental group of the complement of the 3 parabolic arrangement for arbitrary finite reflection group our description is a real analogue to brieskorn s description

stratified sampling for feature subspace selection in random forests for high dimensional data for high dimensional data a large portion of features are often not informative of the class of the objects random forest algorithms tend to use a simple random sampling of features in building their decision trees and consequently select many subspaces that contain few if any informative features in this paper we propose a stratified sampling method to select the feature subspaces for random forests with high dimensional data the key idea is to stratify features into two groups one group will contain strong informative features and the other weak informative features then for feature subspace selection we randomly select features from each group proportionally the advantage of stratified sampling is that we can ensure that each subspace contains enough informative features for classification in high dimensional data testing on both synthetic data and various real data sets in gene classification image categorization and face recognition data sets consistently demonstrates the effectiveness of this new method the performance is shown to better that of state of the art algorithms including svm the four variants of random forests rf ert enrich rf and oblique rf and nearest neighbor nn algorithms

decompositions of the polyhedral product functor with applications to moment angle complexes and related spaces this article gives a natural decomposition of the suspension of a generalized moment angle complex or partial product space which arises as the polyhedral product functor described below the introduction and application of the smash product moment angle complex provides a precise identification of the stable homotopy type of the values of the polyhedral product functor one direct consequence is an analysis of the associated cohomology for the special case of the complements of certain subspace arrangements the geometrical decomposition implies the homological decomposition in earlier work of others as described below because the splitting is geometric an analogous homological decomposition for a generalized moment angle complex applies for any homology theory implied therefore is a decomposition for the stanley reisner ring of a finite simplicial complex and natural generalizations

quantile regression forests random forests were introduced as a machine learning tool in breiman 2001 and have since proven to be very popular and powerful for high dimensional regression and classification for regression random forests give an accurate approximation of the conditional mean of a response variable it is shown here that random forests provide information about the full conditional distribution of the response variable not only about the conditional mean conditional quantiles can be inferred with quantile regression forests a generalisation of random forests quantile regression forests give a non parametric and accurate way of estimating conditional quantiles for high dimensional predictor variables the algorithm is shown to be consistent numerical examples suggest that the algorithm is competitive in terms of predictive power

using motion from orthographic views to verify 3 d point matches the specific problem addressed by the authors is how to detect the true match of a fourth point from among candidate matches in a situation in which three points have already been matched the two sets of points to be matched are both subject to measurement errors the depth error is more dominant than errors in the other two coordinates however the exact statistical distribution of the measurement errors is not known the authors present a new method for solving the problem the method is based on the technique of motion analysis using orthographic views it discards the noisy z depth coordinates and uses only the x and y coordinates of the points to verify the match the effect of depth errors on the motion estimate is completely prevented results show that this method is substantially more effective than previous methods that use all three coordinates

a novel consistent random forest framework bernoulli random forests random forests rfs are recognized as one type of ensemble learning method and are effective for the most classification and regression tasks despite their impressive empirical performance the theory of rfs has yet been fully proved several theoretically guaranteed rf variants have been presented but their poor practical performance has been criticized in this paper a novel rf framework is proposed named bernoulli rfs brfs with the aim of solving the rf dilemma between theoretical consistency and empirical performance brf uses two independent bernoulli distributions to simplify the tree construction in contrast to the rfs proposed by breiman the two bernoulli distributions are separately used to control the splitting feature and splitting point selection processes of tree construction consequently theoretical consistency is ensured in brf i e the convergence of learning performance to optimum will be guaranteed when infinite data are given importantly our proposed brf is consistent for both classification and regression the best empirical performance is achieved by brf when it is compared with state of the art theoretical consistent rfs this advance in rf research toward closing the gap between theory and practice is verified by the theoretical and experimental studies in this paper

the cohomology rings of complements of subspace arrangements the ring structure of the integral cohomology of complements of real linear subspace arrangements is considered while the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of goresky and macpherson we describe the multiplicative structure in terms of the intersection poset the dimension function and orientations of the participating subspaces for the class of arrangements without pairs of intersections of codimension one in particular this yields a description of the integral cohomology ring of complex arrangements conjectured by yuzvinsky for general real arrangements a weaker result is obtained the approach is geometric and the methods are elementary

on representations and k theory of the braid groups let be the fundamental group of the complement of a k 1 hyperplane arrangement such as artin s pure braid group or more generally a homologically toroidal group as defined below the triviality of bundles arising from orthogonal representations of is characterized completely as follows an orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups furthermore the subgroup of elements in the complex k theory of b which arises from complex unitary representations of is shown to be trivial in the case of real k theory the subgroup of elements which arises from real orthogonal representations of is an elementary abelian 2 group which is characterized completely in terms of the first two stiefel whitney classes of the representation in addition quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the jacobi identity for certain choices of poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized heisenberg groups these cohomology relations correspond to non trivial spin representations of the pure braid groups which give rise to trivial bundles

a classification framework for anomaly detection one way to describe anomalies is by saying that anomalies are not concentrated this leads to the problem of finding level sets for the data generating density we interpret this learning problem as a binary classification problem and compare the corresponding classification risk with the standard performance measure for the density level problem in particular it turns out that the empirical classification risk can serve as an empirical performance measure for the anomaly detection problem this allows us to compare different anomaly detection algorithms empirically i e with the help of a test set furthermore by the above interpretation we can give a strong justification for the well known heuristic of artificially sampling labeled samples provided that the sampling plan is well chosen in particular this enables us to propose a support vector machine svm for anomaly detection for which we can easily establish universal consistency finally we report some experiments which compare our svm to other commonly used methods including the standard one class svm

consistency of random forests and other averaging classifiers in the last years of his life leo breiman promoted random forests for use in classification he suggested using averaging as a means of obtaining good discrimination rules the base classifiers used for averaging are simple and randomized often based on random samples from the data he left a few questions unanswered regarding the consistency of such rules in this paper we give a number of theorems that establish the universal consistency of averaging rules we also show that some popular classifiers including one suggested by breiman are not universally consistent

computational topology of equivariant maps from spheres to complements of arrangements the problem of the existence of an equivariant map is a classical topological problem ubiquitous in topology and its applications many problems in discrete geometry and combinatorics have been reduced to such a question and many of them resolved by the use of equivariant obstruction theory a variety of concrete techniques for evaluating equivariant obstruction classes are introduced discussed and illustrated by explicit calculations the emphasis is on d 2n equivariant maps from spheres to complements of arrangements motivated by the problem of finding a 4 fan partition of 2 spherical measures where d 2n is the dihedral group one of the technical highlights is the determination of the d 2n module structure of the homology of the complement of the appropriate subspace arrangement based on the geometric interpretation for the generators of the homology groups of arrangements

poset fiber theorems suppose that f p q is a poset map whose fibers f 1 qless than or equal to q are sufficiently well connected our main result is a formula expressing the homotopy type of p in terms of q and the fibers several fiber theorems from the literature due to babson baclawski and quillen are obtained as consequences or special cases homology cohen macaulay and equivariant versions are given and some applications are discussed

bivariate quantile smoothing splines it has long been recognized that the mean provides an inadequate summary whereas the set of quantiles can supply a more complete description of a sample we introduce bivariate quantile smoothing splines which belong to the space of bilinear tensor product splines as nonparametric estimators for the conditional quantile functions in a two dimensional design space the estimators can be computed by using standard linear programming techniques and can further be used as building blocks for conditional quantile estimations in higher dimensions for moderately large data sets we recommend penalized bivariate b splines as approximate solutions we use real and simulated data to illustrate the methodology proposed

matching 3 d line segments with applications to multiple object motion estimation a two stage algorithm for matching line segments using three dimensional data is presented in the first stage a tree search based on the orientation of the line segments is applied to establish potential matches the sign ambiguity of line segments is fixed by a simple congruency constraint in the second stage a hough clustering technique based on the position of line segments is applied to verify potential matches any paired line segments of a match that cannot be brought to overlap by the translation determined by the clustering are removed from the match unlike previous methods this algorithm combats noise more effectively and ensures the global consistency of a match while the original motivation for the algorithm is multiple object motion estimation from stereo image sequences the algorithm can also be applied to other domains such as object recognition and object model construction from multiple views

the random subspace method for constructing decision forests much of previous attention on decision trees focuses on the splitting criteria and optimization of tree sizes the dilemma between overfitting and achieving maximum accuracy is seldom resolved a method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity the classifier consists of multiple trees constructed systematically by pseudorandomly selecting subsets of components of the feature vector that is trees constructed in randomly chosen subspaces the subspace method is compared to single tree classifiers and other forest construction methods by experiments on publicly available datasets where the method s superiority is demonstrated we also discuss independence between trees in a forest and relate that to the combined classification accuracy

quantile smoothing splines summary although nonparametric regression has traditionally focused on the estimation of conditional mean functions nonparametric estimation of conditional quantile functions is often of substantial practical interest we explore a class of quantile smoothing splines defined as solutions to

combinatorial stratification of complex arrangements we present a method for discretizing complex hyperplane arrange ments by encoding their topology into a finite partially ordered set of sign vectors this is used in the following ways 1 a general method is given for constructing regular cell complexes having the homotopy type of the complement of the arrangement 2 for the case of complexified arrangements this specializes to the con struction of salvetti s we study the combinatorial structure of complexified arrangements and the salvetti complex in some detail 3 this general method simultaneously produces cell decompositions of the singularity link this link is shown to have the homotopy type of a wedge of d spheres for arrangements in cd d 4 4 the homology of the link and the cohomology of the complement are computed in terms of explicit bases which are matched by alexander duality this gives a new more elementary and more generally valid proof for the brieskorn orlik solomon theorem and some related results 5 our setup leads to a more general notion of 2 pseudoarrangements which can be thought of as topologically deformed complex arrangements re taining only the essential topological and combinatorial structure we show that all of the above remains true in this generality except for the sign patterns of the orlik solomon relations department of mathematics royal institute of technology s10044 stockholm sweden e mail address andersb nada kth se institut fur mathematik universitat augsburg d 8900 augsburg germany current address institut mittag leffler auravagen 17 s 18262 djursholm sweden address after 1 april 1992 konrad zuse zentrum fur informationstechnik berlin heilbronner str 10 w 1000 berlin 31 germany e mail addresses ziegler ml kva se ziegler zib berlin de this content downloaded from 157 55 39 211 on fri 24 jun 2016 06 44 45 utc all use subject to http about jstor org terms