a rotation group whose subspace arrangement is not from a real reflection group abstract we exhibit a family of real rotation groups whose subspace arrangements are not contained in that of any real reflection group answering a question of martino and singh

on the cohen macaulay property of modular invariant rings abstract if v is a faithful module for a finite group g over a field of characteristic p then the ring of invariants need not be cohen macaulay if p divides the order of g in this article the cohomology of g is used to study the question of cohen macaulayness of the invariant ring one of the results is a classification of all groups for which the invariant ring with respect to the regular representation is cohen macaulay moreover it is proved that if p divides the order of g then the ring of vector invariants of sufficiently many copies of v is not cohen macaulay a further result is that if g is a p group and the invariant ring is cohen macaulay then g is a bireflection group i e it is generated by elements which fix a subspace of v of codimension at most 2

sparse additive models summary we present a new class of methods for high dimensional non parametric regression and classification called sparse additive models our methods combine ideas from sparse linear modelling and additive non parametric regression we derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size sparse additive models are essentially a functional version of the grouped lasso of yuan and lin they are also closely related to the cosso model of lin and zhang but decouple smoothing and sparsity enabling the use of arbitrary non parametric smoothers we give an analysis of the theoretical properties of sparse additive models and present empirical results on synthetic and real data showing that they can be effective in fitting sparse non parametric models in high dimensional data

a lasso for hierarchical interactions we add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important we give a precise characterization of the effect of this hierarchy constraint prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator a bound on this estimate reveals the amount of fitting saved by the hierarchy constraint we distinguish between parameter sparsity the number of nonzero coefficients and practical sparsity the number of raw variables one must measure to make a new prediction hierarchy focuses on the latter which is more closely tied to important data collection concerns such as cost time and effort we develop an algorithm available in the r package hiernet and perform an empirical study of our method

the center of the enveloping algebra of the p lie algebras n n n when p divides n abstract let be a reductive lie algebra over an algebraically closed field f with char f p 0 suppose g satisfies jantzen s standard assumptions then the structure of z the center of the enveloping algebra is described by the extended veldkamp s theorem we examine here the deviation of z from this theorem in case or and p n it is shown that veldkamp s description is valid for this implies that friedlander parshall donkin decomposition theorem for holds in case p is good for a semi simple simply connected g excluding if p 2 a 1 factors of g in case or we prove a fiber product theorem for a polynomial extension of z however veldkamp s description mostly fails for and in particular z is not cohen macaulay if n 4 in both cases contrary to a result of kac weisfeiler we show for an odd prime p that and do not generate we also show for that the codimension of the non azumaya locus of z is at least 2 if n 3 and exceeds 2 if n 4 this refutes a conjecture of brown goodearl we then show that z is factorial excluding thus confirming a conjecture of premet tange we also verify humphreys conjecture on the parametrization of blocks in case p is good for g

invariant fields of finite irreducible reflection groups we prove the following result if g is a finite irreducible reflection group defined over a base field k then the invariant field of g is purely transcendental over k even if g is divisible by the characteristic of k it is well known that in the above situation the invariant ring is in general not a polynomial ring so the question whether at least the invariant field is purely transcendental noether s problem is quite natural

the depth of invariant rings and cohomology abstract let g be a finite group acting linearly on a vector space v over a field k of positive characteristic p and let p g be a sylow p subgroup ellingsrud and skjelbred compositio math 41 1980 233 244 proved the lower bound depth k v g min dim v p 2 dim v for the depth of the invariant ring with equality if g is a cyclic p group let us call the pair g v flat if equality holds in the above in this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness for g of order not divisible by p2 it ensues that g v is flat if and only if h1 g k v 0 or dim vp 2 dim v we obtain a formul a for the depth of the invariant ring in the case that g permutes a basis of v and has order not divisible by p2 in this situation g v is usually not flat moreover we introduce the notion of visible flatness of pairs g v and prove that this implies flatness for example the groups sl2 q so3 q su3 q sz q and r q with many interesting representations in defining characteristic are visibly flat in particular if g sl2 q and v is the space of binary forms of degree n or a direct sum of such spaces then g v is flat for all q pr with the exception of a finite number of primes p along the way we obtain results about the buchsbaum property of invariant rings and about the depth of the cohomology modules hi g k v we also determine the support of the positive cohomology h g k v as a module over k v g in the appendix the visibly flat pairs g v of a group with bn pair and an irreducible representation are classified

symmetric powers of modular representations for groups with a sylow subgroup of prime order abstract let v be a representation of a finite group g over a field of characteristic p if p does not divide the group order then molien s formula gives the hilbert series of the invariant ring in this paper we find a replacement of molien s formula which works in the case that g is divisible by p but not by p 2 we also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of v into indecomposables our methods can be applied to determine the depth of the invariant ring without computing any invariants this leads to a proof of a conjecture of the second author on certain invariants of gl 2 p

the structure of a random graph at the point of the phase transition consider the random graph models g n edges m and g n prob edge p with m m n 1 n 1 3 n 2 and p p n 1 an 1 3 n for 1 1 define an i component of a random graph as a component which has exactly i more edges than vertices call an i component with i 1 a complex component for both models we show that when a is constant the expected number of complex components is bounded almost surely a s each of these components if any exist has size of order n2 3 and the maximum value of i is bounded in probability we prove that a s the largest suspended tree in each complex component has size of order n2 3 and deletion of all suspended trees results in a smoothed graph of size of order n 3 with the maximum vertex degree 3 the total number of branching vertices i e of degree 3 is bounded in probability thus each complex component is almost surely topologically equivalent to a 3 regular multigraph of a uniformly bounded size lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3 we find a relatively simple integral formula for the limit distribution of the numbers of complex components which implies in particular that all values of the complexity spectrum have positive limiting probabilities we also answer questions raised by erdos and renyi back in 1960 it is proven that there exists p a the limiting planarity probability with 0 p a 1 p oo 1 p oo 0 in particular g n m g n p resp is almost surely nonplanar iff m n 2 n 213 00 np i n 1 3 oo resp

depth and cohomological connectivity in modular invariant theory the cohomological connectivity of the symmetric algebra s v to be the smallest positive integer m such that h m g s v 6 0 we show that min dimk v p m 1 dimk v is a lower bound for the depth of s v g we characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion in particular we show that if g is p nilpotent and p is cyclic then for any modular representation the depth of s v g is min dimk v p 2 dimk v

loop correlations in random wire models we introduce a family of loop soup models on the hypercubic lattice the models involve links on the edges and random pairings of the link endpoints on the sites we conjecture that loop correlations of distant points are given by poisson dirichlet correlations in dimensions three and higher we prove that in a specific random wire model that is related to the classical xy spin system the probability that distant sites form an even partition is given by the poisson dirichlet counterpart

trace pursuit a general framework for model free variable selection abstractwe propose trace pursuit for model free variable selection under the sufficient dimension reduction paradigm two distinct algorithms are proposed stepwise trace pursuit and forward trace pursuit stepwise trace pursuit achieves selection consistency with fixed p forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality the screening consistency property of forward trace pursuit based on sliced inverse regression is established finite sample performances of trace pursuit and other model free variable selection methods are compared through numerical studies supplementary materials for this article are available online

two dimensional vector invariant rings of abelian p groups abstract for any faithful representation v of a non trivial p group over a field of characteristic p 0 it is known that the ring of vector invariants of m copies of v is not cohen macaulay if m 3 however much less is known about the case m 2 in this paper we show that if m 2 and the group is an abelian p group then the ring of invariants of 2v is a complete intersection in some cases and is not cohen macaulay in most cases as a corollary we obtain that if the field is f p and the ring of invariants of the representation v is a polynomial ring then the ring of invariants of 2v is either a complete intersection or not cohen macaulay

misspecified nonconvex statistical optimization for sparse phase retrieval existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models to address this issue we take a first step towards taming model misspecification by studying the high dimensional sparse phase retrieval problem with misspecified link functions in particular we propose a simple variant of the thresholded wirtinger flow algorithm that given a proper initialization linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions we further provide extensive numerical experiments to support our theoretical findings

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

non cohen macaulay invariant rings of infinite groups abstract we give explicit examples of invariant rings that are not cohen macaulay for all classical groups sl n k gl n k sp 2 n k so n k and o n k where k is an algebraically closed field of positive characteristic we prove that every non trivial unipotent group over k has representations such that the invariant ring is not cohen macaulay

on the depth of invariant rings of infinite groups abstract let k be an algebraically closed field for a finitely generated graded commutative k algebra r let cmdef r dim r depth r denote the cohen macaulay defect of r let g be a linear algebraic group over k that is reductive but not linearly reductive we show that there exists a faithful rational representation v of g which we will give explicitly such that cmdef k v k g k 2 for all k n

cutoff for conjugacy invariant random walks on the permutation group we prove a conjecture raised by the work of diaconis and shahshahani z wahrscheinlichkeitstheorie verwandte geb 57 2 159 179 1981 about the mixing time of random walks on the permutation group induced by a given conjugacy class to do this we exploit a connection with coalescence and fragmentation processes and control the kantorovich distance by using a variant of a coupling due to oded schramm as well as contractivity of the distance recasting our proof in the language of ricci curvature our proof establishes the occurrence of a phase transition which takes the following form in the case of random transpositions at time cn 2 the curvature is asymptotically zero for c le 1 and is strictly positive for c 1

moderately ramified actions in positive characteristic in characteristic 2 and dimension 2 wild mathbb z 2 mathbb z actions on k u v ramified precisely at the origin were classified by artin who showed in particular that they induce hypersurface singularities we introduce in this article a new class of wild quotient singularities in any characteristic p 0 and dimension n ge 2 arising from certain non linear actions of mathbb z p mathbb z on the formal power series ring k u 1 dots u n these actions are ramified precisely at the origin and their rings of invariants in dimension 2 are hypersurface singularities with an equation of a form similar to the form found by artin when p 2 in higher dimension the rings of invariants are not local complete intersection in general but remain quasi gorenstein we establish several structure results for such actions and their corresponding rings of invariants

a phase transition in the random transposition random walk our work is motivated by bourque pevzner s simulation study of the effectiveness of the parsimony method in studying genome rearrangement and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on n elements starting from the identity let d t be the minimum number of transpositions needed to go back to the identity element from the location at time t d t undergoes a phase transition for 0 1 d cn 2 u c n where u is an explicit function satisfying u x x 2 moreover we describe the fluctuations of d cn 2 about its mean at each of the three stages subcritical critical and supercritical the techniques used involve viewing the cycles in the random permutation as a coagulation fragmentation process and relating the behavior to the erd s renyi random graph model

on cohen macaulay rings of invariants abstract we investigate the transfer of the cohen macaulay property from a commutative ring to a subring of invariants under the action of a finite group our point of view is ring theoretic and not a priori tailored to a particular type of group action as an illustration we discuss the special case of multiplicative actions that is actions on group algebras k z n via an action on z n

the poisson dirichlet law is the unique invariant distribution for uniform split merge transformations we consider a markov chain on the space of countable partitions of the interval 0 1 obtained first by size biased sampling twice allowing repetitions and then merging the parts if the sampled parts are distinct or splitting the part uniformly if the same part was sampled twice we prove a conjecture of vershik stating that the poisson dirichlet law with parameter 1 is the unique invariant distribution for this markov chain our proof uses a combination of probabilistic combinatoric and representation theoretic arguments

projections of a general binary model on a logistic regression abstract we consider a general binary model for which conditional probability of success given vector of predictors x equals q 1 t x k t x and a family of possibly misspecified logistic regressions fitted to it in the case when x satisfies linearity condition we show that their algebraic structure is uniquely determined and that the vector corresponding to kullback leibler projection on this family is a linear combination of 1 k this generalizes the known result proved by p ruud for k 1 which says that the true and projected vectors are collinear it also follows that the projected vector has the same direction as the first canonical vector which justifies frequent observations that logistic fit yields well performing classifiers even if misspecification is expected in the special case of additive binary model with multivariate normal predictors and when response function q is a convex combination of univariate responses we show that the variance of t x is not larger than the maximal variance of the projected linear combinations for the corresponding univariate problems in the case of balanced additive logistic model it follows that the contribution of i to is bounded by the corresponding coefficient in the convex representation of response function q

invariants of modular groups for a faithful linear representation of a finite group g over a field of characteristic p we study the ring of invariants we especially study the polynomial and cohen macaulay properties of the invariant ring we first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the hilbert ideal conjecture is true for a class of groups in particular we prove that the conjecture is true for vector invariant rings of abelian reflection p groups then we study the relationships between the invariant ring of g and that of a subgroup of g finally we study the invariant rings of affine groups and show that over a finite field if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group

separating invariants and local cohomology the study of separating invariants is a recent trend in invariant theory for a finite group acting linearly on a vector space a separating set is a set of invariants whose elements separate the orbits of g in some ways separating sets often exhibit better behavior than generating sets for the ring of invariants we investigate the least possible cardinality of a separating set for a given g action our main result is a lower bound that generalizes the classical result of serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections we find these bounds to be sharp in a wide range of examples

robust variable and interaction selection for logistic regression and general index models under the logistic regression framework we propose a forward backward method soda for variable selection with both main and quadratic interaction terms in the forward stage soda adds in predic

the hyperbolic geometry of random transpositions turn the set of permutations of n objects into a graph g n by connecting two permutations that differ by one transposition and let t be the simple random walk on this graph in a previous paper berestycki and durrett in discrete random walks 2005 17 26 showed that the limiting behavior of the distance from the identity at time cn 2 has a phase transition at c 1 here we investigate some consequences of this result for the geometry of g n our first result can be interpreted as a breakdown for the gromov hyperbolicity of the graph as seen by the random walk which occurs at a critical radius equal to n 4 let t be a triangle formed by the origin and two points sampled independently from the hitting distribution on the sphere of radius an for a constant 0 0 whereas it is always o n thick when a 1 4 we also show that the hitting distribution of the sphere of radius an is asymptotically singular with respect to the uniform distribution finally we prove that the critical behavior of this gromov like hyperbolicity constant persists if the two endpoints are sampled from the uniform measure on the sphere of radius an however in this case the critical radius is a 1 log 2

depth and detection in modular invariant theory let g be a finite group acting linearly on a vector space v over a field of characteristic p dividing the group order and let r denote s v we study the r g modules h i g r for i 0 with r g itself as a special case there are lower bounds for depth of h i g r and for depth r g we show that a certain sufficient condition for their attainment due to fleischmann kemper and shank may be modified to give a condition which is both necessary and sufficient we apply our main result to classify the representations of the klein four group for which depth r g attains its lower bound we also use our new condition to show that the if g p q with p a p group and q an abelian p group then the depth of r g attains its lower bound if and only if the depth of r p does so

estimation and inference in generalized additive coefficient models for nonlinear interactions with high dimensional covariates in the low dimensional case the generalized additive coefficient model gacm proposed by xue and yang statist sinica16 2006 1423 1446 has been demonstrated to be a powerful tool for studying nonlinear interaction effects of variables in this paper we propose estimation and inference procedures for the gacm when the dimension of the variables is high specifically we propose a groupwise penalization based procedure to distinguish significant covariates for the large p small n setting the procedure is shown to be consistent for model structure identification further we construct simultaneous confidence bands for the coefficient functions in the selected model based on a refined two step spline estimator we also discuss how to choose the tuning parameters to estimate the standard deviation of the functional estimator we adopt the smoothed bootstrap method we conduct simulation experiments to evaluate the numerical performance of the proposed methods and analyze an obesity data set from a genome wide association study as an illustration

model selection for high dimensional quadratic regression via regularization abstractquadratic regression qr models naturally extend linear models by considering interaction effects between the covariates to conduct model selection in qr it is important to maintain the hierarchical model structure between main effects and interaction effects existing regularization methods generally achieve this goal by solving complex optimization problems which usually demands high computational cost and hence are not feasible for high dimensional data this article focuses on scalable regularization methods for model selection in high dimensional qr we first consider two stage regularization methods and establish theoretical properties of the two stage lasso then a new regularization method called regularization algorithm under marginality principle ramp is proposed to compute a hierarchy preserving regularization solution path efficiently both methods are further extended to solve generalized qr models numerical results are also shown to demonstrate performance of the methods s

modular group actions on algebras and p local galois extensions for finite groups abstract let k be a field of positive characteristic p and let g be a finite group in this paper we study the category ts g of finitely generated commutative k algebras a on which g acts by algebra automorphisms with surjective trace if a k x the ring of regular functions of a variety x then trace surjective group actions on a are characterized geometrically by the fact that all point stabilizers on x are p subgroups or equivalently that a p a is a galois extension for every sylow p group of g we investigate categorical properties of ts g using a version of frobenius reciprocity for group actions on k algebras which is based on tensor induction for modules we also describe projective generators in ts g extending and generalizing the investigations started in 7 8 9 in the case of p groups as an application we show that for an abelian or p elementary group g and k large enough there is always a faithful possibly nonlinear action on a polynomial ring such that the ring of invariants is also a polynomial ring this would be false for linear group actions by a result of serre if a is a normal domain and g aut k a an arbitrary finite group we show that a o p g is the integral closure of k soc a the subalgebra of a generated by the simple kg submodules in a for p solvable groups this leads to a structure theorem on trace surjective algebras generalizing the corresponding result for p groups in 8

brownian excursions critical random graphs and the multiplicative coalescent let b t s 0 s be reflecting inhomogeneous brownian motion with drift t s at time s started with b t 0 0 consider the random graph script g sign n n 1 tn 4 3 whose largest components have size of order n 2 3 normalizing by n 2 3 the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of b t corollary 2 the dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process the states of this process are vectors x of nonnegative real cluster sizes x i and clusters with sizes x i and x j merge at rate x i x j the multiplicative coalescent is shown to be a feller process on l 2 the random graph limit specifies the standard multiplicative coalescent which starts from infinitesimally small clusters at time the existence of such a process is not obvious

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

an invariance principle for certain dependent sequences let x 1 x 2 cdots be a strictly stationary second order sequence which is associated i e is such that any two coordinatewise nondecreasing functions of the x i s of finite variance are nonnegatively correlated if sum j operatorname cov x 1 x j infty then the partial sum processes w n t defined in the usual way so that w n m n x 1 cdots x m me x 1 sqrt n for m 1 2 cdots converge in distribution on c lbrack 0 t rbrack to a wiener process this result is based on two general theorems concerning associated random variables which are of independent interest

on the branch locus of quotients by finite groups and the depth of the algebra of invariants abstract let a bg where b is a noetherian algebra over a field k of characteristic p 0 and g is a finite group such that p divides g we give estimates for the depth of a in terms of the codimension of the branch locus of the extension b a

a generalized levene s scale test for variance heterogeneity in the presence of sample correlation and group uncertainty we generalize levene s test for variance scale heterogeneity between k groups for more complex data when there are sample correlation and group membership uncertainty following a two stage regression framework we show that least absolute deviation regression must be used in the stage 1 analysis to ensure a correct asymptotic k 12 k 1 distribution of the generalized scale gs test statistic we then show that the proposed gs test is independent of the generalized location test under the joint null hypothesis of no mean and no variance heterogeneity consequently we generalize the recently proposed joint location scale gjls test valuable in settings where there is an interaction effect but one interacting variable is not available we evaluate the proposed method via an extensive simulation study and two genetic association application studies

loci in quotients by finite groups pointwise stabilizers and the buchsbaum property let k v g be the invariant ring of a nite linear group g gl v and let gu be the pointwise stabilizer of a subspace u v we prove that the following numbers associated to the invariant ring do not increase if one passes from k v g to k v gu the minimal number of homogeneous generators the maximal degree of the generators the number of syzygies and other betti numbers the complete intersection defect the dierence between depth and dimension and the type from this theorems of steinberg serre nakajima kac and watanabe and the author follow which say that if k v g is a polynomial ring a hypersurface a complete intersection or cohen macaulay then the same is true for k v gu furthermore k v gu inherits the gorenstein property from k v g we give an algorithm which transforms generators of k v g into generators of k v g u letp be one of the properties mentioned above we consider the locus ofp in v g spec k v g and prove that for x2 spec k v with image x 0 in v g the local ring k v g0 has the property p if and only if p holds for the invariant ring k v g x of the point stabilizer using this we prove that the non cohen macaulay locus in v g is either empty or it has dimension at least one and codimension at least 3 from this we deduce that k v g is buchsbaum if and only if it is cohen macaulay this proves a conjecture of campbell et al

interaction pursuit in high dimensional multi response regression via distance correlation feature interactions can contribute to a large proportion of variation in many prediction models in the era of big data the coexistence of high dimensionality in both responses and covariates poses unprecedented challenges in identifying important interactions in this paper we suggest a two stage interaction identification method called the interaction pursuit via distance correlation ipdc in the setting of high dimensional multi response interaction models that exploits feature screening applied to transformed variables with distance correlation followed by feature selection such a procedure is computationally efficient generally applicable beyond the heredity assumption and effective even when the number of responses diverges with the sample size under mild regularity conditions we show that this method enjoys nice theoretical properties including the sure screening property support union recovery and oracle inequalities in prediction and estimation for both interactions and main effects the advantages of our method are supported by several simulation studies and real data analysis

mixing times for random k cycles and coalescence fragmentation chains let sn be the permutation group on n elements and consider a random walk on sn whose step distribution is uniform on k cycles we prove a well known conjecture that the mixing time of this process is 1 k nlog n with threshold of width linear in n our proofs are elementary and purely probabilistic and do not appeal to the representation theory of sn

the evolution of the random reversal graph genomes are typically represented as signed permutations where each integer corresponds to a single gene and its sign indicates the orientation of the gene in this paper we study genome rearrangements via reversals sankoff and blanchette 1999 20 flipping entire segments of genes and changing their orientations this abstraction leads to the notion of the reversal graph in which two signed permutations are neighbors if they differ by one reversal we identify the evolution of multiple genomes within the random reversal graph i e the probability space consisting of subgraphs over signed permutations obtained by selecting edges reversals with independent probability n which is an important framework for the analysis of genome rearrangements in the sense of relating the reversal rate n with the global evolution of genomes our main result shows that the structure of the random reversal graph changes dramatically at n 1 n 1 2 for n 1 n 1 2 the random graph consists of components of size at most o n ln n a s and for 1 n 1 2 there emerges a unique largest component of size 2 n n a s where is the survival probability of a certain branching process this giant component is furthermore dense in the reversal graph

correlation pursuit forward stepwise variable selection for index models summary a stepwise procedure correlation pursuit cop is developed for variable selection under the sufficient dimension reduction framework in which the response variable y is influenced by the predictors through an unknown function of a few linear combinations of them unlike linear stepwise regression cop does not impose a special form of relationship such as linear between the response variable and the predictor variables the cop procedure selects variables that attain the maximum correlation between the transformed response and the linear combination of the variables various asymptotic properties of the cop procedure are established and in particular its variable selection performance under a diverging number of predictors and sample size is investigated the excellent empirical performance of the cop procedure in comparison with existing methods is demonstrated by both extensive simulation studies and a real example in functional genomics