the capelli identity tube domains and the generalized laplace transform the key result in this paper is a proof of a large class of identities generalizing the capelli identity cl the capelli identity is a centerpiece of 19th century invariant theory it asserts the equality of two differential operators on the n2 dimensional space m n r of n x n real matrices let e denote the ijth elementary matrix let xii denote the linear functions on m n r dual to e and write 3 for the partial differential operator a ax then this identity says

bases for some reciprocity algebras ii abstract we construct bases for the stable branching algebras for the symmetric pairs gl n o n o n m o n o m and sp 2 n gl n

standard monomial theory for g2 let g be a semi simple algebraic group and b a borel subgroup let x be a schubert variety in g b let l be an ample line bundle on g b as well as its restriction to x a standard monomial theory for schubert varieties in g b is developed in 9 11 8 10 as a generalization of the classical hodge young theory cf 3 4 this theory consists in the construction of a characteristic free basis for h0 x l this theory is extended to schubert varieties in the infinite dimensional flag variety ln b in 13 see also 12 in this paper we extend the theory to schubert varieties in the infinite dimensional flag variety p2n b

rates of convergence for bayes and maximum likelihood estimation for mixture of normal densities we study the rates of convergence of the maximum likelihood estimator mle and posterior distribution in density estimation problems where the densities are location or location scale mixtures of normal distributions with the scale parameter lying between two positive numbers the true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having sub gaussian tails we obtain bounds for hellinger bracketing entropies for this class and from these bounds we deduce the convergence rates of sieve mles in hellinger distance the rate turns out to be log n n where k 1 is a constant that depends on the type of mixtures and the choice of the sieve next we consider a dirichlet mixture of normals as a prior on the unknown density we estimate the prior probability of a certain kullback leibler type neighborhood and then invoke a general theorem that computes the posterior convergence rate in terms the growth rate of the hellinger entropy and the concentration rate of the prior the posterior distribution is also seen to converge at the rate log n n in where k now depends on the tail behavior of the base measure of the dirichlet process

species sampling models consistency for the number of species this paper considers species sampling models using constructions that arise from bayesian nonparametric prior distributions a discrete random measure used to generate a species sampling model can have either a countable infinite number of atoms which has been the emphasis in the recent literature or a finite number of atoms k while allowing k to be assigned a prior probability distribution on the positive integers it is the latter class of model we consider here due to the interpretation of k as the number of species we demonstrate the consistency of the posterior distribution of k as the sample size increases copyright 2013 oxford university press

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

asymptotic behaviour of the posterior distribution in overfitted mixture models abstract summary we study the asymptotic behaviour of the posterior distribution in a mixture model when the number of components in the mixture is larger than the true number of components a situation which is commonly referred to as an overfitted mixture we prove in particular that quite generally the posterior distribution has a stable and interesting behaviour since it tends to empty the extra components this stability is achieved under some restriction on the prior which can be used as a guideline for choosing the prior some simulations are presented to illustrate this behaviour

toric degeneration of schubert varieties and gelfand tsetlin polytopes this note constructs the flat toric degeneration of the manifold fln of flags in cn due to gonciulea and lakshmibai transform groups 1 3 1996 215 as an explicit git quotient of the grobner degeneration due to knutson and miller grobner geometry of schubert polynomials ann math 2 to appear this implies that schubert varieties degenerate to reduced unions of toric varieties associated to faces indexed by rc graphs reduced pipe dreams in the gelfand tsetlin polytope our explicit description of the toric degeneration of fln provides a simple explanation of how gelfand tsetlin decompositions for irreducible polynomial representations of gln arise via geometric quantization

relevant statistics for bayesian model choice the choice of the summary statistics that are used in bayesian inference and in particular in approximate bayesian computation algorithms has bearings on the validation of the resulting inference those statistics are nonetheless customarily used in approximate bayesian computation algorithms without consistency checks we derive necessary and sufficient conditions on summary statistics for the corresponding bayes factor to be convergent namely to select the true model asymptotically those conditions which amount to the expectations of the summary statistics differing asymptotically under the two models are quite natural and can be exploited in approximate bayesian computation settings to infer whether or not a choice of summary statistics is appropriate via a monte carlo validation

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

testing the order of a model using locally conic parametrization population mixtures and stationary arma processes in this paper we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model we give two major applications the case where the number of populations has to be tested in a mixture and the case of stationary arma p q processes where the order p q has to be tested we give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model in the case of order selection for armas the asymptotic distribution is invariant with respect to the parameters generating the process a locally conic parametrization is a key tool in deriving the limiting distributions it allows one to discover the deep similarity between the two problems

convergence of latent mixing measures in finite and infinite mixture models this paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models using transportation distances i e wasserstein metrics the relationship between wasserstein distances on the space of mixing measures and f divergence functionals such as hellinger and kullback leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions convergence in wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed convergence rates of posterior distributions for latent mixing measures are established for both finite mixtures of multivariate distributions and infinite mixtures based on the dirichlet process

quivers invariants and quotient correspondence abstract this paper studies the geometric and algebraic aspects of the moduli spaces of quivers of fence type we first provide two quotient presentations of the quiver varieties and interpret their equivalence as a generalized gelfand macpherson correspondence next we introduce parabolic quivers and extend the above from the actions of reductive groups to the actions of parabolic subgroups interestingly the above geometry finds its natural counterparts in the representation theory as the branching rules and transfer principle in the context of the reciprocity algebra the last half of the paper establishes this connection

chern class formulas for quiver varieties our goal in this paper is to prove a formula for the general degeneracy locus xr associated to an oriented quiver of type an if we are given a sequence of vector bundles and vector bundle maps e0 1 e1 2 e2 y y eny1 n en on an algebraic variety x and a collection r rij 0 i j n of non negative integers there is a degeneracy locus xr xr e xr e de ned by xr fx 2 x j rank ei x ej x rij 8 i jg 1 1

young tableaux gelfand patterns and branching rules for classical groups abstract young tableaux and gelfand patterns which combinatorially describe characters of irreducible finite dimensional representations of complex classical lie groups are presented some branching rules and or character recursions are given explicit combinatorial descriptions these recursions are derived from manipulations of determinantal character formulas

a survey of search directions in interior point methods for linear programming a basic characteristic of an interior point algorithm for linear programming is the search direction many papers on interior point algorithms only give an implicit description of the search direction in this report we derive explicit expressions for the search directions used in many well known algorithms comparing these explicit expressions gives a good insight into the similarities and differences between the various algorithms moreover we give a survey of projected gradient and newton directions for all potential and barrier functions this is done both for the affine and projective variants

nonparametric bayes modeling of populations of networks abstractreplicated network data are increasingly available in many research fields for example in connectomic applications interconnections among brain regions are collected for each patient under study motivating statistical models which can flexibly characterize the probabilistic generative mechanism underlying these network valued data available models for a single network are not designed specifically for inference on the entire probability mass function of a network valued random variable and therefore lack flexibility in characterizing the distribution of relevant topological structures we propose a flexible bayesian nonparametric approach for modeling the population distribution of network valued data the joint distribution of the edges is defined via a mixture model that reduces dimensionality and efficiently incorporates network information within each mixture component by leveraging latent space representations the formulation leads to an efficient gibbs sampler and provides simple and c

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

a bayesian information criterion for singular models summary we consider approximate bayesian model choice for model selection problems that involve models whose fisher information matrices may fail to be invertible along other competing submodels such singular models do not obey the regularity conditions underlying the derivation of schwarz s bayesian information criterion bic and the penalty structure in bic generally does not reflect the frequentist large sample behaviour of the marginal likelihood although large sample theory for the marginal likelihood of singular models has been developed recently the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning guided by examples such as determining the number of components in mixture models the number of factors in latent factor models or the rank in reduced rank regression we propose a resolution to this paradox and give a practical extension of bic for singular model selection problems

bases for some reciprocity algebras i for a complex vector space v let p v be the algebra of polynomial functions on v in this paper we construct bases for the algebra of all gl n c x gl m1 c x gl m2 c x x gl mr c highest weight vectors in p c n c m where m m 1 m r and mj n for all 1 j r and the algebra of gl n c x gl k c x gl 1 c highest weight vectors in p c n c k c n c l

schubert polynomials and the nilcoxeter algebra abstract schubert polynomials were introduced and extensively developed by lascoux and schutzenberger after an earlier less combinatorial version had been considered by bernstein gelfand and gelfand and demazure we give a new development of the theory of schubert polynomials based on formal computations in the algebra of operators u1 u2 satisfying the relations u2i 0 uiuj ujui if i j 2 and uiui 1ui ui 1uiui 1 we call this algebra the nilcoxeter algebra of the symmetric group s n our development leads to simple proofs of many standard results in particular a symmetry of the stable schubert polynomials fw b an explicit combinatorial formula for schubert polynomials due to billey jockusch and stanley c the cauchy formula for schubert polynomials and d a formula of macdonald for s w 1 1 our main new result is a proof of a conjectured q analogue of d due to macdonald which gives a formula for s w 1 q q2

a cayley hamilton theorem for the skew capelli elements for the central elements of the universal enveloping algebra of the lie algebra gln named the skew capelli elements a cayley hamilton type formula is given its classical counterpart is an elementary formula for two alternating matrices as a byproduct of the main result the description of the skew capelli elements given by k kinoshita and m wakayama explicit capelli identities for skew symmetric matrices proc edinburgh math soc to appear is deduced naturally

theta lifting of unitary lowest weight modules and their associated cycles we consider a reductive dual pair g g in the stable range with g the smaller member and of hermitian symmetric type we study theta lifting of holomorphic nilpotent k c orbits in relation to theta lifting of unitary lowest weight modules of g we determine the associated cycles of all such representations in particular we prove that the multiplicity in the associated cycle is preserved under theta lifting

the gelfand cetlin system and quantization of the complex flag manifolds the construction of angle action variables for collective completely integrable systems is described and the associated bohr sommerfeld sets are determined the quantization method of sniatycki applied to such systems gives formulas for multiplicities for the gelfand cetlin system on complex flag manifolds we show that these formulas give the correct answers for the multiplicities of the associated representations

a survey of the implications of the behavior of the central path for the duality theory of linear programming the literature in the field of interior point methods for linear programming has been almost exclusively algorithmic oriented very few contributions have been made towards the theory of linear programming itself in particular none of them offer a simple self contained introduction to the theory of linear programming and linear inequalities the purpose of this paper is to show that the interior point methodology can be used to introduce the field of linear programming starting from scratch and using only elementary results from calculus and linear algebra we prove that for every value of the barrier parameter the logarithmic barrier function for the primal dual problem has a unique minimizer and that the path of these minimizers the central path converges to a strictly complementary pair of optimal solutions these results were proved more than a decade ago with advanced mathematical arguments our proofs are new they are also simpler and often more natural than the ones currently known they p

erratum to remarks on classical invariant theory although this paper circulated as a preprint for 12 years an error in the argument for the capelli identity went unnoticed until professor m wakayama read the paper in order to review it i thank him for his care and for bringing this error to my attention the error occurs on page 566 the result at issue that det ii is central in gyn is true and classical and the general line of argument is sufficient to prove it but the details of the computation are described incorrectly the identity on line 3 page 566 is correct but the description of the calculation drawn from it is not instead of terms of the form

asymptotics for likelihood ratio tests under loss of identifiability this paper describes the large sample properties of the likelihood ratio test statistic lrts when the parameters characterizing the true null distribution are not unique it is well known that the classical asymptotic theory for the likelihood ratio test does not apply to such problems and the lrts may not have the typical chi squared type limiting distribution this paper establishes a general quadratic approximation of the log likelihood ratio function in a hellinger neighborhood of the true density which is valid with or without loss of identifiability of the true distribution under suitable conditions the asymptotic null distribution of the lrts under loss of identifiability can be obtained by maximizing the quadratic form these results extend the work of chernoff and le cam in particular applications to testing the number of mixture components in finite mixture models are discussed

a reciprocity law and the skew pieri rule for the symplectic group we use the theory of skew duality to show that decomposing the tensor product of k irreducible representations of the symplectic group sp2m sp2m is equivalent to branching from sp2n to sp2n1 sp2nk where n n1 nk are positive integers such that n n1 nk and the njs depend on m as well as the representations in the tensor product using this result and a work of lepowsky we obtain a skew pieri rule for sp2m i e a description of the irreducible decomposition of the tensor product of an irreducible representation of the symplectic group sp2m with a fundamental representation

standard monomial bases and degenerations of som c representations we study explicit weight bases for the finite dimensional representations of so m using a flat degeneration technique we describe the representation spaces via a lattice cone associated with the gelfand tsetlin patterns for gl m combinatorial descriptions of the weight bases for so m relative to those for gl m will be given

the capelli problem for gl m n and the spectrum of invariant differential operators abstract the capelli problem for the symmetric pairs gl gl gl gl o and gl sp is closely related to the theory of jack polynomials and shifted jack polynomials for special values of the parameter see 12 15 14 18 in this paper we extend this connection to the lie superalgebra setting namely to the supersymmetric pairs g k gl m 2 n osp m 2 n and gl m n gl m n gl m n acting on w s 2 c m 2 n and c m n c m n to achieve this goal we first prove that the center of the universal enveloping algebra of the lie superalgebra g maps surjectively onto the algebra pd w g of g invariant differential operators on the superspace w thereby providing an affirmative answer to the abstract capelli problem for w our proof works more generally for gl m n acting on s 2 c m n and is new even for the ordinary cases m 0 or n 0 considered by howe and umeda in 9 we next describe a natural basis d of pd w g that we call the capelli basis using the above result on the abstract capelli problem we generalize the work of kostant and sahi 12 15 20 by showing that the spectrum of d is given by a polynomial c which is characterized uniquely by certain vanishing and symmetry properties we further show that the top homogeneous parts of the eigenvalue polynomials c coincide with the spherical polynomials d which arise as radial parts of k spherical vectors of finite dimensional g modules and which are super analogues of jack polynomials this generalizes results of knop and sahi 14 finally we make a precise connection between the polynomials c and the shifted super jack polynomials of sergeev and veselov 25 for special values of the parameter we show that the two families are related by a change of coordinates that we call the frobenius transform

okounkov bodies and toric degenerations let vardelta be the okounkov body of a divisor d on a projective variety x we describe a geometric criterion for vardelta to be a lattice polytope and show that in this situation x admits a flat degeneration to the corresponding toric variety this degeneration is functorial in an appropriate sense

minimal graded lie algebras and representations of quadratic algebras abstract let g 0 b 0 be a quadratic lie algebra i e a lie algebra g 0 with a non degenerate symmetric invariant bilinear form b 0 and let v be a finite dimensional representation of g 0 we define on g 0 b 0 v v g 0 v a structure of local lie algebra in the sense of kac 4 where the bracket between g 0 and v resp v is given by the representation resp and where the bracket between v and v depends on b 0 and this implies the existence of two z graded lie algebras g m a x g 0 b 0 v and g m i n g 0 b 0 v whose local part is g 0 b 0 v we investigate these graded lie algebras more specifically in the case where g 0 is reductive roughly speaking the map g 0 b 0 v g m i n g 0 b 0 v is a bijection between triplets and a class of graded lie algebras we show that the existence of associated sl 2 triples is equivalent to the existence of non trivial relative invariants on some orbit and we define the graded lie algebras of polynomial type which give rise to some dual airs

a basis for the gln tensor product algebra abstract this paper focuses on the gl n tensor product algebra which encapsulates the decomposition of tensor products of arbitrary irreducible representations of gl n we will describe an explicit basis for this algebra this construction relates directly with the combinatorial description of littlewood richardson coefficients in terms of littlewood richardson tableaux philosophically one may view this construction as a recasting of the littlewood richardson rule in the context of classical invariant theory

multiplicities of points on a schubert variety in a minuscule gp in this paper we prove the results announced in 13 let g be a semi simple simply connected algebraic group defined over an algebraically closed field k let t be a maximal torus b a bore1 subgroup b 3 t let w be the weyl group of g let r resp r be the set of roots resp positive roots relative to t resp b let s be the set of simple roots in r let p be a maximal parabolic subgroup in g with associated fundamental weight w let w be the weyl group of p and wp be the set of minimal representatives of w w for w e wp let e w be the point and x w the schubert variety in g p associated to w in this paper we deter mine the multiplicity m w of x w at e z where e z e x w for all minuscule p s and also for p pgn g being of type c here pun denotes the maximal parabolic subgroup obtained by omitting a the determina tion of m w is done as follows let l be the ample generator of pic g p a basis has been constructed for x w l in terms of standard monomials on x w cf 16 11 i let u be the unipotent subgroup of g generated by u et r rp here r denotes the set of roots of p and u denotes the unipotent subgroup of g associated to tl e r then u e r gives an affme neighborhood of e z in g p let a be the affine algebra of u e z and a a where is the ideal of elements of a that vanish on x w n u e z let m be the maximal ideal in a h corresponding to e r then using the results of 16 111 we obtain a basis of m f this enables us to obtain an inductive formula for f the hilbert polynomial of x w at e t cf corollaries 3 8 and 4 11 and also express m w in terms of m w s x w s being the schubert divisors in x w such that e t e x w cf theorems 3 7 and 4 10 using this we

superbosonization of invariant random matrix ensembles superbosonization is a new variant of the method of commuting and anti commuting variables as used in studying random matrix models of disordered and chaotic quantum systems we here give a concise mathematical exposition of the key formulas of superbosonization conceived by analogy with the bosonization technique for dirac fermions the new method differs from the traditional one in that the superbosonization field is dual to the usual hubbard stratonovich field the present paper addresses invariant random matrix ensembles with symmetry group un on or uspn giving precise definitions and conditions of validity in each case the method is illustrated at the example of wegner s n orbital model superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non gaussian and or non invariant type

nonlinear rescaling vs smoothing technique in convex optimization we introduce an alternative to the smoothing technique approach for constrained optimization as it turns out for any given smoothing function there exists a modification with particular properties we use the modification for nonlinear rescaling nr the constraints of a given constrained optimization problem into an equivalent set of constraints the constraints transformation is scaled by a vector of positive parameters the lagrangian for the equivalent problems is to the correspondent smoothing penalty functions as augmented lagrangian to the classical penalty function or mbfs to the barrier functions moreover the lagrangians for the equivalent problems combine the best properties of quadratic and nonquadratic augmented lagrangians and at the same time are free from their main drawbacks sequential unconstrained minimization of the lagrangian for the equivalent problem in primal space followed by both lagrange multipliers and scaling parameters update leads to a new class of nr multipliers methods which are equivalent to the interior quadratic prox methods for the dual problem we proved convergence and estimate the rate of convergence of the nr multipliers method under very mild assumptions on the input data we also estimate the rate of convergence under various assumptions on the input data in particular under the standard second order optimality conditions the nr method converges with q linear rate without unbounded increase of the scaling parameters which correspond to the active constraints we also established global quadratic convergence of the nr methods for linear programming with unique dual solution we provide numerical results which strongly support the theory

toric degenerations of gelfand cetlin systems and potential functions abstract we define a toric degeneration of an integrable system on a projective manifold and prove the existence of a toric degeneration of the gelfand cetlin system on the flag manifold of type a as an application we calculate the potential function for a lagrangian torus fiber of the gelfand cetlin system

standard monomial theory for flag algebras of gl n and sp 2n let g be the general linear group or the symplectic group over the complex numbers and u be its maximal unipotent subgroup we study standard monomial theory for the ring of regular functions on g u called the flag algebra using the philosophy of grobner bases and sagbi bases combined with classical invariant theory from the realization of the flag algebra in a concrete polynomial setting we obtain explicit standard monomial bases for irreducible representations we also recreate known combinatorics of young tableaux and gelfand tsetlin patterns and toric degenerations of flag varieties from the structure of leading monomials

a generalization of the littlewood richardson rule for the general linear group gz the littlewood richardson rule see 18 193 gives a method to calculate the decomposition of a tensor product of two irreducible g representations the aim of this article is to give a generalization of this rule for all simple simply connected algebraic groups of type a b c d g2 and e we obtain also partial results for g of type f4 e and es the restrictions in the last three cases come from the fact that for the formulation of the decomposition rules we need the notion of a standard young tableau such a notion has been developed by seshadri lakshmibai musili and rajeswari in a series of articles see ll 13 14 16 but not yet for all representations of the last three exceptional groups the advantage of the notion of a young tableau developed by seshadri et al is that it is independent of the type of the group for the convenience of the reader not used to this notion we give first a seperate proof for g sz using the classical notion of a standard young tableau of course for applications the classical notion is much more appropriate in the appendix we give a translation of the notion of a standard young tableau in the sense of seshadri et al into the classical notion of a young tableau for g sp2 and spin in 3 8 we give also such a translation for g g for the other exceptional groups such a translation is not known to the author let p be a dominant weight for g and denote by i the corresponding simple g module we assume that the character of vp is given by ct eycr where the sum has to be taken over all standard young tableaux f of shape p p and v 5 is the weight of the tableau y now by the

bounds for bayesian order identification with application to mixtures the efficiency of two bayesian order estimators is studied by using nonparametric techniques we prove new underestimation and overestimation bounds the results apply to various models including mixture models in this case the errors are shown to be o e an and o log n b p n a b 0 respectively 1 introduction order identification deals with the estimation and test of a structural parameter which indexes the complexity of a model in other words the most economical representation of a random phenomenon is sought this problem is encountered in many situations including mixture models 13 19 with an unknown number of components cluster analysis 9 when the number of clusters is unknown autoregressive models 1 when the process memory is not known this paper is devoted to the study of two bayesian estimators of the order of a model frequentist properties of efficiency are particularly investigated we obtain new efficiency bounds under mild assumptions providing a theoretical answer to the questions raised for instance in 7 see their section 4

on choosing mixture components via non local priors choosing the number of mixture components remains an elusive challenge model selection criteria can be either overly liberal or conservative and return poorly separated components of limited practical use we formalize non local priors nlps for mixtures and show how they lead to well separated components with non negligible weight interpretable as distinct subpopulations we also propose an estimator for posterior model probabilities under local priors and nlps showing that bayes factors are ratios of posterior to prior empty cluster probabilities the estimator is widely applicable and helps to set thresholds to drop unoccupied components in overfitted mixtures we suggest default prior parameters based on multimodality for normal t mixtures and minimal informativeness for categorical outcomes we characterize theoretically the nlp induced sparsity derive tractable expressions and algorithms we fully develop normal binomial and product binomial mixtures but the theory computation and principles hold more generally we observed a serious lack of sensitivity of the bayesian information criterion insufficient parsimony of the akaike information criterion and a local prior and a mixed behaviour of the singular bayesian information criterion we also considered overfitted mixtures their performance was competitive but depended on tuning parameters under our default prior elicitation nlps offered a good compromise between sparsity and power to detect meaningfully separated components

expandable factor analysis bayesian sparse factor models have proven useful for characterizing dependence in multivariate data but scaling computation to large numbers of samples and dimensions is problematic we propose expandable factor analysis for scalable inference in factor models when the number of factors is unknown the method relies on a continuous shrinkage prior for efficient maximum a posteriori estimation of a low rank and sparse loadings matrix the structure of the prior leads to an estimation algorithm that accommodates uncertainty in the number of factors we propose an information criterion to select the hyperparameters of the prior expandable factor analysis has better false discovery rates and true positive rates than its competitors across diverse simulation settings we apply the proposed approach to a gene expression study of ageing in mice demonstrating superior results relative to four competing methods

toric degeneration of branching algebras abstract for each classical symmetric pair g h there is a naturally defined multi graded algebra a g h called the branching algebra for g h which encodes the branching rule from g to h this algebra has a natural family of subalgebras depending on integer parameters for a certain range of the parameters the subalgebras have a particularly simple structure and are called stable branching algebras in this paper we show that the stable branching algebras for eight out of the ten families of classical symmetric pairs are flat deformations of the semigroup algebras of explicitly described lattice cones

on the complexity of following the central path of linear programs by linear extrapolation ii a class of algorithms is proposed for solving linear programming problems withm inequality constraints by following the central path using linear extrapolation with a special adaptive choice of steplengths the latter is based on explicit results concerning the convergence behaviour of newton s method to compute points on the central pathx r r 0 and this allows to estimate the complexity i e the total numbern n r of steps needed to go from an initial pointx r to a final pointx r 0 by an integral of the local weighted curvature of the primal dual path here the central curve is parametrized with the logarithmic penalty parameterr 0 it is shown that for large classes of problems the complexity integral i e the number of stepsn is not greater than constm log r where 1 2 e g 1 4 or 3 8 note that 1 2 gives the complexity of zero order methods we also provide a lower bound for the complexity showing that for some problems the above estimation can hold only for 1 3 as a byproduct many analytical and structural properties of the primal dual central path are obtained there are for instance close relations between the weighted curvature and the logarithmic derivatives of the slack variables the dependence of these quantities on the parameterr is described also related results hold for a family of weighted trajectories into which the central path can be embedded

posterior concentration and fast convergence rates for generalized bayesian learning abstract in this paper we study the learning rate of generalized bayes estimators in a general setting where the hypothesis class can be uncountable and have an irregular shape the loss function can have heavy tails and the optimal hypothesis may not be unique we prove that under the multi scale bernstein s condition the generalized posterior distribution concentrates around the set of optimal hypotheses and the generalized bayes estimator can achieve fast learning rate our results are applied to show that the standard bayesian linear regression is robust to heavy tailed distributions

a capelli harish chandra homomorphism for a real reductive dual pair the capelli identities define a homomorphism c from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group in terms of the harish chandra isomorphism this map involves a shift we view a dual pair as a lie supergroup and offer a construction of the homomorphism c based solely on the harish chandra s radial component maps thus we provide a geometric interpretation of the shift

stable branching rules for classical symmetric pairs we approach the problem of obtaining branching rules from the point of view of dual reductive pairs specifically we obtain a stable branching rule for each of 10 classical families of symmetric pairs in each case the branching multiplicities are expressed in terms of littlewood richardson coefficients some of the formulas are classical and include for example littlewood s restriction rule as a special case

integrable systems toric degenerations and okounkov bodies let x be a smooth projective variety of dimension n over mathbb c equipped with a very ample hermitian line bundle mathcal l in the first part of the paper we show that if there exists a toric degeneration of x satisfying some natural hypotheses which are satisfied in many settings then there exists a surjective continuous map from x to the special fiber x 0 which is a symplectomorphism on an open dense subset u from this we are then able to construct a completely integrable system on x in the sense of symplectic geometry more precisely we construct a collection of real valued functions h 1 ldots h n on x which are continuous on all of x smooth on an open dense subset u of x and pairwise poisson commute on u moreover our integrable system in fact generates a hamiltonian torus action on u in the second part we show that the toric degenerations arising in the theory of newton okounkov bodies satisfy all the hypotheses of the first part of the paper in this case the image of the moment map mu h 1 ldots h n x rightarrow mathbb r n is precisely the newton okounkov body delta delta r v associated to the homogeneous coordinate ring r of x and an appropriate choice of a valuation v on r our main technical tools come from algebraic geometry differential kahler geometry and analysis specifically we use the gradient hamiltonian vector field and a subtle generalization of the famous ojasiewicz gradient inequality for real valued analytic functions since our construction is valid for a large class of projective varieties x this manuscript provides a rich source of new examples of integrable systems we discuss concrete examples including elliptic curves flag varieties of arbitrary connected complex reductive groups spherical varieties and weight varieties

lagrangian fibers of gelfand cetlin systems abstract a gelfand cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a gelfand cetlin polytope motivated by the study of nishinou nohara ueda 24 on the floer theory of gelfand cetlin systems we provide a detailed description of topology of gelfand cetlin fibers in particular we prove that any fiber over an interior point of an r dimensional face of the gelfand cetlin polytope is an isotropic submanifold and is diffeomorphic to t r n for some smooth manifold n and t r s 1 r we also prove that such n s are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration we also devise an algorithm of reading off lagrangian fibers from the combinatorics of the ladder diagram

skew pieri algebras of the general linear group let v be an irreducible polynomial representation of the general linear group gln gln c and let 1 q be nonnegative integers less than or equal to n we call a description of the irreducible decomposition of the tensor product v 1 cn q cn an iterated skew pieri rule for gln in this paper we define a family of complex algebras whose structure encodes an iterated skew pieri rule for gln and we call these algebras iterated skew pieri algebras our main goal is to construct a basis for each of these algebras thereby giving explicit highest weight vectors in the above tensor product let v be an irreducible polynomial representation of the general linear group gln gln c and let 1 q be nonnegative integers less than or equal to n we call a description of the irreducible decomposition of the tensor product v 1 cn q cn an iterated skew pieri rule for gln in this paper we define a family of complex algebras whose structure encodes an iterated skew pieri rule for gln and we call these algebras iterated skew pieri algebras our main goal is to construct a basis for each of these algebras thereby giving explicit highest weight vectors in the above tensor product

on tangent spaces to schubert varieties ii abstract we prove the results on the tangent spaces to schubert varieties announced in v lakshmibai math res lett 2 1995 473 477 for g classical we give two descriptions of the tangent space to a schubert variety at id the first description is in terms of the root system and the second one is in terms of multiplicities of certain weights in the fundamental representations of g

integrability of vector and multivector fields associated with interior point methods for linear programming in the feasible region of a linear programming problem a number of desirably good directions have been defined in connexion with various interior point methods each of them determines a contravariant vector field in the region whose only stable critical point is the optimum point some interior point methods incorporate a two or higher dimensional search which naturally leads us to the introduction of the corresponding contravariant multivector field we investigate the integrability of those multivector fields i e whether a contravariantp vector field isxp forming is enveloped by a family ofxq s q p or envelops a family ofxq s q p in j a schouten s terminology wherexq is aq dimensional manifold immediate consequences of known facts are 1 the directions hitherto proposed arex1 forming with the optimum point of the linear programming problem as the stable accumulation point and 2 there is anx2 forming contravariant bivector field for which the center path is the critical submanifold most of the meaningfulp vector fields withp 3 are notxp forming in general though they envelop that bivector field this observation will add another circumstantial evidence that the bivector field has a kind of invariant significance in the geometry of interior point methods for linear programming for a kind of appendix it is noted that if we have several objectives i e in the case of multiobjective linear programs extension to higher dimensions is easily obtained

polytopes associated to demazure modules of symmetrizable kac moody algebras of rank two let 1 2 be the two fundamental weights of a symmetrizable kac moody algebra g of rank two hence necessarily affine or finite and an element of the weyl group in this paper we construct polytopes p 1 p 2 rl and a linear map rl h such that for any dominant weight k1 1 k2 2 we have char e e e x where the sum is over all the integral points x of the polytope k1p 1 k2p 2 furthermore we show that there exists a flat deformation of the schubert variety s into the toric variety defined by p 1 p 2

on bayesian analysis of mixtures with an unknown number of components with discussion new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

on the bayes optimality of f measure maximizers the f measure which has originally been introduced in information retrieval is nowadays routinely used as a performance metric for problems such as binary classification multi label classification and structured output prediction optimizing this measure is a statistically and computationally challenging problem since no closed form solution exists adopting a decision theoretic perspective this article provides a formal and experimental analysis of different approaches for maximizing the f measure we start with a bayes risk analysis of related loss functions such as hamming loss and subset zero one loss showing that optimizing such losses as a surrogate of the f measure leads to a high worst case regret subsequently we perform a similar type of analysis for f measure maximizing algorithms showing that such algorithms are approximate while relying on additional assumptions regarding the statistical distribution of the binary response variables furthermore we present a new algorithm which is not only computationally efficient but also bayes optimal regardless of the underlying distribution to this end the algorithm requires only a quadratic with respect to the number of binary responses number of parameters of the joint distribution we illustrate the practical performance of all analyzed methods by means of experiments with multi label classification problems

arithmetic cohen macaulayness and arithmetic normality for schubert varieties let g be a semisimple simply connected chevalley group over a field k let t be a maximal torus b a borel subgroup d t and w the weyl group of g relative to t let p be a maximal parabolic subgroup of classical type cf 7 8 i e a maximal parabolic subgroup such that if w is the associated fundamental weight then i w av i 1 2 w c a is 2 for all roots ae being a w invariant scalar product on x t q where x t character group of t for t e w wp let x t denote the schubert variety in g p associated to t namely x t is the zariski closure btp mod p in g p of the bruhat cell endowed with the canonical reduced structure for the canonical projective embedding g p in p h g p l let rt denote the homogeneous co ordinate ring of x t then the main result of this paper is given by

a non crossing standard monomial theory abstract the second author has introduced non crossing tableaux objects whose non nesting analogues are semi standard young tableaux we relate non crossing tableaux to gelfand tsetlin patterns and develop the non crossing analogue of standard monomial theory leclerc and zelevinsky s weakly separated sets are special cases of non crossing tableaux and we suggest that non crossing tableaux may help illuminate the theory of weakly separated sets

new schur function series this paper presents some new product identities for certain summations of schur functions these identities are generalizations of some famous identities known to littlewood and appearing in macdonald s book we refer to these identities as the littlewood type formulas in addition analogues for summations of characters of the other classical groups are also given the littlewood type formulas in this paper are separated into two classes the rational schur function series and the generalized schur function series an application of a rational schur function series to the infinite product representation of the elliptic theta functions is also given we prove these littlewood type formulas using the cauchy binet formula the cauchy binet formula is a basic but powerful tool applicable in the present context which can be derived from our pfaffian formula as we explain q 1998 academic press

bayesian cumulative shrinkage for infinite factorizations the dimension of the parameter space is typically unknown in a variety of models that rely on factorizations for example in factor analysis the number of latent factors is not known and has to be inferred from the data although classical shrinkage priors are useful in such contexts increasing shrinkage priors can provide a more effective approach that progressively penalizes expansions with growing complexity in this article we propose a novel increasing shrinkage prior called the cumulative shrinkage process for the parameters that control the dimension in overcomplete formulations our construction has broad applicability and is based on an interpretable sequence of spike and slab distributions which assign increasing mass to the spike as the model complexity grows using factor analysis as an illustrative example we show that this formulation has theoretical and practical advantages relative to current competitors including an improved ability to recover the model dimension an adaptive markov chain monte carlo algorithm is proposed and the performance gains are outlined in simulations and in an application to personality data

convergence rates of posterior distributions for non i i d observations we consider the asymptotic behavior of posterior distributions and bayes estimators based on observations which are required to be neither independent nor identically distributed we give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion we then specialize our results to independent nonidentically distributed observations markov processes stationary gaussian time series and the white noise model we apply our general results to several examples of infinite dimensional statistical models including nonparametric regression with normal errors binary regression poisson regression an interval censoring model whittle estimation of the spectral density of a time series and a nonlinear autoregressive model

classes of hypercomplex polynomials of discrete variable based on the quasi monomiality principle with the aim of derive a quasi monomiality formulation in the context of discrete hypercomplex variables one will amalgamate through a clifford algebraic structure of signature 0 n the umbral calculus framework with lie algebraic symmetries the exponential generating function egf carrying the continuum dirac operator d j 1 n e j x j together with the lie algebraic representation of raising and lowering operators acting on the lattice h z n is used to derive the corresponding hypercomplex polynomials of discrete variable as appell sets with membership on the space clifford vector valued polynomials some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the poisson charlier polynomials are introduced certain applications from the view of interpolation theory and integral transforms are also discussed

a class of polynomial variable metric algorithms for linear optimization in the paper the behaviour of interior point algorithms is analyzed by using a variable metric method approach a class of polynomial variable metric algorithms is given achieving o n l iterations for solving a canonical form linear optimization problem with respect to a wide class of riemannian metrics wheren is the number of dimensions and a fixed value it is shown that the vector fields of several interior point algorithms for linear optimization is the negative riemannian gradient vector field of a linear a potential or a logarithmic barrier function for suitable riemannian metrics

toric degenerations of grassmannians and schubert varieties from matching field tableaux abstract we study grobner degenerations of grassmannians and the schubert varieties inside them we provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in the sense of sturmfels and zelevinsky in 31 we prove that these ideals are all quadratically generated and they yield a sagbi basis of the plucker algebra this leads to a new family of toric degenerations of grassmannians moreover we apply our results to construct a family of grobner degenerations of schubert varieties inside grassmannians we provide a complete characterization of toric ideals among these degenerations in terms of the combinatorics of matching fields permutations and semi standard tableaux