the bunch kaufman factorization of symmetric matrices signature similar to sign regular matrices abstract the bunch kaufman pivoting strategy is a most commonly used method in practice to factor symmetric indefinite matrices however this method in general may destroy the band structure of banded matrices and its growth factor bound of 2 57 n 1 can almost be attained in this paper we investigate the behavior of the method for factorizing symmetric indefinite matrices that are signature similar to sign regular matrices with signature the growth factor bound depending on flips in the signature is derived for factorizing such banded matrices if the bunch kaufman pivoting strategy is modified as suggested by sorensen and van loan then not only do the same growth factor bounds hold but also the bandwidth is nicely preserved

gaussian elimination is stable for the inverse of a diagonally dominant matrix let b e m n c be a row diagonally dominant matrix i e i b ii b ij i 1 n where 0 i 1 i 1 n with a max 1 i n i we show that no pivoting is necessary when gaussian elimination is applied to a b 1 moreover the growth factor for a does not exceed 1 the same results are true with row diagonal dominance being replaced by column diagonal dominance

quasi ldu factorization of nonsingular totally nonpositive matrices abstract let a a ij r n n be a nonsingular totally nonpositive matrix in this paper we describe some properties of these matrices when a11 0 and obtain a characterization in terms of the quasi ldu factorization of a where l is a block lower triangular matrix d is a diagonal matrix and u is a unit upper triangular matrix

total nonpositivity of nonsingular matrices in this paper nonsingular totally nonpositive matrices are studied and new characterizations are provided in terms of the signs of minors with consecutive initial rows or consecutive initial columns these characterizations extend an existing characterization that uses some restrictive hypotheses

some q analogues of the carter payne theorem we prove a q analogue of the carter payne theorem for the two special cases cor responding to moving an arbitrary number of nodes between adjacent rows or moving one node between an arbitrary number of rows as a consequence we show that these homomorphism spaces are one dimensional when q 6 1 we apply these results to complete the classification of the reducible specht modules for the hecke algebras of the symmetric groups when q 6 1 our methods can also be used to determine certain other pairs of specht modules between which there is a homomorphism in particular we describe the homomorphism space homh s n s for an arbitrary partition s although widely conjectured no q analogue of the full carter payne theorem is known in this paper we prove such an analogue in two important special cases namely when b a 1 or when 1 combinatorially this corresponds to moving an arbitrary number of nodes between adjacent rows or moving one node between an arbitrary number of rows it turns out that our proof has interesting implications we will turn our attention to the classification of the reducible specht modules for the algebras hf q sn when q 6 1 recent work see section 5 3 has resulted in the completion of this classification when q 1 combining our main result theorem 4 5 4 with previously published work we prove that a specht module is reducible if and only if it is e p reducible as defined in definition 5 1 we will also use our discussion of semistandard homomorphisms to obtain a description of the specht modules which contain a submodule isomorphic to the trivial module s n see

spectral regularization algorithms for learning large incomplete matrices we use convex relaxation techniques to provide a sequence of regularized low rank solutions for large scale matrix completion problems using the nuclear norm as a regularizer we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm our algorithm soft impute iteratively replaces the missing elements with those obtained from a soft thresholded svd with warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter the computationally intensive part of our algorithm is in computing a low rank svd of a dense matrix exploiting the problem structure we show that the task can be performed with a complexity of order linear in the matrix dimensions our semidefinite programming algorithm is readily scalable to large matrices for example soft impute takes a few hours to compute low rank approximations of a 106 x 106 incomplete matrix with 107 observed entries and fits a rank 95 approximation to the full netflix training set in 3 3 hours our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state of the art techniques

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

on schur complements of sign regular matrices of order k abstract the issue regarding schur complements of sign regular matrices is rather subtle it is known that the class of totally nonnegative matrices is not closed under arbitrary schur complementation in this paper we demonstrate how schur complements of sign regular matrices of order k are sign regular of a certain order in particular some results for totally nonnegative and totally nonpositive matrices are provided as our corollaries

poisson matrix recovery and completion we extend the theory of low rank matrix recovery and completion to the case when poisson observations for a linear combination or a subset of the entries of a matrix are available which arises in various applications with count data we consider the usual matrix recovery formulation through maximum likelihood with proper constraints on the matrix m of size d 1 by d 2 and establish theoretical upper and lower bounds on the recovery error our bounds for matrix completion are nearly optimal up to a factor on the order of cal o log d 1 d 2 these bounds are obtained by combining techniques for recovering sparse vectors with compressed measurements in poisson noise those for analyzing low rank matrices as well as those for one bit matrix completion davenport 1 bit matrix completion information and inference information and inference vol 3 no 3 pp 189 223 sep 2014 although these two problems are different in nature the adaptation requires new techniques exploiting properties of the poisson likelihood function and tackling the difficulties posed by the locally sub gaussian characteristic of the poisson distribution our results highlight a few important distinctions of the poisson case compared to the prior work including having to impose a minimum signal to noise requirement on each observed entry and a gap in the upper and lower bounds we also develop a set of efficient iterative algorithms and demonstrate their good performance on synthetic examples and real data

main effects and interactions in mixed and incomplete data frames a mixed data frame mdf is a table collecting categorical numerical and count observations the use of mdf is widespread in statistics and the applications are numerous from abundance data in ec

on the extension of some total positivity inequalities abstract some inequalities recently obtained for totally positive matrices are extended to more general classes of matrices among these larger classes of matrices sign consistent sign regular and matrices with signed bidiagonal decomposition are considered the class of matrices with a signed bidiagonal decomposition contains the nonsingular totally positive matrices and their inverses

a stable test for strict sign regularity a stable test to check if a given matrix is strictly sign regular is provided among other nice properties we prove that it has an optimal growth factor the test is compared with other alternative tests appearing in the literature and its advantages are shown

characterizations of jacobi sign regular matrices abstract several characterizations of jacobi nonsingular sign regular matrices are presented moreover a stable test of o n elementary operations is obtained to check if an n n jacobi nonsingular matrix is sign regular

q tensor space and q weyl modules we obtain the irreducible representations of the q schur algebra motivated by the fact that these representations give all the irreducible representations of gln q in the nondescribing characteristic the irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction the theory of polynomial representations of general linear groups is equivalent to the representation theory of schur algebras see green s book 5 and the bibliography therein in 4 we defined q analogues of schur algebras when q 1 these are the usual schur algebras and when q is a prime power representations of q schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case including all irreducible representations of these groups and important information about decomposition numbers it is natural to ask what features of the classical schur algebras have qanalogues in this paper we define q analogues of tensor space of weyl modules and of weight spaces thereby generalizing the main reslllts which appear in green s book 5 for example we classify the irreducible modules for qschur algebras we determine bases for q weyl modules compatible with weight spaces and we give results on composition multiplicities of the irreducible modules in q weyl modules the proofs are largely self contained so by specializing q to 1 we recover the corresponding results in 5 this paper therefore is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics 1 the q schur algebra let r be a natural number let r be an integral domain and let q be a unit in r we denote the symmetric group on r letters by er the hecke algebra z is the r free r algebra with basis twlw e er where the multiplication is determined by the following rule if a i i 1 is a basic transposition in received by the editors august 15 1989 1980 mathematics subject classification 198 5 revision primary 1 6a64 1 6a6 5 secondary 20c30 this research was supported in part by nsf grant no dms 8802290 the authors gratefully acknowledge support received from nato grant no 0222 87 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

factorizing complex symmetric matrices with positive definite real and imaginary parts complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for lu factorization this result adds to the classes of matrix for which it is known to be safe not to pivot in lu factorization block mathrm ldl t factorization with the pivoting strategy of bunch and kaufman is also considered and it is shown that for such matrices only 1 times 1 pivots are used and the same growth factor bound of 2 holds but that interchanges that destroy band structure may be made the latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in linpack and lapack

matrix completion by least square low rank and sparse self representations abstract conventional matrix completion methods are generally based on rank minimization these methods assume that the given matrix is of low rank and the data points are drawn from a single subspace of low dimensionality therefore they are not effective in completing matrices where the data are drawn from multiple subspaces in this paper we establish a novel matrix completion framework that is based on self representation specifically least square low rank and sparse self representations based matrix completion algorithms are provided the underlying idea is that one data point can be efficiently reconstructed by other data points belonging to a common subspace where the missing entries are recovered so as to fit the common subspace the proposed algorithms actually maximize the weighted correlations among the columns of a given matrix we prove that the proposed algorithms are approximations for rank minimization of the incomplete matrix in addition they are able to complete high rank or even full rank matrices when the data are drawn from multiple subspaces comparative studies are conducted on synthetic datasets natural image inpainting tasks temperature prediction task and collaborative filtering tasks the results show that the proposed algorithms often outperform other state of the art algorithms in various tasks

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

composite self concordant minimization we propose a variable metric framework for minimizing the sum of a self concordant function and a possibly non smooth convex function endowed with an easily computable proximal operator we theoretically establish the convergence of our framework without relying on the usual lipschitz gradient assumption on the smooth part an important highlight of our work is a new set of analytic step size selection and correction procedures based on the structure of the problem we describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numerically on both synthetic and real data

sign structure preserves for m banded factorizations of sign regular matrices abstract a square matrix a a ij is called m banded if a ij 0 for all i j m motivated by the results of rainey and habetler j w rainey g j habetler tridiagonalization of completely nonnegative matrices math comp 26 1972 121 128 we provide m banded factorizations of sign regular matrices that preserve their sign structures as applications some eigenvalue interlacing properties and majorization relations are obtained for certain sign regular matrices

noisy matrix completion understanding statistical guarantees for convex relaxation via nonconvex optimization this paper studies noisy low rank matrix completion given partial and noisy entries of a large low rank matrix the goal is to estimate the underlying matrix faithfully and efficiently arguably o

on nonsingular sign regular matrices abstract an m n matrix a is sign regular if for each k 1 k min m n all k k submatrices of a have determinant with the same nonstrict sign the zero pattern of nonsingular sign regular matrices is analyzed it is proved that the number of zero entries which can appear in a nonsingular sign regular matrix depends on its signature a matrix is totally nonpositive if all its minors are nonpositive a test for recognizing nonsingular totally nonpositive matrices is also provided

combined matrices of almost strictly sign regular matrices abstract the combined matrix is a very useful concept for many applications almost strictly sign regular assr matrices form an important structured class of matrices with two possible zero patterns which are either type i staircase or type ii staircase we prove that under an irreducibility condition the pattern of zero and nonzero entries of an assr matrix is preserved by the corresponding combined matrix without the irreducibility condition it is proved that type i and type ii staircases are still preserved illustrative numerical examples are included

restricted strong convexity and weighted matrix completion optimal bounds with noise we consider the matrix completion problem under a form of row column weighted entrywise sampling including the case of uniformentrywise sampling as a special case we analyze the associated random observation operator and prove that with high probability it satisfies a form of restricted strong convexity with respect to weighted frobenius norm using this property we obtain as corollaries a number of error bounds on matrix completion in the weighted frobenius norm under noisy sampling and for both exact and near low rank matrices our results are based on measures of the spikiness and low rankness of matrices that are less restrictive than the incoherence conditions imposed in previous work our technique involves an m estimator that includes controls on both the rank and spikiness of the solution and we establish non asymptotic error bounds in weighted frobenius norm for recovering matrices lying with lq balls of bounded spikiness using information theoretic methods we show that no algorithm can achieve better estimates up to a logarithmic factor over these same sets showing that our conditions on matrices and associated rates are essentially optimal

a test for strict sign regularity a characterization of strict sign regular matrices is obtained it is given in an algorithmic form which allows the easy construction of a test to check if a matrix is strictly sign regular the test is based on the neville elimination of several submatrices of a and requires less computational work than those which can be derived from other previously known characterizations of these matrices

variation diminishing transformations a direct approach to total positivity and its statistical applications abstract karlin has shown that the variation diminishing property possessed by totally positive distributions allows a unified and straightforward presentation of many basic properties of hypothesis tests his work in fact reveals that total positivity is equivalent to a suitably formulated variation diminishing property this largely expository article begins with the latter concept and consequently gives a more direct account of the theory this approach avoids the extensive mathematical preliminaries previously required and isolates the more important statistical property

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

the irreducible representations of the alternating group which remain irreducible in characteristic p let p be an odd prime and an the alternating group of degree n we determine which ordinary irreducible representations of u n remain irreducible in characteristic p verifying the author s conjecture from 2010 given the preparatory work done in 2010 our task is to determine which self conjugate partitions label specht modules for the symmetric group in characteristic p having exactly two composition factors this is accomplished through the use of the robinson brundan kleshchev i restriction functors together with known results on decomposition numbers for the symmetric group and additional results on the mullineux map and homomorphisms between specht modules

a matricial description of neville elimination with applications to total positivity abstract the neville elimination process used by the authors in some previous papers in connection with totally positive matrices is studied in detail in the case of nonsingular matrices a wide class of matrices is found where neville elimination has a lower computational cost than gauss elimination finally some new characterizations are obtained for strictly totally positive and nonsingular totally positive matrices in terms of their neville elimination and that of their inverses

combined matrices of sign regular matrices abstract the combined matrix of a nonsingular matrix a is the hadamard entry wise product c a a a 1 t since each row and column sum of c a is equal to one the combined matrix is doubly stochastic when it is nonnegative in this work we study the nonnegativity of the combined matrix of sign regular matrices based upon their signature in particular a few coordinates of the signature e of a play a crucial role in determining whether or not c a is nonnegative

some new decomposable specht modules abstract we present with proof a new family of decomposable specht modules for the symmetric group in characteristic 2 these specht modules are labelled by partitions of the form a 3 1 b and are the first new examples found for thirty years our method of proof is to exhibit summands isomorphic to irreducible specht modules by constructing explicit homomorphisms between specht modules

homomorphisms between specht modules in positive characteristic the specht modules s corresponding to partitions are not necessarily irreducible and understanding their structure is vital to understanding the representation theory of the symmetric group in this paper we address the related problem of finding the spaces of homomorphisms between specht modules indeed in 2 carter and payne showed that the space of homomorphisms from s to s is non zero for certain pairs of partitions and where the young diagram for is obtained from that for by moving several nodes from one row to another we also consider partitions of this type and by explicitly examining certain combinations of semi standard homomorphisms we are able to give a constructive proof of the carter payne theorem and to generalise it

an algorithm for semistandardising homomorphisms abstract suppose is a partition of n and a composition of n and let s m denote the specht module and permutation module defined by dipper and james for the iwahori hecke algebra h n of the symmetric group w n we give an explicit fast algorithm for expressing a tableau homomorphism a s m as a linear combination of semistandard homomorphisms along the way we provide a utility result related to removing rows from tableaux