minors of the moore penrose inverse abstract let q k n 1 k 1 1 k n denote the strictly increasing sequences of k elements from 1 n for q k n we denote by a the submatrix of a with rows indexed by columns by the submatrix obtained by deleting the rows and columns is denoted by a lsqb rsqb for nonsingular a r n n the jacobi identity relates the minors of the inverse a 1 to those of a det a 1 lsqb rsqb 1 k i 1 i k i 1 i det a lsqb rsqb det a for any q k n we generalize the jacobi identity to matrices a r m n r expressing the minors of the moore penrose inverse a in terms of the minors of the maximal nonsingular submatrices a ij of a in our notation det a lsqb rsqb 1 vol 2 a i j n det a ij a a ij for any q k m q k n 1 k r where vol 2 a denotes the sum of squares of determinants of r r submatrices of a we apply our results to questions concerning the nonnegativity of principal minors of the moore penrose inverse

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

an extension of chebfun to two dimensions an object oriented matlab system is described that extends the capabilities of chebfun to smooth functions of two variables defined on rectangles functions are approximated to essentially machine precision by using iterative gaussian elimination with complete pivoting to form chebfun2 objects representing low rank approximations operations such as integration differentiation function evaluation and transforms are particularly efficient global optimization the singular value decomposition and rootfinding are also extended to chebfun2 objects numerical applications are presented

new studies of randomized augmentation and additive preprocessing abstract a standard gaussian random matrix hereafter referred to just as gaussian matrix has full rank with probability 1 and is well conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from the square shape and becomes more rectangular if we append sufficiently many gaussian rows or columns to any normalized and possibly rank deficient or ill conditioned matrix then the augmented matrix has full rank with probability 1 and is well conditioned with a probability close to 1 we specify and prove these properties of augmentation and extend them to additive preprocessing that is to adding a product of two rectangular gaussian matrices by applying our randomization techniques to a matrix that has numerical rank r we accelerate the known algorithms for the approximation of its trailing singular spaces associated with all its positive singular values except for the r largest values our algorithms use much fewer random parameters and run much faster when various random sparse and structured preprocessors replace gaussian empirically the outputs of the resulting algorithms are as accurate as the outputs under gaussian preprocessing our novel duality techniques provides formal support so far missing for these empirical observations and opens door to de randomization of our preprocessing and to further acceleration and simplification of our algorithms by using more efficient sparse and structured preprocessors our techniques and our progress can be applied to various other fundamental matrix computations such as the celebrated low rank approximation of a matrix by means of random sampling

improved bound for rank revealing lu factorizations abstract in many applications it is necessary to determine the rank or numerical rank of a matrix many of these situations involve matrices that are very large order or that are sparse or that may undergo some form of modification rank k update row or column appended or removed in these cases the singular value decomposition s cost may be prohibitively high or the decomposition may not be computationally feasible especially for large sparse problems we thus examine the theoretical merits of rank revealing lu rrlu factorizations we find that in those cases where the nullity is small and the gap is well defined an rrlu factorization could be a very useful tool

random multipliers numerically stabilize gaussian and block gaussian elimination proofs and an extension to low rank approximation abstract we study two applications of standard gaussian random multipliers at first we prove that with a probability close to 1 such a multiplier is expected to numerically stabilize gaussian elimination with no pivoting as well as block gaussian elimination then by extending our analysis we prove that such a multiplier is also expected to support low rank approximation of a matrix without customary oversampling our test results are in good accordance with this formal study the results remain similar when we replace gaussian multipliers with random circulant or toeplitz multipliers which involve fewer random parameters and enable faster multiplication we formally support the observed efficiency of random structured multipliers applied to approximation but we still continue our research in the case of elimination we specify a narrow class of unitary inputs for which gaussian elimination with no pivoting is numerically unstable and then prove that with a probability close to 1 a gaussian random circulant multiplier does not fix numerical stability problems for such inputs we also prove that the power of the random circulant preprocessing increases if we also include random permutations

three results in connection with inverse matrices abstract first we show that the moore penrose solution of an arbitrary system of linear equations is a convex combination of the solutions of all uniquely solvable partial systems the other two results concern the elements of inverse toeplitz band matrices namely the asymptotic behavior of a determinant appearing in a formula of d s meek and a modification of a formula of w d hoskins and p j ponzo for matrices with binomial coefficients in the limit case

cholesky decomposition of a positive semidefinite matrix with known kernel the cholesky decomposition of a symmetric positive semidefinite matrix a is a useful tool for solving the related consistent system of linear equations or evaluating the action of a generalized inverse especially when a is relatively large and sparse to use the cholesky decomposition effectively it is necessary to identify reliably the positions of zero rows or columns of the factors and to choose these positions so that the nonsingular submatrix of a of the maximal rank is reasonably conditioned the point of this note is to show how to exploit information about the kernel of a to accomplish both tasks the results are illustrated by numerical experiments

rank revealing gaussian elimination by the maximum volume concept abstract a gaussian elimination algorithm is presented that reveals the numerical rank of a matrix and identifies a square nonsingular submatrix of maximum dimension the bounds on the singular values of the submatrix and its schur complement are similar to the best known bounds for rank revealing lu factorization but in contrast to existing methods the algorithm does not make use of the normal matrix an implementation for dense matrices is described whose computational cost is roughly twice the cost of an lu factorization with complete pivoting because of its flexibility in choosing pivot elements the algorithm is amenable to implementation with blocked memory access and for sparse matrices

notes on the wasserstein metric in hilbert spaces let x y be a pair of hilbert valued random variables for which the wasserstein distance between the marginal distributions is reached we prove that the mapping omega rightarrow x omega y omega is increasing in a certain sense moreover if y satisfies a nondegeneration condition we can take x t y with t monotone in the sense of zarantarello we apply these results to obtain a proof of the central limit theorem clt in hilbert spaces which does not make use of the clt for real valued random variables

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

on the existence and computation of rank revealing lu factorizations abstract by exploring properties of schur complements this paper presents bounds on the existence of rank revealing lu factorizations that are comparable with those of rank revealing qr factorizations the new bounds provide substantial improvement over previously derived bounds this paper also proposes two algorithms using gaussian elimination with a block pivoting strategy to select a subset of columns from a given matrix which has a guaranteed relatively large smallest singular value each of these two algorithms is faster than its orthogonal counterpart for dense matrices if implemented appropriately these algorithms are faster than the corresponding rank revealing qr methods even when the orthogonal matrices are not explicitly updated based on these two algorithms an algorithm using only gaussian elimination for computing rank revealing lu factorizations is introduced

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

computation of generalized equivariant cohomologies of kac moody flag varieties abstract in 1998 goresky kottwitz and macpherson showed that for certain projective varieties x equipped with an algebraic action of a complex torus t the equivariant cohomology ring h t x can be described by combinatorial data obtained from its orbit decomposition in this paper we generalize their theorem in three different ways first our group g need not be a torus second our space x is an equivariant stratified space along with some additional hypotheses on the attaching maps third and most important we allow for generalized equivariant cohomology theories e g instead of h t for these spaces we give a combinatorial description of e g x as a subring of e g f i where the f i are certain invariant subspaces of x our main examples are the flag varieties g p of kac moody groups g with the action of the torus of g in this context the f i are the t fixed points and e g is a t equivariant complex oriented cohomology theory such as h t k t or mu t we detail several explicit examples

optimal solution of a monge kantorovitch transportation problem abstract suppose that p and q are probabilities on a separable banach space e it is known that if p q satisfies certain regularity conditions and a random variable x has law p then there exists a function f e e such that the function f x has the law q and the random pair x f x is an optimal coupling for the monge kantorovitch problem in this paper we provide an approximation of the function f when the law q is discrete thenwe extend this main result to any law q the proofs are based on a relationship between optimal couplings and nonlinear equations

a fast and stable algorithm for downdating the singular value decomposition abstract in this paper we modify a classical downdating svd algorithm and reduce its complexity significantly we use a structured low rank approximation algorithm to compute an hierarchically semiseparable hss matrix approximation to the eigenvector matrix of a diagonal matrix plus rank one modification the complexity of our downdating algorithm is analyzed we further show that the structured low rank approximation algorithm is backward stable numerous experiments have been done to show the efficiency of our algorithm for some matrices with large dimensions our algorithm can be much faster than that using plain matrix matrix multiplication routine in intel mkl in both sequential and parallel cases

low rank approximation of a sparse matrix based on lu factorization with column and row tournament pivoting in this paper we present an algorithm for computing a low rank approximation of a sparse matrix based on a truncated lu factorization with column and row permutations we present various approaches for determining the column and row permutations that show a trade off between speed versus deterministic probabilistic accuracy we show that if the permutations are chosen by using tournament pivoting based on qr factorization then the obtained truncated lu factorization with column row tournament pivoting lu crtp satisfies bounds on the singular values which have similarities with the ones obtained by a communication avoiding rank revealing qr factorization experiments on challenging matrices show that lu crtp provides a good low rank approximation of the input matrix and it is less expensive than the rank revealing qr factorization in terms of computational and memory usage costs while also minimizing the communication cost we also compare the computational complexity of our algorithm with randomized a

motivic decompositions of twisted flag varieties and representations of hecke type algebras motivated by the motivic galois group and the kostant kumar results on equivariant cohomology of flag varieties we provide a uniform de scription of motivic direct sum decompositions with integer coefficients of versal flag varieties in terms of integer representations of the associated affine nil hecke algebra h more generally we establish an equivalence between the h motivic subcat egory generated by the motive of e b and the category of projective modules of the associated rational algebra d of push pull operators where e is a tor sor for a split semisimple linear algebraic group g over a field k b is a borel subgroup of g h is an algebraic oriented cohomology theory in the sense of levine morel e g chow ring ch or an algebraic cobordism the algebra d can be think of as an integer analogue of the hopf algebra of the h motivic galois group of e b as an application taking h ch and specializing the coefficients to the finite field fp we obtain that p modular projective representations of d h are generated by an irreducible h module corresponding to the generalized rost voevodsky motive for g p

mechanism free domain decomposition the simulation of three dimensional 3d structural dynamics on massively parallel platforms places stringent requirements on the existing software infrastructure a constrained and nonlinear graph partitioning problem that arises in scalable iterative substructuring methods such as finite element tearing and interconnecting feti methods is identified new sufficient criteria on a partition are presented that ensure the applicability of feti methods and improve the associated preconditioner one dimensional finite elements in 3d structures are treated by an encapsulation method the techniques are demonstrated on complex finite element model problems

a volume associated with m x n matrices abstract let a mxnr with nonzero singular values 1 2 r the volume of a vol a is defined as zero if r 0 and otherwise vola ri 1 i or equivalently vola det2aij summing over all r x r nonsingularsubmatrices aij the matrix volume vol a generalizes the absolute value of the determinant from nonsingular to arbitrary matrices any r dimensional unit cube in r at is mapped under a into a parallelepiped of volume vol a this paper gives properties and applications of vol a in particular the moore penrose inverse of a is a convex combination of ordinary inverses of its maximal nonsingular submatrices a i j ij or is a convex combination of moore penrose inverses of its maximal full rank submatrices a i i j j where the convex weights are proport ional to the squares of volumes of the corresponding submatrices since a a the matrix a is a convex combination of inverses of submatrices of a interestingly the weights of these convex combinations are the above weights in transposed position

efficient algorithms for computing a strong rank revealing qr factorization given anm n matrixm withm n it is shown that there exists a permutation fi and an integer k such that the qr factorization myi q ak ckbk reveals the numerical rank of m the k k upper triangular matrix ak is well conditioned ilckll2 is small and bk is linearly dependent on ak with coefficients bounded by a low degree polynomial in n existing rank revealing qr rrqr algorithms are related to such factorizations and two algorithms are presented for computing them the new algorithms are nearly as efficient as qr with column pivoting for most problems and take o ran2 floating point operations in the worst case

on maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices abstract the problem of finding a k k submatrix of maximum volume of a matrix a is of interest in a variety of applications for example it yields a quasi best low rank approximation constructed from the rows and columns of a we show that such a submatrix can always be chosen to be a principal submatrix not only for symmetric semidefinite matrices but also for diagonally dominant matrices then we analyze the low rank approximation error returned by a greedy method for volume maximization cross approximation with complete pivoting our bound for general matrices extends an existing result for symmetric semidefinite matrices and yields new error estimates for diagonally dominant matrices in particular for doubly diagonally dominant matrices the error is shown to remain within a modest factor of the best approximation error we also illustrate how the application of our results to cross approximation for functions leads to new and better convergence results

rank revealing lu factorizations abstract we consider permutations of any given squared matrix and the generalized lu r factorization of the permuted matrix that reveals the rank deficiency of the matrix chan has considered the case with nearly rank deficiency equal to one this paper extends his results to the case with nearly rank deficiency greater than one two applications in constrained optimization are given we are primarily interested in the existence of such factorizations in addition to the theories we also present an efficient two pass rank revealing lu r algorithm

strong rank revealing lu factorizations abstract for any m n matrix a we introduce a definition of strong rank revealing lu rrlu factorization related to the definition presented by pan but with some extensions similar to the notion of strong rank revealing qr factorization developed in the joint work of gu and eisenstat a pivoting strategy based on the idea of local maximum volumes is introduced and a proof of the existence of strong rrlu is given

on solutions to the mass transfer problem this paper studies the monge kantorovich mass transfer mt problem on metric spaces with possibly unbounded cost function conditions are given under which the mt problem is solvable and furthermore an optimal solution can be obtained as the weak limit of a sequence of optimal solutions to suitably approximating mt problems

randomized lu decomposition using sparse projections abstract a fast algorithm that approximates a low rank lu decomposition is presented in order to achieve a low complexity the algorithm uses sparse random projections combined with fft based random projections the asymptotic approximation error of the algorithm is analyzed and a theoretical error bound is presented finally numerical examples illustrate that for a similar approximation error the sparse lu algorithm is faster than recent state of the art methods the algorithm is completely parallelizable and can fully run on a gpu the performance is tested on a gpu card showing a significant speed up improvement in the running time in comparison to a sequential execution

on selecting a maximum volume sub matrix of a matrix and related problems given a matrix email protected r m x n n vectors in m dimensions we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible this notion turned out to be important in low rank approximations to matrices and rank revealing qr factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways in this paper from a complexity theoretic point of view we propose four related problems in which we try to find a sub matrix email protected r m x k of a given matrix email protected r m x n such that i s m a x c the largest singular value of c is minimum ii s m i n c the smallest singular value of c is maximum iii k c email protected m a x c s m i n c the condition number of c is minimum and iv the volume of the parallelepiped defined by the column vectors of c is maximum we establish the np hardness of these problems and further show that they do not admit ptas we then study a natural greedy heuristic for the maximum volume problem and show that it has approximation ratio 2 o k l o g k our analysis of the greedy heuristic is tight to within a logarithmic factor in the exponent which we show by explicitly constructing an instance for which the greedy heuristic is 2 w k from optimal when a has unit norm columns a related problem is to select the maximum number of vectors with a given volume we show that if the optimal solution selects k columns then greedy will select w k logk columns providing a logk approximation

cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure the direct methods for the solution of systems of linear equations with a symmetric positive semidefinite sps matrix a usually comprise the cholesky decomposition of a nonsingular diagonal block a of a and effective evaluation of the action of a generalized inverse of the corresponding schur complement in this note we deal with both problems paying special attention to the stiffness matrices of floating structures without mechanisms we present a procedure which first identifies a well conditioned positive definite diagonal block a of a then decomposes a by the cholesky decomposition and finally evaluates a generalized inverse of the schur complement s of a the schur complement s is typically very small so the generalized inverse can be effectively evaluated by the singular value decomposition svd if the rank of a or a lower bound on the nonzero eigenvalues of a are known then the svd can be implemented without any epsilon moreover if the kernel of a is known then the svd can be replaced by effective regularization the results of numerical experiments show that the proposed method is useful for effective implementation of the feti based domain decomposition methods copyright 2011 john wiley sons ltd

subspace iteration randomization and singular value problems a classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank the truncated singular value decomposition svd is known to provide the best such approximation for any given fixed rank however the svd is also known to be very costly to compute among the different approaches in the literature for computing low rank approximations randomized algorithms have attracted researchers attention recently due to their surprising reliability and computational efficiency in different application areas typically such algorithms are shown to compute with very high probability low rank approximations that are within a constant factor from optimal and are known to perform even better in many practical situations in this paper we present a novel error analysis that considers randomized algorithms within the subspace iteration framework and show with very high probability that highly accurate low rank approximations as well as singular values ca

on the existence and computation of lu factorizations with small pivots let a be an n by n matrix which may be singular with a one dimensional null space and consider the lu factorization of a when a is exactly singular we show conditions under which a pivoting strategy will produce a zero n th pivot when a is not singular we show conditions under which a pivoting strategy will produce an nth pivot that is o g or o k a where is the smallest singular value of a and k a is the condition number of a these conditions are expressed in terms of the elements of a in general but reduce to conditions on the elements of the singular vectors corresponding to when a is nearly or exactly singular they can be used to build a 2 pass factorization algorithm which is guaranteed to produce a small n th pivot for nearly singular matrices as an example we exhibit an lu factorization of the n by n upper triangular matrix

single pass randomized algorithms for lu decomposition abstract in this paper we present some single pass randomized algorithms to compute lu decomposition these algorithms need only one pass over the original matrix and hence are very suitable for extremely large and high dimensional matrix stored outside of core memory or generated in a streaming fashion rigorous error bounds and complexity of these algorithms are provided numerical experiments show that these single pass algorithms have the similar accuracy and runtime excluding the cost of matrix transfer compared with the state of the art randomized algorithms for lu decomposition

solving linear systems of equations with randomization augmentation and aggregation abstract seeking a basis for the null space of a rectangular and possibly rank deficient and ill conditioned matrix we apply randomization augmentation and aggregation to reduce our task to computations with well conditioned matrices of full rank our algorithms avoid pivoting and orthogonalization preserve matrix structure and sparseness and in the case of an ill conditioned input perform only a small part of the computations with high accuracy we extend the algorithms to the solution of nonhomogeneous nonsingular ill conditioned linear systems of equations whose matrices have small numerical nullities our estimates and experiments show dramatic progress versus the customary matrix algorithms where the input matrices are rank deficient or ill conditioned our study can be of independent technical interest we extend the known results on conditioning of random matrices to randomized preconditioning estimate the condition numbers of randomly augmented matrices and link augmentation to aggregation as well as homogeneous to nonhomogeneous linear systems of equations

computing 2 4 and 2 3 inverses by using the sherman morrison formula a finite recursive procedure for computing 2 4 generalized inverses and the analogous recursive procedure for computing 2 3 generalized inverses of a given complex matrix are presented the starting points of both introduced methods are general representations of these classes of generalized inverses these representations are formed using certain matrix products which include the moore penrose inverse or the usual inverse of a symmetric matrix product and the sherman morrison formula for the inverse of a symmetric rank one matrix modification the computational complexity of the methods is analyzed defined algorithms are tested on randomly generated matrices as well as on test matrices from the matrix computation toolbox

strong duality of the monge kantorovich mass transfer problem in metric spaces this paper studies the monge kantorovich mass transfer mt problem on metric spaces and with an unbounded cost function conditions are given under which the strong duality condition holds that is mt and its dual mt are both solvable and their optimal values coincide

numerically safe gaussian elimination with no pivoting abstract gaussian elimination with no pivoting and block gaussian elimination are attractive alternatives to the customary but communication intensive gaussian elimination with partial pivoting 1 provided that the computations proceed safely and numerically safely that is run into neither division by 0 nor numerical problems empirically safety and numerical safety of genp have been consistently observed in a number of papers where an input matrix was pre processed with various structured multipliers chosen ad hoc our present paper provides missing formal support for this empirical observation and explains why it was elusive so far namely we prove that genp is numerically unsafe for a specific class of input matrices in spite of its pre processing with some well known and well tested structured multipliers but we also prove that genp and bge are safe and numerically safe for the average input matrix pre processed with any nonsingular and well conditioned multiplier this should embolden search for sparse and structured multipliers and we list and test some new classes of them we also seek randomized pre processing that universally that is for all input matrices supports i safe genp and bge with probability 1 and or ii numerically safe genp and bge with a probability close to 1 we achieve goal i with a gaussian structured multiplier and goal ii with a gaussian unstructured multiplier and alternatively with gaussian structured augmentation we consistently confirm all these formal results with our tests of genp for benchmark inputs we have extended our approach to other fundamental matrix computations and keep working on further extensions

a coproduct structure on the formal affine demazure algebra in the present paper we generalize the coproduct structure on nil hecke rings introduced and studied by kostant and kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law we then construct an algebraic model of the t equivariant oriented cohomology of the variety of complete flags

geometric representations of the formal affine hecke algebra abstract for any formal group law there is a formal affine hecke algebra defined by hoffnung malagon lopez savage zainoulline coming from this formal group law there is also an oriented cohomology theory we identify the formal affine hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the steinberg variety as a consequence this algebra acts on the corresponding cohomology of the springer fibers this generalizes the action of classical affine hecke algebra on the k theory of the springer fibers constructed by lusztig we also give a residue interpretation of the formal affine hecke algebra which generalizes the residue construction of ginzburg kapranov vasserot when the formal group law comes from a 1 dimensional algebraic group

fast direct methods for gaussian processes a number of problems in probability and statistics can be addressed using the multivariate normal gaussian distribution in the one dimensional case computing the probability for a given mean and variance simply requires the evaluation of the corresponding gaussian density in the n dimensional setting however it requires the inversion of an n times n covariance matrix c as well as the evaluation of its determinant det c in many cases such as regression using gaussian processes the covariance matrix is of the form c sigma 2 i k where k is computed using a specified covariance kernel which depends on the data and additional parameters hyperparameters the matrix c is typically dense causing standard direct methods for inversion and determinant evaluation to require mathcal o n 3 work this cost is prohibitive for large scale modeling here we show that for the most commonly used covariance functions the matrix c can be hierarchically factored into a product of block low rank updates of the identity matrix yielding an mathcal o n log 2 n algorithm for inversion more importantly we show that this factorization enables the evaluation of the determinant det c permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining k our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single cpu core the combination of nearly optimal scaling in terms of problem size with high performance computing resources will permit the modeling of previously intractable problems we illustrate the performance of the scheme on standard covariance kernels

rank revealing qr factorizations and the singular value decomposition t chan has noted that even when the singular value decomposition of a matrix a is known it is still not obvious how to find a rank revealing qr factorization rrqr of a if a has numerical rank deficiency this paper offers a constructive proof of the existence of the rrqr factorization of any matrix a of size m x n with numerical rank r the bounds derived in this paper that guarantee the existence of rrqr are all of order f i in comparison with chan s 0 2 it has been known for some time that if a is only numerically rank one deficient then the column permutation l7 of a that guarantees a small rnn in the qr factorization of a n can be obtained by inspecting the size of the elements of the right singular vector of a corresponding to the smallest singular value of a to some extent our paper generalizes this well known result we consider the interplay between two important matrix decompositions the singular value decomposition and the qr factorization of a matrix a in particular we are interested in the case when a is singular or nearly singular it is well known that for any a e r m x n a real matrix with rn rows and n columns where without loss of generality we assume rn n there are orthogonal matrices u and v such that where c is a diagonal matrix with nonnegative diagonal elements we assume that a 2 a2 2 2 on 2 0 the decomposition 0 1 is the singular value decomposition svd of a and the ai are the singular values of a the columns of v are the right singular vectors of a and the columns of u are the left singular vectors of a mathematically in terms of the singular values received december 1 1990 revised february 8 199 1 199 1 mathematics subject classification primary 65f30 15a23 15a42 15a15