non negative matrix factorization ill posedness and a geometric algorithm non negative matrix factorization nmf has been proposed as a mathematical tool for identifying the components of a dataset however popular nmf algorithms tend to operate slowly and do not always identify the components which are most representative of the data in this paper an alternative algorithm for performing nmf is developed using the geometry of the problem the computational costs of the algorithm are explored and it is shown to successfully identify the components of a simulated dataset the development of the geometric algorithm framework illustrates the ill posedness of the nmf problem and suggests that nmf is not sufficiently constrained to be applied successfully outside of a particular class of problems

basic soules matrices and their applications abstract in this paper we study the properties of the basic soules matrices in r n n which are a special subclass of the n n soules matrices generated via the basic soules basis the basic soules basis has the sign pattern n and it corresponds to the vector e n r n of all 1 s the basic soules matrices are up to a multiple by a positive scalar symmetric and doubly stochastic we begin by investigating the permanents of basic soules matrices next for a nonsingular basic soules matrix a r n n we show that the matrix a a 1 which is known to be a nonsingular m matrix has a basic soules basis of eigenvectors furthermore we obtain explicit formulas for the eigenvalues of a a 1 in terms of the eigenvalues of a finally let a r n n be a basic soules matrix of spectral radius 1 and set q i a by investigating the sign pattern of the off diagonal entries of the group inverse q of q we determine when the perron root is a concave function in each of the off diagonal entries at a

tutorial on seismic interferometry part 2 underlying theory and new advances in the 1990s the method of time reversed acoustics was developed this method exploits the fact that the acoustic wave equation for a lossless medium is invariant for time reversal when ultrasonic responses recorded by piezoelectric transducers are reversed in time and fed simultaneously as source signals to the transducers they focus at the position of the original source even when the medium is very complex in seismic interferometry the time reversed responses are not physically sent into the earth but they are convolved with other measured responses the effect is essentially the same the time reversed signals focus and create a virtual source which radiates waves into the medium that are subsequently recorded by receivers a mathematical derivation based on reciprocity theory formalizes this principle the crosscorrelation of responses at two receivers integrated over different sources gives the green s function emitted by a virtual source at the position of one of the receivers and observed by the other receiver this green s function representation for seismic interferometry is based on the assumption that the medium is lossless and nonmoving recent developments circumventing these assumptions include interferometric representations for attenuating and or moving media as well as unified representations for waves and diffusion phenomena bending waves quantum mechanical scattering potential fields elastodynamic electromagnetic poroelastic and electroseismic waves significant improvements in the quality of the retrieved green s functions have been obtained with interferometry by deconvolution a trace by trace deconvolution process compensates for complex source functions and the attenuation of the medium interferometry by multidimensional deconvolution also compensates for the effects of one sided and or irregular illumination

computing nonnegative rank factorizations abstract the existence of nonnegative generalized inverses in terms of nonnegative rank factorizations is considered an algorithm is presented which computes a nonnegative rank factorization of a nonnegative matrix when a nonnegative 1 inverse exists

using underapproximations for sparse nonnegative matrix factorization nonnegative matrix factorization consists in approximately factorizing a nonnegative data matrix by the product of two low rank nonnegative matrices it has been successfully applied as a data analysis technique in numerous domains e g text mining image processing microarray data analysis collaborative filtering etc we introduce a novel approach to solve nmf problems based on the use of an underapproximation technique and show its effectiveness to obtain sparse solutions this approach based on lagrangian relaxation allows the resolution of nmf problems in a recursive fashion we also prove that the underapproximation problem is np hard for any fixed factorization rank using a reduction of the maximum edge biclique problem in bipartite graphs we test two variants of our underapproximation approach on several standard image datasets and show that they provide sparse part based representations with low reconstruction error our results are comparable and sometimes superior to those obtained by two standard sparse nonnegative matrix factorization techniques

a rank one reduction formula and its applications to matrix factorizations let a in r m times n denote an arbitrary matrix if x in r n and y in r m are vectors such that omega y t ax ne 0 then the matrix b a omega 1 axy t a a has rank exactly one less than the rank of a this wedderburn rank one reduction formula is easy to prove yet the idea is so powerful that perhaps all matrix factorizations can be derived from it the formula also appears in places such as the positive definite secant updates bfgs and dfp as well as the abs methods by repeatedly applying the formula to reduce ranks a biconjugation process analogous to the gram schmidt process with oblique projections can be developed this process provides a mechanism for constructing factorizations such as text ldm t qr and svd under a common framework of a general biconjugate decomposition v t au omega that is diagonal and nonsingular two characterizations of biconjugation provide new insight into the lanczos method and its breakdown one characterization shows that

nonnegative matrix factorization and i divergence alternating minimization abstract in this paper we consider the nonnegative matrix factorization nmf problem given an elementwise nonnegative matrix v r m n find for assigned k nonnegative matrices w r m k and h r k n such that v wh exact nontrivial nonnegative factorizations do not always exist hence it is interesting to pose the approximate nmf problem the criterion which is commonly employed is i divergence between nonnegative matrices the problem becomes that of finding for assigned k the factorization wh closest to v in i divergence an iterative algorithm em like for the construction of the best pair w h has been proposed in the literature in this paper we interpret the algorithm as an alternating minimization procedure a la csiszar tusnady and investigate some of its stability properties nmf is widespreading as a data analysis method in applications for which the positivity constraint is relevant there are other data analysis methods which impose some form of nonnegativity we discuss here the connections between nmf and archetypal analysis

graph dual regularization non negative matrix factorization for co clustering low rank matrix factorization is one of the most useful tools in scientific computing data mining and computer vision among of its techniques non negative matrix factorization nmf has received considerable attention due to producing a parts based representation of the data recent research has shown that not only the observed data are found to lie on a nonlinear low dimensional manifold namely data manifold but also the features lie on a manifold namely feature manifold in this paper we propose a novel algorithm called graph dual regularization non negative matrix factorization dnmf which simultaneously considers the geometric structures of both the data manifold and the feature manifold we also present a graph dual regularization non negative matrix tri factorization algorithm dnmtf as an extension of dnmf moreover we develop two iterative updating optimization schemes for dnmf and dnmtf respectively and provide the convergence proofs of our two optimization schemes experimental results on uci benchmark data sets several image data sets and a radar hrrp data set demonstrate the effectiveness of both dnmf and dnmtf

on the geometric interpretation of the nonnegative rank the nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank one factors needed to reconstruct it exactly the problem of determining this rank and computing the corresponding nonnegative factors is difficult however it has many potential applications e g in data mining graph theory and computational geometry in particular it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program in this paper we introduce and study a related quantity called the restricted nonnegative rank we show that computing this quantity is equivalent to a problem in polyhedral combinatorics and fully characterize its computational complexity this in turn sheds new light on the nonnegative rank problem and in particular allows us to provide new improved lower bounds based on its geometric interpretation we apply these results to slack matrices and linear euclidean distance matrices and obtain counter examples to two conjectures of beasly and laffey namely we show that the nonnegative rank of linear euclidean distance matrices is not necessarily equal to their dimension and that the rank of a matrix is not always greater than the nonnegative rank of its square

a note on upper bounds on the cp rank abstract hanna and laffey gave an upper bound on the cp rank of a completely positive matrix in terms of its rank and the number of zeros in a full rank principal submatrix this bound for the case that the matrix is positive was improved by barioli and berman in this paper a new straightforward proof of both results is given and the same approach is used to improve hanna and laffey s bound in the case that the matrix has a zero entry

on ternary quotients of cubic hecke algebras we prove that the quotient of the group algebra of the braid group introduced by funar commun math phys 173 513 558 1995 collapses in characteristic distinct from 2 in characteristic 2 we define several quotients of it which are connected to the classical hecke and birman wenzl murakami quotients but which admit in addition a symmetry of order 3 we also establish conditions on the possible markov traces factorizing through it

nonnegative matrix factorization for spectral data analysis data analysis is pervasive throughout business engineering and science very often the data to be analyzed is nonnegative and it is often preferable to take this constraint into account in the analysis process here we are concerned with the application of analyzing data obtained using astronomical spectrometers which provide spectral data which is inherently nonnegative the identification and classification of space objects that cannot be imaged in the normal way with telescopes is an important but difficult problem for tracking thousands of objects including satellites rocket bodies debris and asteroids in orbit around the earth in this paper we develop an effective nonnegative matrix factorization algorithm with novel smoothness constraints for unmixing spectral reflectance data for space object identification and classification purposes promising numerical results are presented using laboratory and simulated datasets

estimating and correcting the amplitude radiation pattern of a virtual source in the virtual source vs method we crosscorrelate seismic recordings at two receivers to create a new data set as if one of these receivers were a virtual source and the other a receiver we focus on the amplitudes and kinematics of vs data generated by an array of active sources at the surface and recorded by an array of receivers in a borehole the quality of the vs data depends on the radiation pattern of the virtual source which in turn is controlled by the spatial aperture of the surface source distribution theory suggests that when the receivers are surrounded by multi component sources completely filling a closed surface then the virtual source has an isotropic radiation pattern and vs data possess true amplitudes in practical applications limited sourceaperture and deployment of a single source type create an anisotropic radiation pattern of the virtual source leading to distorted amplitudes this pattern can be estimated by autocorrelating the spatial fourier transform of the downgoing wavefield in the special case of a laterally invariant medium the vs data can be improved by deconvolving the vs data with the estimated amplitude radiation pattern in the frequency wavenumber domain this operation alters the amplitude spectrum but not the phase of the data we can also steer the virtual source by assigning it a new desired amplitude radiation pattern given sufficient illumination exists in the desired directions alternatively time gating the downgoing wavefield before crosscorrelation already common practice in implementing the vs method can improve the radiation characteristics of a virtual source

complex reflection groups braid groups hecke algebras presentations a la coxeter are given for all irreducible nite com plex re ection groups they provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra hecke algebras we introduce monodromy representations of the braid groups which factorize through the hecke algebras extending results of cherednik opdam kohno and others summary introduction complex re ection groups and their presentations a background from complex re ection groups b presentations braid groups and their diagrams a generalities about hyperplane complements b generalities about the braid groups c the braid diagrams proofs of the main theorems for the braid groups b de e r a notation and prerequisites b computation of b de e r and of its center for d c computation of b e e r and of its center hecke algebras a background from di erential equations and monodromy b a family of monodromy representations of the braid group c hecke algebras diagrams and tables appendix generators of the monodromy around a divisor appendix tables to mathematics subject classi cation primary g we thank jean michel peter orlik pierre vogel for useful conversations and the isaac newton institute for its hospitality while the last version of this manuscript was written up the second named author gratefully acknowledges nancial support by the fondation alexander von humboldt for his stays in paris michel brou e gunter malle raphael rouquier

a maximal cubic quotient of the braid algebra i abstract we study a quotient of the group algebra of the braid group in which the artin generators satisfy a cubic relation this quotient is maximal among the ones satisfying such a cubic relation we also investigate the proper quotients of it that appear in the realm of quantum groups and describe another maximal quotient related to the usual hecke algebras finally we describe the connection between this algebra and a quotient of the algebra of horizontal chord diagrams introduced by vogel we prove that these two are isomorphic for n 5

probability matrices non negative rank and parameterization of mixture models abstract in this paper we parameterize non negative matrices of sum one and rank at most two using the least possible number of parameters we also show how this parameterization relates to a class of statistical models known in probability and statistics as mixture models for contingency tables in particular we show how to use this parameterization to make some optimization problems computationally easier

green s function representations for seismic interferometry the term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations the first version of this principle was derived by claerbout 1968 who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response for an arbitrary 3d inhomogeneous lossless medium it follows from rayleigh s reciprocity theorem and the principle of time reversal invariance that the acoustic green s function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points the integral is along sources on an arbitrarily shaped surface enclosing these points no assumptions are made with respect to the diffusivity of the wavefield the rayleigh betti reciprocity theorem leads to a similar representation of the elastodynamic green s function when a part of the enclosing surface is the earth s free surface the integral needs only to be evaluated over the remaining part of the closed surface in practice not all sources are equally important the main contributions to the reconstructed green s function come from sources at stationary points when the sources emit transient signals a shaping filter can be applied to correct for the differences in source wavelets when the sources are uncorrelated noise sources the representation simplifies to a direct crosscorrelation of wavefield observations at two points similar as in methods that retrieve green s functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface

retrieving the green s function in an open system by cross correlation a comparison of approaches l we compare two approaches for deriving the fact that the green s function in an arbitrary inhomogeneous open system can be obtained by cross correlating recordings of the wave field at two positions one approach is based on physical arguments exploiting the principle of time reversal invariance of the acoustic wave equation the other approach is based on rayleigh s reciprocity theorem using a unified notation we show that the result of the time reversal approach can be obtained as an approximation of the result of the reciprocity approach

lattice extensions of hecke algebras abstract we investigate the extensions of the hecke algebras of finite complex reflection groups by lattices of reflection subgroups that we introduced for some of them in our previous work on the yokonuma hecke algebras and their connections with artin groups when the hecke algebra is attached to the symmetric group and the lattice contains all reflection subgroups then these algebras are the diagram algebras of braids and ties of aicardi and juyumaya we prove a structure theorem for these algebras generalizing a result of espinoza and ryom hansen from the case of the symmetric group to the general case we prove that these algebras are symmetric algebras at least when w is a coxeter group and in general under the trace conjecture of broue malle and michel

a geometric treatment of generalized inverses and semigroups of nonnegative matrices abstract the purpose of this paper is to provide a unified treatment from the geometric viewpoint of the following closely related aspects of nonnegative matrices nonnegative matrices with nonnegative generalized inverses of various kinds nonnegative rank factorization regular elements green s relations and maximal subgroups of the semigroups of nonnegative matrices stochastic matrices column stochastic matrices and doubly stochastic matrices

three observations on the determinantal range abstract let a c m n the algebra of n n complex matrices the set of complex numbers c a det a ucu u u i n is the c determinantal range of a in this note it is proved that c a is an elliptical disc for a c m 2 a necessary and sufficient condition for c a to be a line segment is given when a and c are normal matrices with pairwise distinct eigenvalues the linear operators l that satisfy the linear preserver property c a c l a for all a c m n are characterized

the nonnegative rank of a matrix hard problems easy solutions using elementary linear algebra we develop a technique that leads to solutions of two widely known problems on nonnegative matrices first we give a short proof of the result by vavasis stating that the nonnegative rank of a matrix is np hard to compute this proof is essentially contained in the paper by jiang and ravikumar who discussed this topic in different terms fifteen years before the work of vavasis second we present a solution of the cohen rothblum problem on rational nonnegative factorizations which was posed in 1993 and remained open until now

nonnegative matrix factorization of a correlation matrix we present a dedicated algorithm for the nonnegative factorization of a correlation matrix from an application in financial engineering we look for a low rank approximation the origin of the problem is discussed in some detail next to the description of the algorithm we prove by means of a counter example that an exact nonnegative decomposition of a general positive semidefinite matrix is not always available

a reverse hadamard inequality abstract let u be an n n unitary matrix with determinant equal to 1 let a be an n n real matrix with rank a 2 and entries satisfying aij 1 for 1 i j n then it follows that det a u 1 this reversal of the hadamard inequality can be obtained easily from an old result of fiedler in this article we present a different proof of this fact and discuss its ramifications

the sign representation for shephard groups shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes for a shephard group we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes this representation turns out to have many equivalent guises making it analogous to the sign representation of a finite coxeter group one of these guises is up to a twist the cohomology of the milnor fiber for the isolated singularity at 0 in the hypersurface defined by any homogeneous invariant of minimal degree

a note on the computation of the cp rank abstract the purpose of this note is to address the computational question of determining whether or not a square nonnegative matrix over the reals is completely positive and finding its cp rank when it is we show that these questions can be resolved by finite algorithms and we provide non polynomial complexity bounds on the number of arithmetic boolean operations that these algorithms require we state several open questions including the existence of polynomial algorithms to resolve the above problems and availability of algorithms for addressing complete positivity over the rationals and over 0 1 matrices

on the complexity of nonnegative matrix factorization nonnegative matrix factorization nmf has become a prominent technique for the analysis of image databases text databases and other information retrieval and clustering applications the problem is most naturally posed as continuous optimization in this report we define an exact version of nmf then we establish several results about exact nmf i that it is equivalent to a problem in polyhedral combinatorics ii that it is np hard and iii that a polynomial time local search heuristic exists

the determinantal conjecture and hadamard type inequalities abstract the first part of this paper presents an approach to a possible salvation of an idea advanced by papers of bebiano merikoski da providencia and virtanen for the solution of the oliveira marcus determinantal conjecture let a 1 a n and b 1 b n be given complex numbers and define the vertex points v n j 1 a j b j c for a permutation let a and b be normal matrices with eigenvalues a 1 a n and b 1 b n respectively the oliveira marcus conjecture asks whether the complex number det a b is necessarily in the convex hull of the n vertex points the authors mentioned above had hoped that for every unitary matrix u the n k n k nonegative matrix with entries det u i j 2 might be representable as convex combinations of the subset of matrices stemming from permutation matrices unfortunately this turns out not to be the case the second part shows how inequalities of hadamard type can be used to further this program at least in the special case n 4 the oliveira marcus conjecture is reduced in case n 4 to a geometrical problem relating only to the disposition of the vertex points

the difference between 5 5 doubly nonnegative and completely positive matrices abstract the convex cone of n n completely positive cp matrices and its dual cone of copositive matrices arise in several areas of applied mathematics including optimization every cp matrix is doubly nonnegative dnn i e positive semidefinite and component wise nonnegative and it is known that for n 4 only every dnn matrix is cp in this paper we investigate the difference between 5 5 dnn and cp matrices defining a bad matrix to be one which is dnn but not cp we i design a finite procedure to decompose any n n dnn matrix into the sum of a cp matrix and a bad matrix which itself cannot be further decomposed ii show that every bad 5 5 dnn matrix is the sum of a cp matrix and a single bad extreme matrix and iii demonstrate how to separate bad extreme matrices from the cone of 5 5 cp matrices

on nonnegative factorization of matrices abstract it is shown that a sufficient condition for a nonnegative real symmetric matrix to be completely positive is that the matrix is diagonally dominant

sparse and unique nonnegative matrix factorization through data preprocessing nonnegative matrix factorization nmf has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part based representation however nmf has the drawback of being highly ill posed that is there typically exist many different but equivalent factorizations in this paper we introduce a completely new way to obtaining more well posed nmf problems whose solutions are sparser our technique is based on the preprocessing of the nonnegative input data matrix and relies on the theory of m matrices and the geometric interpretation of nmf this approach provably leads to optimal and sparse solutions under the separability assumption of donoho and stodden 2003 and for rank three matrices makes the number of exact factorizations finite we illustrate the effectiveness of our technique on several image data sets

orthogonal bases that lead to symmetric nonnegative matrices abstract in a paper dating back to 1983 soules constructs from a positive vector x an orthogonal matrix r which has the property that for any nonnegative diagonal matrix with nonincreasing diagonal entries the matrix r rt has all its entries nonnegative independently fiedler in 1988 showed that any symmetric irreducible nonsingular matrix whose powers are all m matrices and hence an mma matrix in the language of friedland hershkowitz and schneider must have an orthogonal matrix of eigenvectors r which has similar properties to those of r here for a given positive n vector x we investigate the structure of all orthogonal matrices r for which for any nonnegative diagonal matrix as above the matrices r rt are nonnegative up to a permutation of its columns each such r corresponds to a binary tree whose vertices are subsets of the set 1 2 n with the property that each vertex has either no successor or exactly two disjoint successors for such orthogonal matrices r and such nonsingular diagonal matrices we show that the set of matrices of the form r rt and the set of inverse mma matrices i e matrices whose inverses are mma matrices coincide using this result we establish a relation between strictly ultrametric matrices and inverse mma matrices finally we show that the qr factorization of r rt for certain such r s has a special sign pattern

some remarks on a conjecture of de oliveira abstract let u n be the group of the unitary n n matrices let a diag 1 n b diag 1 n where 1 n 1 n are complex numbers a class of normal matrices satisfying the inclusion det a ubu u n co n i 1 i i s n where co is the convex hull of and s n is the symmetric group of degree n is obtained

the nonnegative rank factorizations of nonnegative matrices abstract let a p m n r the set of all m n nonnegative matrices having the same rank r for matrices a in p m n n we introduce the concepts of a has only trivial nonnegative rank factorizations and a can have nontrivial nonnegative rank factorizations correspondingly the set p m n n is divided into two disjoint subsets p 1 and p 2 such that p 1 p 2 p m n n it happens that the concept of a has only trivial nonnegative rank factorizations is a generalization of a is prime in p n n n we characterize the sets p 1 and p 2 some of our results generalize some theorems in the paper of daniel j richman and hans schneider 9

svd based initialization a head start for nonnegative matrix factorization we describe nonnegative double singular value decomposition nndsvd a new method designed to enhance the initialization stage of nonnegative matrix factorization nmf nndsvd can readily be combined with existing nmf algorithms the basic algorithm contains no randomization and is based on two svd processes one approximating the data matrix the other approximating positive sections of the resulting partial svd factors utilizing an algebraic property of unit rank matrices simple practical variants for nmf with dense factors are described nndsvd is also well suited to initialize nmf algorithms with sparse factors many numerical examples suggest that nndsvd leads to rapid reduction of the approximation error of many nmf algorithms

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

the bmm symmetrising trace conjecture for the exceptional 2 reflection groups of rank 2 abstract we prove the symmetrising trace conjecture of broue malle and michel for the generic hecke algebra associated to the exceptional irreducible complex reflection group g 13 our result completes the proof of the conjecture for the exceptional 2 reflection groups of rank 2

the marcus de oliveira conjecture bilinear forms and cones abstract the well known determinantal cinjecture of de oliveira and marcus omc confines the determinant det x y of the sum of normaln n matricesx y to a certain region in the complex plane even the subconjecture obtained by specializing it ton 4 x hermitian andy normal is still open we view the subconjecture as a special case of an assertion concerning a certain family of bilinear forms ofr16 c16 and give a method that may prove useful for establishing it for many of such matrix pairs independent of their spectrum in particular we apply it successfully in the case of a prominent unitary similarity of drury s threatening omc unfortunately we find the assertion extended naturally to pairs of complex arguments to be false and the ideas outlined inapplicable for the general omc n 4 case we also report on some computer experiments formulate omc n 4 as a statement about cones and find it would be implied by establishing the emptiness of certain semialgebraic sets defined by systems of quadratic and linear relations

a counterexample to a question of merikoski and virtanen on the compounds of unitary matrices abstract we provide a counterexample to a question of merikoski and virtanen relating to the determinantal conjecture of marcus and de oliveira

non negative matrix factorization revisited uniqueness and algorithm for symmetric decomposition non negative matrix factorization nmf has found numerous applications due to its ability to provide interpretable decompositions perhaps surprisingly existing results regarding its uniqueness properties are rather limited and there is much room for improvement in terms of algorithms as well uniqueness aspects of nmf are revisited here from a geometrical point of view both symmetric and asymmetric nmf are considered the former being tantamount to element wise non negative square root factorization of positive semidefinite matrices new uniqueness results are derived e g it is shown that a sufficient condition for uniqueness is that the conic hull of the latent factors is a superset of a particular second order cone checking this condition is shown to be np complete yet this and other results offer insights on the role of latent sparsity in this context on the computational side a new algorithm for symmetric nmf is proposed which is very different from existing ones it alternates between procrustes rotation and projection onto the non negative orthant to find a non negative matrix close to the span of the dominant subspace simulation results show promising performance with respect to the state of art finally the new algorithm is applied to a clustering problem for co authorship data yielding meaningful and interpretable results

matrices satisfying a conjecture of g n de oliveira on determinants abstract it is shown that the determinant of the sum of a positive definite hermitian matrix and a skew hermitian matrix with eigenvalues j and i j j 1 n respectively is contained in the convex hull of the points j 1 n j i j s n as usual s n denotes the symmetric group of degree n this result proves in a particular case a conjecture of g n de oliveira on determinants

nonnegative ranks decompositions and factorizations of nonnegative matrices the nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank one matrices into which the matrix can be decomposed additively such decompositions are useful in diverse scientific disciplines we obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm

graph regularized nonnegative matrix factorization for data representation matrix factorization techniques have been frequently applied in information retrieval computer vision and pattern recognition among them nonnegative matrix factorization nmf has received considerable attention due to its psychological and physiological interpretation of naturally occurring data whose representation may be parts based in the human brain on the other hand from the geometric perspective the data is usually sampled from a low dimensional manifold embedded in a high dimensional ambient space one then hopes to find a compact representation which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure in this paper we propose a novel algorithm called graph regularized nonnegative matrix factorization gnmf for this purpose in gnmf an affinity graph is constructed to encode the geometrical information and we seek a matrix factorization which respects the graph structure our empirical study shows encouraging results of the proposed algorithm in comparison to the state of the art algorithms on real world problems

computing symmetric nonnegative rank factorizations an algorithm is described for the nonnegative rank factorization nrf of some completely positive cp matrices whose rank is equal to their cp rank the algorithm can compute the symmetric nrf of any nonnegative symmetric rank r matrix that contains a diagonal principal submatrix of that rank and size with leading cost o rm2 operations in the dense case the algorithm is based on geometric considerations and is easy to implement the implications for matrix graphs are also discussed

on the convergence of multiplicative update algorithms for nonnegative matrix factorization nonnegative matrix factorization nmf is useful to find basis information of nonnegative data currently multiplicative updates are a simple and popular way to find the factorization however for the common nmf approach of minimizing the euclidean distance between approximate and true values no proof has shown that multiplicative updates converge to a stationary point of the nmf optimization problem stationarity is important as it is a necessary condition of a local minimum this paper discusses the difficulty of proving the convergence we propose slight modifications of existing updates and prove their convergence techniques invented in this paper may be applied to prove the convergence for other bound constrained optimization problems

some remarks on a conjecture of boyle and handelman abstract boyle and handelman have conjectured that whenever a is an n n nonnegative matrix with rank a r and perron root 1 the inequality det i ta n r r 1r holds for all real numbers satisfying 1 we introduce an analogous conjecture involving nonnegative central class functions on the permutation group sn the analogue of the rank condition in this context is a condition on the support of the nonabelian fourier transform of the central function we are able to establish that both conjectures are true in case 2r n

computation of vector sublattices and minimal lattice subspaces of rk applications in finance abstract in this article we carry out a computational study on polyrakis algorithms i a polyrakis finite dimensional lattice subspaces of c and curves of r n trans am math soc 348 1996 2793 2810 i a polyrakis minimal lattice subspaces trans am math soc 351 1999 4183 4203 these algorithms can be utilised for the determination of the vector sublattice and the minimal lattice subspace generated by a finite set of positive vectors of r k it is demonstrated that the findings of our work can be very useful in the field of economics especially in completion by options of security markets and portfolio insurance

a dual braid monoid for the free group abstract we construct a quasi garside monoid structure for the free group this monoid should be thought of as a dual braid monoid for the free group generalising the constructions by birman ko lee and by the author of new garside monoids for artin groups of spherical type

multistability of α divergence based nmf algorithms in this paper the multistability of a class of amari s a divergence based nonnegative matrix factorization learning algorithms is analyzed the analysis results show that invariant sets for the update algorithms can be constructed in these invariant sets the non convergence of the discussed algorithms can be guaranteed based on lyapunov s stability theorem the local convergence of this class of learning algorithms is proved in the domain of their update rules in the simulation the analysis results are applied to image representation experiment results demonstrate that selecting suitable initial data for different applications of these nonnegative matrix factorization algorithms is very important

on a conjecture of marcus and de oliveira abstract let x y be two normal n n matrices over c with respective spectra x 1 x n and y 1 y n marcus and de oliveira conjectured that the determinant of x y lies in the convex hull generated by the n complex points n i 1 x i y i where ranges over the symmetric group s n we prove the conjecture for the case that n 2 of the y i are equal first the problem is reduced to the question of nonnegative solvability of a certain system of linear equations then the required nonnegative solvability is demonstrated by involving a measure theoretic generalization of the marriage lemma our main result forms a positive counterpart to a recent example given by drury

on reduced rank nonnegative matrix factorization for symmetric nonnegative matrices abstract let v r m n be a nonnegative matrix the nonnegative matrix factorization nnmf problem consists of finding nonnegative matrix factors w r m r and h r r n such that v wh lee and seung proposed two algorithms one of which finds nonnegative w and h such that v wh f is minimized after examining the case in which r 1 about which a complete characterization of the solution is possible we consider the case in which m n and v is symmetric we focus on questions concerning when the best approximate factorization results in the product wh being symmetric and on cases in which the best approximation cannot be a symmetric matrix finally we show that the class of positive semidefinite symmetric nonnegative matrices v generated via a soules basis admit for every 1 r rank v a nonnegative factorization wh which coincides with the best approximation in the frobenius norm to v in r n n of rank not exceeding r an example of applications in which nnmf factorizations for nonnegative symmetric matrices v arise is video and other media summarization technology where v is obtained from a distance matrix

on a theorem of wielandt and the compounds of unitary matrices let a and b be normal endomorphisms with prescribed eigenvalues defined on a finite dimensional unitary space a celebrated theorem of wielandt states that the eigenvalues of a b are forced to lie in a certain subset of the complex plane the primary objective of this article is to extend wielandt s result to give information about the joint distribution of the eigenvalues of a b the main tool in establishing this extension is a result on the compounds of unitary matrices if u is an n n unitary matrix then birkhoff s famous result on doubly stochastic matrices is often applied to write the matrix ujk 2 jk as a convex combination of permutation matrices the natural generalization of this process to the compound of a unitary matrix is known to fail here we show that it succeeds if one considers only a certain restricted subset of entries

the nmf problem and lattice subspaces abstract suppose that a is a nonnegative n m real matrix the nmf problem is the determination of two nonnegative real matrices f v so that a f v with intermediate dimension p smaller than m i n n m in this article we present a general method for the determination of two nonnegative real factors f v of a based on the theory of lattice subspaces and positive bases

structured nonnegative matrix factorization with applications to hidden markov realization and clustering abstract in this paper we study the structured nonnegative matrix factorization problem given a square nonnegative matrix p decompose it as p vav with v and a nonnegative matrices and with the dimension of a as small as possible we propose an iterative approach that minimizes the kullback leibler divergence between p and vav subject to the nonnegativity constraints on a and v with the dimension of a given the approximate structured decomposition p vav is closely related to the approximate symmetric decomposition p vv it is shown that the approach for finding an approximate structured decomposition can be adapted to solve the symmetric decomposition problem approximately finally we apply the nonnegative decomposition vav to the hidden markov realization problem and to the clustering of data vectors based on their distance matrix

the determinant of the sum of two normal matrices with prescribed eigenvalues abstract given complex numbers 1 n 1 n what can we say about the determinant of a b where a b is an n n normal matrix with eigenvalues 1 n 1 n some partial answers are offered to this question

modeling of wave propagation in inhomogeneous media we present a methodology providing a new perspective on modeling and inversion of wave propagation satisfying time reversal invariance and reciprocity in generally inhomogeneous media the approach relies on a representation theorem of the wave equation to express the green function between points in the interior as an integral over the response in those points due to sources on a surface surrounding the medium following a predictable initial computational effort green s functions between arbitrary points in the medium can be computed as needed using a simple cross correlation algorithm

double soules pairs and matching soules bases abstract we consider two generalizations of the notion of a soules basis matrix a pair of nonsingular n n matrices p q is called a double soules pair if the first columns of p and q are positive pq t i and p q t is nonnegative for every n n nonnegative diagonal matrix with nonincreasing diagonal elements in a paper by chen han and neumann an algorithm for generating such pairs was given here we characterize all such pairs and discuss some implications of the characterization we also consider pairs of matrices u v such that u and v each consists of k orthonormal columns the first of which is positive and u v is nonnegative for every k k nonnegative diagonal matrix with nonincreasing diagonal elements we call such pairs matching soules pairs we characterize all such pairs and make some observations regarding the nonnegative matrices and generalized inverses generated by them

universal deformations of the finite quotients of the braid group on 3 strands abstract we prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively have finite rank this is a special case of a conjecture by broue malle and rouquier for the generic hecke algebra of an arbitrary complex reflection group exploring the consequences of this case we prove that we can determine completely the irreducible representations of this braid group of dimension at most 5 thus recovering a classification of tuba and wenzl in a more general framework

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

a bound for the determinant of certain hadamard products and for the determinant of the sum of two normal matrices abstract we establish an upper bound for the absolute value of the determinant of the hadamard elementwise product of a unitary matrix and a general complex matrix in the case that the general matrix is fixed and the unitary matrix is allowed to vary this estimate is best possible as a corollary we obtain an upper bound for the absolute value of the determinant of the sum of two normal matrices with specified eigenvalues this corollary supports the determinantal conjecture of marcus and de oliveira

the validity of the marcus de oliveira conjecture for essentially hermitian matrices abstract let x y be matrices of spectra x 1 x n and y 1 y n respectively for a wide class of pairs of essentially hermitian matrices x y we prove that the determinant of x y belongs to the convex hull of the n points n i 1 x i y i s n

some notes on de oliveira s determinantal conjecture abstract let a b u n x n a diag a j b diag b j u unitary d det a u b u h z j 1 n a j b j s n de oliveira s conjecture d is a convex combination of the z s is shown to be true if all the z s are collinear the question of representing u m 2 u m is the m th compound of u as a convex combination of the p m s is also studied here and 2 are understood elementwise and p denotes the permutation matrix with rows corresponding to

a cubic defining algebra for the links gould polynomial abstract we define a finite dimensional cubic quotient of the group algebra of the braid group endowed with a essentially unique markov trace which affords the links gould invariant of knots and links we investigate several of its properties and state several conjectures about its structure

a nonconvex splitting method for symmetric nonnegative matrix factorization convergence analysis and optimality symmetric nonnegative matrix factorization symnmf has important applications in data analytics problems such as document clustering community detection and image segmentation in this paper we propose a novel nonconvex variable splitting method for solving symnmf the proposed algorithm is guaranteed to converge to the set of karush kuhn tucker kkt points of the nonconvex symnmf problem furthermore it achieves a global sublinear convergence rate we also show that the algorithm can be efficiently implemented in parallel further sufficient conditions are provided that guarantee the global and local optimality of the obtained solutions extensive numerical results performed on both synthetic and real datasets suggest that the proposed algorithm converges quickly to a local minimum solution

non negative matrix factorization with fixed row and column sums in this short note we focus on the use of the generalized kullback leibler kl divergence in the problem of non negative matrix factorization nm f we will show that when using the generalized kl divergence as cost function for nivif the row sums and the column sums of the original matrix are preserved in the approximation we will use this special characteristic in several approximation problems c 2007 elsevier inc all rights reserved

monoids in the fundamental groups of the complement of logarithmic free divisors in c3 abstract we study monoids generated by certain zariski van kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in c 3 listed by sekiguchi these monoids admit positive homogeneous presentations theorem 1 five of them are artin monoids and eight of them are free abelian monoids the remaining four monoids are not gausian and hence are neither garside nor artin theorem 2 however we introduce the concept of fundamental elements for positive homogeneously presented monoids and we show that all 17 monoids possess fundamental elements theorem 3 as an application of the study of monoids we solve some decision problems for the fundamental groups except in three cases

on a recent determinantal inequality abstract queiro and kovacec have recently established an inequality for the modulus of the determinant of the sum of two normal matrices their inequality involves the eigenvalues of these two matrices in this note we discuss the equality case and obtain a result which supports a long standing conjecture due to marcus

on the quotients of cubic hecke algebras between the rank 3 quotients of cubic hecke algebras there is essentially one of maximal dimension we prove it has a unique markov trace having values in a torsion module therefore the description of a markov trace on the cubic hecke algebra corresponding tox 3 1 and having the parameters 1 1 is derived thus we obtain a numerical link invariant of finite degree and define a whole sequence of 3rd order vassiliev invariants

non negative matrix factorization algorithm for the deconvolution of one dimensional chromatograms in chromatogram analysis overlapped chromatograms are difficult to analyze if they are not resolved the conventional multivariate resolution techniques do not give accurate results when the chromatograms are severely overlapped in this work ml nmfdiv modified non negative matrix factorization nmf with divergence objective algorithm has been proposed for the separation of severely overlapped chromatograms of acetone and acrolein mixture before applying nmf principal component analysis pca is applied to determine number of components in the mixture taken most of the nmf algorithms used so far for chromatogram separation do not converge to a stable limit point and no uniqueness in the results to get unique results instead of random initialization three different initialization methods namely robust initialization nndsvd non negative double singular value decomposition based initialization and efa evolving factor analysis based initializations have been used in this work and the performances are compared the multiplicative update of already existing nmfdiv algorithm has been modified and proposed in this work as ml nmfdiv nmfdiv with modified multiplicative update for overlapped chromatogram separation to improve the convergence the proposed ml nmfdiv algorithm is applied on the simulated and experimental chromatograms obtained for acetone and acrolein mixture the results of proposed ml nmfdiv are compared with existing multivariate curve resolution alternating least square mcr als method

knot polynomials and vassiliev s invariants a fundamental relationship is established between jones knot invariants and vassiliev s knot invariants since vassiliev s knot invariants have a firm grounding in classical topology one obtains as a result a first step in understanding the jones polynomial by topological methods

projective nonnegative graph embedding we present in this paper a general formulation for nonnegative data factorization called projective nonnegative graph embedding pnge which 1 explicitly decomposes the data into two nonnegative components favoring the characteristics encoded by the so called intrinsic and penalty graphs respectively and 2 explicitly describes how to transform each new testing sample into its low dimensional nonnegative representation in the past such a nonnegative decomposition was often obtained for the training samples only e g nonnegative matrix factorization nmf and its variants nonnegative graph embedding nge and its refined version multiplicative nonnegative graph embedding mnge those conventional approaches for out of sample extension either suffer from the high computational cost or violate the basic nonnegative assumption in this work pnge offers a unified solution to out of sample extension problem and the nonnegative coefficient vector of each datum is assumed to be projected from its original feature representation with a universal nonnegative transformation matrix a convergency provable multiplicative nonnegative updating rule is then derived to learn the basis matrix and transformation matrix extensive experiments compared with the state of the art algorithms on nonnegative data factorization demonstrate the algorithmic properties in convergency sparsity and classification power

efficient and non convex coordinate descent for symmetric nonnegative matrix factorization given a symmetric nonnegative matrix a symmetric nonnegative matrix factorization symnmf is the problem of finding a nonnegative matrix h usually with much fewer columns than a such that a approx hh t symnmf can be used for data analysis and in particular for various clustering tasks unlike standard nmf which is traditionally solved by a series of quadratic convex subproblems we propose to solve symnmf by directly solving the nonconvex problem namely minimize vert a hh t vert 2 which is a fourth order nonconvex problem in this paper we propose simple and very efficient coordinate descent schemes which solve a series of fourth order univariate subproblems exactly we also derive convergence guarantees for our methods and show that they perform favorably compared to recent state of the art methods on synthetic and real world datasets especially on large and sparse input matrices

on the nonnegative rank of euclidean distance matrices abstract the euclidean distance matrix for n distinct points in r r is generically of rank r 2 it is shown in this paper via a geometric argument that its nonnegative rank for the case r 1 is generically n