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

fast nonnegative matrix factorization an active set like method and comparisons nonnegative matrix factorization nmf is a dimension reduction method that has been widely used for numerous applications including text mining computer vision pattern discovery and bioinformatics a mathematical formulation for nmf appears as a nonconvex optimization problem and various types of algorithms have been devised to solve the problem the alternating nonnegative least squares anls framework is a block coordinate descent approach for solving nmf which was recently shown to be theoretically sound and empirically efficient in this paper we present a novel algorithm for nmf based on the anls framework our new algorithm builds upon the block principal pivoting method for the nonnegativity constrained least squares problem that overcomes a limitation of the active set method we introduce ideas that efficiently extend the block principal pivoting method within the context of nmf computation our algorithm inherits the convergence property of the anls framework and can easily be extended to other constrained nmf formulations extensive computational comparisons using data sets that are from real life applications as well as those artificially generated show that the proposed algorithm provides state of the art performance in terms of computational speed

interior and boundary continuity of weak solutions of degenerate parabolic equations in this paper we consider degenerate parabolic equations of the form bf u t diva x t u u b x t u u d 0 where a and b are respectively vector and scalar valued baire functions defined on u x r1 x r where u is an open subset of rn x t the functions a and b are subject to natural structural inequalities sufficiently general conditions are allowed on the relation b c r1 x rl so that the porus medium equation and the model for the two phase stefan problem can be considered the main result of the paper is that weak solutions of are continuous throughout u in the event that u q x 0 t where q is an open set of r it is also shown that a weak solution is continuous at xo to c aq x 0 t provided xo is a regular point for the laplacian on u 0 introduction in this paper we consider the problem of continuity of weak solutions of parabolic equations of the form at 1 tf3 u diva x t u ux b x t u ux 3 0 where a and b are respectively vector and scalar valued baire functions defined on u x r x rn and u is an open subset of r n x t throughout this paper u will take the form qt q x 0 t where q is an open bounded subset of rn the functions a and b are required to satisfy the following structural inequalities ia x t u p i co p 12 c1 x t i u 12 c2 x t where ao bo and c0 are constants with ao and bo nonnegative and c0 0 the remaining coefficients are nonnegative measurable functions which for simplicity of exposition we will assume to be bounded thus we assume there is a nonnegative constant k such that 3 a x t k b1 x t k ci x t k our results remain valid if the coefficients lie in those appropriate lebesgue classes which have been considered by previous authors e g lsu t throughout we received by the editors march 28 1981 1980 mathematics subject classification primary 35k10 35k15 secondary 35k20 35k65 research supported in part by a grant from the national science foundation d1982 american mathematical society 0002 9947 81 0000 0376 05 00

efficient nonnegative matrix factorization via projected newton method nonnegative matrix factorization nmf is a popular decomposition technique in pattern analysis document clustering image processing and related fields in this paper we propose a fast nmf algorithm via projected newton method pnm first we propose pnm to efficiently solve a nonnegative least squares problem which achieves a quadratic convergence rate under appropriate assumptions second in the framework of an alternating optimization method we adopt pnm as an essential subroutine to efficiently solve the nmf problem moreover by exploiting the low rank assumption of nmf we make pnm very suitable for solving nmf efficiently empirical studies on both synthetic and real world text and image data demonstrate that pnm is quite efficient to solve nmf compared with several state of the art algorithms

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

definition and properties of supersolutions to the porous medium equation we study a wide class of supersolutions of the porous medium equation these supersolutions are defined as lower semi continuous functions obeying the comparison principle we show that they have a spatial sobolev gradient and give sharp summab ility exponents we also study pointwise behaviour

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

the obstacle problem for the porous medium equation we prove existence results for the obstacle problem related to the porous medium equation for sufficiently regular obstacles we find continuous solutions whose time derivative belongs to the dual of a parabolic sobolev space we also employ the notion of weak solutions and show that for more general obstacles such a weak solution exists the latter result is a consequence of a stability property of weak solutions with respect to the obstacle

non negative matrix factorization with sparseness constraints non negative matrix factorization nmf is a recently developed technique for finding parts based linear representations of non negative data although it has successfully been applied in several applications it does not always result in parts based representations in this paper we show how explicitly incorporating the notion of sparseness improves the found decompositions additionally we provide complete matlab code both for standard nmf and for our extension our hope is that this will further the application of these methods to solving novel data analysis problems

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

on some sets of generators of finite groups abstract in several papers finite groups with fixed cardinalities of sets of generators are investigated and sometimes are named b groups groups with all subgroups being b groups are called groups with the basis property in the first part of this paper we correct and generalize some characterizations of groups with the basis property in the second part we consider groups satisfying analogous conditions but only for sets of generators of prime power orders the class of groups introduced here is much larger then the class of groups with the basis property for example it contains all nilpotent groups

the petrovskiĭ criterion and barriers for degenerate and singular p parabolic equations in this paper we obtain sharp petrovski criteria for the p parabolic equation both in the degenerate case p 2 and the singular case 1 p 2 we also give an example of an irregular boundary point at which there is a barrier thus showing that regularity cannot be characterized by the existence of just one barrier

improving non negative matrix factorizations through structured initialization in this paper we explore a recent iterative compression technique called non negative matrix factorization nmf several special properties are obtained as a result of the constrained optimization problem of nmf for facial images the additive nature of nmf results in a basis of features such as eyes noses and lips we explore various methods for efficiently computing nmf placing particular emphasis on the initialization of current algorithms we propose using spherical k means clustering to produce a structured initialization for nmf we demonstrate some of the properties that result from this initialization and develop an efficient way of choosing the rank of the low dimensional nmf representation

boundary regularity for the porous medium equation we study the boundary regularity of solutions to the porous medium equation u t delta u m in the degenerate range m 1 in particular we show that in cylinders the dirichlet problem with positive continuous boundary data on the parabolic boundary has a solution which attains the boundary values provided that the spatial domain satisfies the elliptic wiener criterion this condition is known to be optimal and it is a consequence of our main theorem which establishes a barrier characterization of regular boundary points for general not necessarily cylindrical domains in bf r n 1 one of our fundamental tools is a new strict comparison principle between sub and superparabolic functions which makes it essential for us to study both nonstrict and strict perron solutions to be able to develop a fruitful boundary regularity theory several other comparison principles and pasting lemmas are also obtained in the process we obtain a rather complete picture of the relation between sub superparabolic functions and weak sub supersolutions

on the dirichlet problem for the nonlinear diffusion equation in non smooth domains abstract we study the dirichlet problem for the parabolic equation u t u m m 0 in a bounded non cylindrical and non smooth domain r n 1 n 2 existence and boundary regularity results are established we introduce a notion of parabolic modulus of left lower or left upper semicontinuity at the points of the lateral boundary manifold and show that the upper or lower holder condition on it plays a crucial role for the boundary continuity of the constructed solution the holder exponent 1 2 is critical as in the classical theory of the one dimensional heat equation u t u xx

well posedness of the dirichlet problem for the non linear diffusion equation in non smooth domains we investigate the dirichlet problem for the parablic equation u t u m m 0 in a non smooth domain r n 1 n 2 in a recent paper u g abdulla j math anal appl 260 2 2001 384 403 existence and boundary regularity results were established in this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial boundary data in particular we prove l 1 contraction estimation in general non smooth domains

nonnegative matrix factorization for rapid recovery of constituent spectra in magnetic resonance chemical shift imaging of the brain we present an algorithm for blindly recovering constituent source spectra from magnetic resonance mr chemical shift imaging csi of the human brain the algorithm which we call constrained nonnegative matrix factorization cnmf does not enforce independence or sparsity instead only requiring the source and mixing matrices to be nonnegative it is based on the nonnegative matrix factorization nmf algorithm extending it to include a constraint on the positivity of the amplitudes of the recovered spectra this constraint enables recovery of physically meaningful spectra even in the presence of noise that causes a significant number of the observation amplitudes to be negative we demonstrate and characterize the algorithm s performance using sup 31 p volumetric brain data comparing the results with two different blind source separation methods bayesian spectral decomposition bsd and nonnegative sparse coding nnsc we then incorporate the cnmf algorithm into a hierarchical decomposition framework showing that it can be used to recover tissue specific spectra given a processing hierarchy that proceeds coarse to fine we demonstrate the hierarchical procedure on sup 1 h brain data and conclude that the computational efficiency of the algorithm makes it well suited for use in diagnostic work up

a boundary modulus of continuity for a class of singular parabolic equations abstract parabolic equations describing diffusion phenomena with change of phase are considered it is demonstrated that weak solutions are continuous up to the parabolic boundary of the domain of definition the continuity is quantitatively described by a modulus determined a priori only in terms of the data

projected gradient methods for nonnegative matrix factorization nonnegative matrix factorization nmf can be formulated as a minimization problem with bound constraints although bound constrained optimization has been studied extensively in both theory and practice so far no study has formally applied its techniques to nmf in this letter we propose two projected gradient methods for nmf both of which exhibit strong optimization properties we discuss efficient implementations and demonstrate that one of the proposed methods converges faster than the popular multiplicative update approach a simple matlab code is also provided