lifts of convex sets and cone factorizations in this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices we show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual this generalizes a theorem of yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix symmetric lifts of convex sets can also be characterized similarly when the cones live in a family our results lead to the definition of the rank of a convex set with respect to this family we present results about this rank in the context of cones of positive semidefinite matrices our methods provide new tools for understanding cone lifts of convex sets

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

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

galois objects with normal bases for free hopf algebras of prime degree suppose r is a commutative ring and h a commutative cocommutative hopf algebra which is a free r module of prime rank the set of isomorphism classes of r algebras which are galois h objects forms an abelian group see chase and sweedler a in this paper we show that when h is an algebra over a special subring of the p adic integers containing a full set of p 1 st roots of unity then the subring consisting of those galois objects which are isomorphic to h as h module h hom h r is isomorphic to ue r u r where the notation u r means units of r congruent to 1 modx and 5 is a p 1 st root of an element b of r which will be seen to characterize the hopf algebra h the paper is organised as follows we briefly review notation related to hopf algebras integrals and galois objects in section 0 in section 1 we present the description of hopf algebras of order p given by tate and oort in their paper group schemes of prime order s such a hopf algebra h is associated with a pair of constants a b in fact hz r x x ax and h x r x xp bx where ab wi some unit of r in section 2 we describe the h module structure of h and the h module structure of a galois h object s and we state what it means for s to have a normal basis s z h as h module in section 3 we assume r contains a p l st root of 6 we show rz pz c h from which it follows that s contains a kummer order c modp s with s r and sis si i

noether s theorem for hopf orders in group algebras let k be a local field with valuation ring r of residue character istic p containing a primitive pth root of unity fp we state an analog to noether s theorem for modules over i hopf algebras and use induction tech niques to deduce a criterion for this analog to hold we then construct a family of noncommutative hopf algebras which satisfy the criterion if l k is a galois extension of number fields with group g and rings of integers s r then in a natural way l is a a g module and s is an rg module the normal basis theorem asserts that l kg as kg modu es the analogous result does not generally hold for s and rg even after localizing r in 1931 e noether n showed that locally for all primes p of r s rg as left 5g modules if and only if the extension l k is tame that is for every prime p of r the ramification index of any prime p of s lying over p is relatively prime to the characteristic of the residue field r p in fact the following are equivalent 1 s rg locally as 5g modules 2 l k is tame 3 the trace map s r is surjective 4 5 is a projective 5g module in 1986 l childs and s hurley c h generalized the notions of tameness and local normal basis to modules where h is an 5 hopf algebra their definitions specialized to those above when it rg and they obtained in par ticular the result that when h is commutative s h locally as modules if and only if s is tame waterhouse w notes that the childs hurley notions of tameness do not imply the existence of local normal basis when h is noncommutative even for group rings over fields the counterexample is not entirely satisfactory in that any natural number theoretic example would have normal basis over the quotient field in accordance with a result of kreimer and cook k c note that the childs hurley results assume commutativity of h yet noether s theorem holds for arbitrary finite groups accordingly there seems reason to believe that analogous results should hold for at least some non commutative h other than group algebras we say that noether s theorem

learning hidden markov models using nonnegative matrix factorization the baum welch algorithm together with its derivatives and variations has been the main technique for learning hidden markov models hmms from observational data we present an hmm learning algorithm based on the nonnegative matrix factorization nmf of higher order markovian statistics that is structurally different from the baum welch and its associated approaches the described algorithm supports estimation of the number of recurrent states of an hmm and iterates the nmf algorithm to improve the learned hmm parameters numerical examples are provided as well

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

maximal tame extensions over hopf orders in rings of integers of p adic number fields abstract let k k be a cyclic totally ramified kummer extension of degree pn with galois group g let o and o be the rings of integers in k and k respectively for n 1 f bertrandias and m j ferton determined the ring end o g o of o g endomorphisms of o and l n childs constructed the maximal order s in o whose ring of o g endomorphisms is a hopf order a hopf order whose linear dual is a larson order in kg is called a dual larson order in this paper the generators of a dual larson order are given we determine end o g o and obtain a maximal dual larson order contained in end o g o we construct a hopf galois extension s in o whose ring h s of o g endomorphisms is a dual larson order we affirm an order s in o with a dual larson order h s is a tame h s extension and show that s is maximal in such orders s

hopf galois kummer theory of formal groups let f g be an isogeny between finite n dimensional formal groups defined over r the valuation ring of some field extension k of qp let h be the hopf algebra which arises from this isogeny for such h we classify gal h the group of 7 galois objects let ai be the maximal ideal of r and let p f k denote the n tuples of m under the group operation induced by f our main result is the construction of an isomorphism from the cokernel of p f to gal where p f is the induced map from p f k to p g k in geometric language gal describes the group of isomorphism classes of principal homogeneous spaces for spec over spec geometric methods have been used by mazur to establish the above isomorphism but the proof is nonconstructive a geometric approach has also provided a formula for the cardinality of gal we give an alternative derivation of this result using formal group techniques introduction given a field k and a finite abelian group g a question in classical field theory is to determine gal k g the isomorphism classes of galois field extensions of k with group g when k has characteristic zero and contains a primitive nth root of unity classical kummer theory asserts that gal k cn u k u k n where u k denotes the units of k and cn is the cyclic group of order n furthermore there is an explicit isomorphism from u k u k n to gal cn given by sending u g u k mod u k n to k x xn u in their development of hopf galois theory chase and sweedier 1 provide criteria for an action or co action of a cocommutative hopf algebra on an exten sion of rings to be galois an algebra s which admits a galois co action by an hopf algebra h is called an galois object given a ring r and a cocommu tative hopf algebra a basic question in this theory is to determine gal the group of isomorphism classes of galois objects when k is a field and h is the dual of the group ring kg g a finite group gal becomes gal k g determining gal is also of great interest because in geometric language this describes the group of isomorphism classes of principal homogeneous spaces for spec over spec we make the conventions that all rings are commutative all formal groups are commutative and all hopf algebras are abelian commutative and cocommu tative an hopf algebra is called finite if it is finitely generated and projective as an module

on fuzzy congruence of a near ring module the aim of this paper is to introduce fuzzy submodule and fuzzy congruence of an r module near ring module to obtain the correspondence between fuzzy congruences and fuzzy submodules of an r module to define quotient r module of an r module over a fuzzy submodule and to obtain correspondence between fuzzy congruences of an r module and fuzzy congruences of quotient r module over a fuzzy submodule of an r module

fast and robust recursive algorithmsfor separable nonnegative matrix factorization in this paper we study the nonnegative matrix factorization problem under the separability assumption that is there exists a cone spanned by a small subset of the columns of the input nonnegative data matrix containing all columns which is equivalent to the hyperspectral unmixing problem under the linear mixing model and the pure pixel assumption we present a family of fast recursive algorithms and prove they are robust under any small perturbations of the input data matrix this family generalizes several existing hyperspectral unmixing algorithms and hence provides for the first time a theoretical justification of their better practical performance

sequential dimensionality reduction for extracting localized features abstract linear dimensionality reduction techniques are powerful tools for image analysis as they allow the identification of important features in a data set in particular nonnegative matrix factorization nmf has become very popular as it is able to extract sparse localized and easily interpretable features by imposing an additive combination of nonnegative basis elements nonnegative matrix underapproximation nmu is a closely related technique that has the advantage to identify features sequentially in this paper we propose a variant of nmu that is particularly well suited for image analysis as it incorporates the spatial information that is it takes into account the fact that neighboring pixels are more likely to be contained in the same features and favors the extraction of localized features by looking for sparse basis elements we show that our new approach competes favorably with comparable state of the art techniques on synthetic facial and hyperspectral image data sets

relative galois module structure of rings of integers and elliptic functions iii in this paper we consider the problem of describing the galois module structure of rings of integers of abelian extensions of a quadratic imaginary number field k in certain cases by use of elliptic functions we are able to exhibit specific generators of such rings of integers as galois modules the general problem of finding galois integral generators by means of evaluation of transcendental functions has been considered by a number of mathematicians our results represent the first major advance outside the cyclotomic case we recall that the principal cyclotomic results are the hilbert speiser theorem and more generally the following result of leopoldt cf le theorem 1 let e denote an abelian extension of q let f gal e q

a new view of fuzzy hypernear rings in this survey article we first introduce the concept of quasi coincidence of a fuzzy interval value with an interval valued fuzzy set this concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we consider the interval valued q fuzzy sub hypernear rings hyperideals of a hypernear ring and hence a generalization of a fuzzy sub near ring ideal is given some related properties of fuzzy hypernear rings are described finally we consider the concept of implication based interval valued fuzzy sub hypernear rings hyperideals in a hypernear ring in particular the implication operators in lukasiewicz system of continuous valued logic are discussed

worst case results for positive semidefinite rank we present various worst case results on the positive semidefinite psd rank of a nonnegative matrix primarily in the context of polytopes we prove that the psd rank of a generic n n dimensional polytope with v v vertices is at least nv frac 1 4 nv 14 improving on previous lower bounds for polygons with v v vertices we show that psd rank cannot exceed 4 left lceil v 6 right rceil 4v 6 which in turn shows that the psd rank of a p times q p q matrix of rank three is at most 4 left lceil min p q 6 right rceil 4min p q 6 in general a nonnegative matrix of rank k 1 atopwithdelims 2 k 12 has psd rank at least k k and we pose the problem of deciding whether the psd rank is exactly k k using geometry and bounds on quantifier elimination we show that this decision can be made in polynomial time when k k is fixed

the lower and upper approximations in a hypergroup this paper presents a relationship between rough sets and hypergroup theory we analyze the lower and upper approximations of a subset with respect to an invertible subhypergroup and we consider some particular situations moreover the notion of a rough subhypergroup is introduced finally fuzzy rough subhypergroups are introduced and characterized

positive semidefinite rank let m in mathbb r p times q m rp q be a nonnegative matrix the positive semidefinite rank psd rank of m is the smallest integer k for which there exist positive semidefinite matrices a i b j ai bj of size k times k k k such that m ij mathrm trace a i b j mij trace aibj the psd rank has many appealing geometric interpretations including semidefinite representations of polyhedra and information theoretic applications in this paper we develop and survey the main mathematical properties of psd rank including its geometry relationships with other rank notions and computational and algorithmic aspects

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

robustness analysis of preconditioned successive projection algorithm for general form of separable nmf problem abstract the successive projection algorithm spa has been known to work well for separable nonnegative matrix factorization nmf problems arising in applications such as topic extraction from documents and endmember detection in hyperspectral images one of the reasons is in that the algorithm is robust to noise gillis and vavasis showed in 8 that a preconditioner can further enhance its noise robustness the proof rested on the condition that the dimension d and factorization rank r in the separable nmf problem coincide with each other however it may be unrealistic to expect that the condition holds in separable nmf problems appearing in actual applications in such problems d is usually greater than r this paper shows without the condition d r that the preconditioned spa is robust to noise

monogenic hopf algebras over discrete valuation rings with low ramification abstract let k be an extension of q p with absolute ramification index 1 e p 1 let r o k and let k be the residue field of r we show that any monogenic finite abelian local hopf algebra with local dual lifts to r and we construct all such lifts for h an r hopf algebra arising from one of these lifts we realize spech as the kernel of an isogeny of one dimensional formal groups we then obtain a complete list of fields l for which l k is an h k galois object

fuzzy γ hypernear rings the concept of c hypernear rings which is a generalization of c near rings and hypernear rings is introduced some related properties of c hypernear rings are described in particular three isomorphism theorems of c hypernear rings are established by using this new idea we consider the fuzzy hyperideals of c hypernear rings finally the fundamental relations between c hypernear rings and c near rings are discussed

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

a flexible and efficient algorithmic framework for constrained matrix and tensor factorization we propose a general algorithmic framework for constrained matrix and tensor factorization which is widely used in signal processing and machine learning the new framework is a hybrid between alternating optimization ao and the alternating direction method of multipliers admm each matrix factor is updated in turn using admm hence the name ao admm this combination can naturally accommodate a great variety of constraints on the factor matrices and almost all possible loss measures for the fitting computation caching and warm start strategies are used to ensure that each update is evaluated efficiently while the outer ao framework exploits recent developments in block coordinate descent bcd type methods which help ensure that every limit point is a stationary point as well as faster and more robust convergence in practice three special cases are studied in detail non negative matrix tensor factorization constrained matrix tensor completion and dictionary learning extensive simulations and experiments with real data are used to showcase the effectiveness and broad applicability of the proposed framework

pawlak s approximations in γ semihypergroups the c semihypergroup is a generalization of a semigroup a generalization of a semihypergroup and a generalization of a c semigroup in this paper we discuss the roughness of c subsemihypergroups and c hyperideals bi c hyperideals in c semihypergroups in addition we consider rough sets with respect to the idempotent regular relations in a quotient c semihypergroup the upper approximations with respect to green s equivalence relations are derived in the context of c subsemihypergroups and c hyperideals

sparse and non negative bss for noisy data non negative blind source separation bss has raised interest in various fields of research as testified by the wide literature on the topic of non negative matrix factorization nmf in this context it is fundamental that the sources to be estimated present some diversity in order to be efficiently retrieved sparsity is known to enhance such contrast between the sources while producing very robust approaches especially to noise in this paper we introduce a new algorithm in order to tackle the blind separation of non negative sparse sources from noisy measurements we first show that sparsity and non negativity constraints have to be carefully applied on the sought after solution in fact improperly constrained solutions are unlikely to be stable and are therefore sub optimal the proposed algorithm named ngmca non negative generalized morphological component analysis makes use of proximal calculus techniques to provide properly constrained solutions the performance of ngmca compared to other state of the art algorithms is demonstrated by numerical experiments encompassing a wide variety of settings with negligible parameter tuning in particular ngmca is shown to provide robustness to noise and performs well on synthetic mixtures of real nmr spectra

some 0 1 polytopes need exponential size extended formulations we prove that there are 0 1 polytopes p subseteq mathbb r n that do not admit a compact lp formulation more precisely we show that for every n there is a set x subseteq 0 1 n such that conv x must have extension complexity at least 2 n 2 cdot 1 o 1 in other words every polyhedron q that can be linearly projected on conv x must have exponentially many facets in fact the same result also applies if conv x is restricted to be a matroid polytope conditioning on mathbf np not subseteq mathbf p poly our result rules out the existence of a compact formulation for any mathbf np hard optimization problem even if the formulation may contain arbitrary real numbers

a multilevel approach for nonnegative matrix factorization nonnegative matrix factorization nmf the problem of approximating a nonnegative matrix with the product of two low rank nonnegative matrices has been shown to be useful in many applications such as text mining image processing and computational biology in this paper we explain how algorithms for nmf can be embedded into the framework of multilevel methods in order to accelerate their initial convergence this technique can be applied in situations where data admit a good approximate representation in a lower dimensional space through linear transformations preserving nonnegativity several simple multilevel strategies are described and are experimentally shown to speed up significantly three popular nmf algorithms alternating nonnegative least squares multiplicative updates and hierarchical alternating least squares on several standard image datasets

monogenic hopf algebras and local galois module theory let k be an unramified extension of qp and denote the ring of integers of k by r ok let h be an r hopf algebra with monogenic dual h we realize h as the kernel of an isogeny of one dimensional formal groups this allows us to give a complete list of fields l for which l k is h k hopf galois and s ol is a free h module

robust near separable nonnegative matrix factorization using linear optimization nonnegative matrix factorization nmf has been shown recently to be tractable under the separability assumption under which all the columns of the input data matrix belong to the convex cone generated by only a few of these columns bittorf recht re and tropp factoring nonnegative matrices with linear programs nips 2012 proposed a linear programming lp model referred to as hottopixx which is robust under any small perturbation of the input matrix however hottopixx has two important drawbacks i the input matrix has to be normalized and ii the factorization rank has to be known in advance in this paper we generalize hottopixx in order to resolve these two drawbacks that is we propose a new lp model which does not require normalization and detects the factorization rank automatically moreover the new lp model is more flexible significantly more tolerant to noise and can easily be adapted to handle outliers and other noise models finally we show on several synthetic data sets that it outperforms hottopixx while competing favorably with two state of the art methods

uniqueness of nonnegative tensor approximations we show that for a nonnegative tensor a best nonnegative rank r approximation is almost always unique its best rank one approximation may always be chosen to be a best nonnegative rank one approximation and the set of nonnegative tensors with nonunique best rank one approximations forms an algebraic hypersurface we show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor that does not satisfy this equation is guaranteed to have a unique best rank one approximation we also establish an analogue for real or nonnegative symmetric tensors in addition we prove a singular vector variant of the perron frobenius theorem for positive tensors and apply it to show that a best nonnegative rank r approximation of a positive tensor can never be obtained by deflation as an aside we verify that the euclidean distance ed discriminants of the segre variety and the veronese variety are hypersurfaces and give defining equations of these ed discriminants

topological hypergroupoids hypergroups are generalizations of groups if this binary operation is taken to be multivalued then we arrive at a hypergroup the motivation for generalization of the notion of group resulted naturally from various problems in non commutative algebra another motivation for such an investigation came from geometry in various branches of mathematics we encounter important examples of topologico algebraical structures like topological groupoids groups rings fields etc in this contribution various kinds of continuity of hyperoperations will be introduced namely pseudocontinuous strongly pseudocontinuous and continuous hyperoperations further the relationship between them is studied our aim is to generalize the concept of topological groupoid to topological hypergroupoid

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 fuzzy subnear rings and ideals abstract the aim of this paper is to introduce the notion of a fuzzy subnear ring to study fuzzy ideals of a near ring and to give some properties of fuzzy prime ideals of a near ring

normal fuzzy r subgroups in near rings in this paper we introduce the notion of a normal fuzzy r subgroup in a near ring and investigate some related properties

non parametric identification and estimation of the number of components in multivariate mixtures this article analyzes the identiability of the number of components in k variate mcomponent nite mixture models in which each component distribution has independent marginals including models in latent class analysis without making parametric assumptions on the component distributions we investigate how one can identify the number of components from the distribution function of the observed data when k 2 a lower bound on the number of components m is nonparametrically identiable from the rank of a matrix constructed from the distribution function of the observed variables building on this identication condition we develop a procedure to consistently estimate a lower bound on the number of components

online nonnegative matrix factorization with outliers we propose a unified and systematic framework for performing online nonnegative matrix factorization in the presence of outliers our framework is particularly suited to large scale data we propose two solvers based on projected gradient descent and the alternating direction method of multipliers we prove that the sequence of objective values converges almost surely by appealing to the quasi martingale convergence theorem we also show the sequence of learned dictionaries converges to the set of stationary points of the expected loss function almost surely in addition we extend our basic problem formulation to various settings with different constraints and regularizers we also adapt the solvers and analyses to each setting we perform extensive experiments on both synthetic and real datasets these experiments demonstrate the computational efficiency and efficacy of our algorithms on tasks such as parts based basis learning image denoising shadow removal and foreground background separation

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

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

finding approximately rank one submatrices with the nuclear norm and ell_1 norm we propose a convex optimization formulation with the nuclear norm and ell 1 norm to find a large approximately rank one submatrix of a given nonnegative matrix we develop optimality conditions for the formulation and characterize the properties of the optimal solutions we establish conditions under which the optimal solution of the convex formulation has a specific sparse structure finally we show that under certain hypotheses with high probability the approach can recover the rank one submatrix even when it is corrupted with random noise and inserted as a submatrix into a much larger random noise matrix

the valuative condition and r hopf algebra orders in kc p 3 let k be a finite extension of qp endowed with the p adic valuation and let r be its ring of integers in this paper we give a complete classification of r hopf algebra orders in the group ring kcp3 for an arbitrary hopf order h we show that either the group scheme sph or the group scheme sph with h the linear dual can be viewed as the middle term of the baer product of a distinguished extension with a generically trivial extension using this characterization we then compute algebra generators for either h or h

on fuzzy isomorphism theorems of γ hyperrings in this paper we first derive three isomorphism theorems and a jordan holder theorem of c hyperrings by fuzzy c hyperideals we establish three fuzzy isomorphism theorems of c hyperrings

monogenic hopf orders and associated orders of valuation rings abstract let r be a complete discrete valuation ring of mixed characteristic with perfect residue field and let h be a finite local commutative r hopf algebra we consider when there exists a finite extension of the field of fractions of r whose valuation ring is a galois h object if this occurs then h is monogenic conversely if h is also cocommutative and h is monogenic then there exists a valuation ring which is a galois h object to prove this result we represent h as the kernel of an isogeny of a special type between formal groups over r we deduce that if a is a finite abelian r hopf algebra such that both a and its dual are local then a is the associated order of a valuation ring if and only if the dual of a is monogenic

semidefinite programming based preconditioning for more robust near separable nonnegative matrix factorization nonnegative matrix factorization nmf under the separability assumption can provably be solved efficiently even in the presence of noise and has been shown to be a powerful technique in document classification and hyperspectral unmixing this problem is referred to as near separable nmf and requires that there exists a cone spanned by a small subset of the columns of the input nonnegative matrix approximately containing all columns in this paper we propose a preconditioning based on the minimum volume ellipsoid and semidefinite programming making the input matrix well conditioned this in turn can improve significantly the performance of near separable nmf algorithms which is illustrated on the popular successive projection algorithm spa the new preconditioned spa is provably more robust to noise and outperforms spa on several synthetic data sets we also show how an active set method allows us to apply the preconditioning on large scale real world hyperspectral images

efficient nonnegative matrix factorization by dc programming and dca in this letter we consider the nonnegative matrix factorization nmf problem and several nmf variants two approaches based on dc difference of convex functions programming and dca dc algorithm are developed the first approach follows the alternating framework that requires solving at each iteration two nonnegativity constrained least squares subproblems for which dca based schemes are investigated the convergence property of the proposed algorithm is carefully studied we show that with suitable dc decompositions our algorithm generates most of the standard methods for the nmf problem the second approach directly applies dca on the whole nmf problem two algorithms one computing all variables and one deploying a variable selection strategy are proposed the proposed methods are then adapted to solve various nmf variants including the nonnegative factorization the smooth regularization nmf the sparse regularization nmf the multilayer nmf the convex convex hull nmf and the symmetric nmf we also show that our algorithms include several existing methods for these nmf variants as special versions the efficiency of the proposed approaches is empirically demonstrated on both real world and synthetic data sets it turns out that our algorithms compete favorably with five state of the art alternating nonnegative least squares algorithms

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

algebraic boundary of matrices of nonnegative rank at most three understanding the boundary of the set of matrices of nonnegative rank at most r is important for applications in nonconvex optimization the zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible we give a minimal generating set for the ideal of each irreducible component in fact this generating set is a grobner basis with respect to the graded reverse lexicographic order this solves a conjecture by robeva sturmfels and the last author

the real nonnegative inverse eigenvalue problem is np hard abstract a list of complex numbers is realizable if it is the spectrum of a nonnegative matrix in 1949 suleimanova posed the nonnegative inverse eigenvalue problem niep the problem of determining which lists of complex numbers are realizable the version for reals of the niep rniep asks for realizable lists of real numbers in the literature there are many sufficient conditions or criteria for lists of real numbers to be realizable we will present an unified account of these criteria then we will see that the decision problem associated to the rniep is np hard and we will study the complexity for the decision problems associated to known criteria

counting hopf galois structures on non abelian galois field extensions let l be a field which is a galois extension of the field k with galois group g greither and pareigis gp87 showed that for many g there exist k hopf algebras h other than the group ring kg which make l into an h hopf galois extension of k or a galois h object in the sense of chase and sweedler cs69 using galois descent they translated the problem of determining the hopf galois structures on l k into one which depends only on the galois group g by this translation they showed for example that any galois extension with non abelian g admits at least one non classical hopf galois structure byott by96 further translated the problem to a more amenable group theoretic problem and showed that a galois extension l k of fields with group g has a unique hopf galois structure namely that by kg iff n the order of g is a burnside number that is is coprime to n euler s phi function of n this implies that g is cyclic of square free order

accelerated multiplicative updates and hierarchical als algorithms for nonnegative matrix factorization nonnegative matrix factorization nmf is a data analysis technique used in a great variety of applications such as text mining image processing hyperspectral data analysis computational biology and clustering in this letter we consider two well known algorithms designed to solve nmf problems the multiplicative updates of lee and seung and the hierarchical alternating least squares of cichocki et al we propose a simple way to significantly accelerate these schemes based on a careful analysis of the computational cost needed at each iteration while preserving their convergence properties this acceleration technique can also be applied to other algorithms which we illustrate on the projected gradient method of lin the efficiency of the accelerated algorithms is empirically demonstrated on image and text data sets and compares favorably with a state of the art alternating nonnegative least squares algorithm

a study on fuzzy interior ideals of γ semigroups in this paper we fuzzify the notions of interior ideals and characteristic interior ideals of a c semigroup and investigate some of their basic properties finally a characterization of a simple c semigroup in terms of fuzzy interior ideals is obtained

subtractive clustering for seeding non negative matrix factorizations non negative matrix factorization is a multivariate analysis method which is proven to be useful in many areas such as bio informatics molecular pattern discovery pattern recognition document clustering and so on it seeks a reduced representation of a multivariate data matrix into the product of basis and encoding matrices possessing only non negative elements in order to learn the so called part based representations of data all algorithms for computing non negative matrix factorization are iterative therefore particular emphasis must be placed on a proper initialization of nmf because of its local convergence the problem of selecting appropriate starting matrices becomes more complex when data possess special meaning as in document clustering in this paper we propose the adoption of the subtractive clustering algorithm as a scheme to generate initial matrices for non negative matrix factorization algorithms comparisons with other commonly adopted initializations of non negative matrix factorization algorithms have been performed and the proposed scheme reveals to be a good trade off between effectiveness and speed moreover the effectiveness of the proposed initialization to suggest a number of basis for nmf when data distances are estimated is illustrated when nmf is used for solving clustering problems where the number of groups in which the data are grouped is not known a priori the influence of a proper rank factor on the interpretability and the effectiveness of the results are also discussed

fixed points of the em algorithm and nonnegative rank boundaries matrices of nonnegative rank r represent mixtures of r independent distributions likelihood inference for this model leads to problems in real algebraic geometry that are addressed here for the first time we characterize the fixed point locus of expectation maximization and we study the boundary of the space of matrices with nonnegative rank at most 3 both of these are algebraic varieties with many irreducible components

a unified convergence analysis of the multiplicative update algorithm for regularized nonnegative matrix factorization the multiplicative update mu algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization nmf problems under a wide range of divergences and regularizers however theoretical convergence guarantees have only been derived for a few special divergences without regularization in this work we provide a conceptually simple self contained and unified proof for the convergence of the mu algorithm applied on nmf with a wide range of divergences and regularizers our main result shows the sequence of iterates i e pairs of basis and coefficient matrices produced by the mu algorithm converges to the set of stationary points of the nonconvex nmf optimization problem our proof strategy has the potential to open up new avenues for analyzing similar problems in machine learning and signal processing

factoring a band matrix over a semiring abstract matrix factorization problems over various semirings naturally arise in different contexts of modern pure and applied mathematics these problems are very hard in general and cause computational difficulties in applications we give a survey of what is known on the algorithmic complexity of boolean tropical nonnegative and positive semidefinite factorizations and we examine the behavior of the corresponding rank functions on matrices of bounded bandwidth we show that the boolean tropical and t norm versions of matrix factorization become polynomial time solvable when restricted to this class of matrices and we also show that the nonnegative rank of a tridiagonal matrix is easy to compute we recall several open problems from earlier papers on the topic and formulate a number of further questions

classification of the hopf galois structures on prime power radical extensions abstract let p be an odd prime and n a positive integer and let k be a field of characteristic zero let k k w with w p n a k where a is such that k k p n and let r denote the largest integer between 0 and n such that k k p r k p r where p r denotes a primitive p r th root of unity the extension k k is separable but not necessarily normal and by greither and pareigis is h galois with h a k hopf algebra form of a group ring kn where k is the normal closure of k k h is said to be almost classical if n k k the result is that if r n then there are p r hopf galois structures on k k for which the associated group n is cyclic of order p n of these p min r n r are almost classical and the rest are non almost classical when r n there are p n 1 h galois structures for which n c p n of which only one is almost classical finally we show that these are the only structures possible that is for this class of extensions n must be cyclic

integral hopf galois structures on degree p2 extensions of p adic fields abstract let l k be a totally ramified normal extension of p adic fields of degree p2 we investigate the behavior of the valuation ring o l in the various hopf galois structures on l k specifically we determine when o l is hopf galois with respect to a hopf order in the corresponding hopf algebra when this occurs o l is necessarily a free module over this hopf order we also determine which hopf orders can arise in this way for cyclic extensions l k of degree p2 l n childs has shown under certain restrictions on the ramification numbers that if o l is hopf galois with respect to a hopf order in one of the hopf galois structures on l k then the same is true in all p hopf galois structures on l k we show that this no longer holds if the ramification conditions are relaxed or if elementary abelian extensions of degree of p2 are considered we illustrate our results with a special family of kummer extensions and with certain extensions arising from lubin tate formal groups

tame realisable classes over hopf orders abstract letobe the ring of algebraic integers in a number fieldkand letgbe a finite abelian group mcculloh has characterised the classes in the locally free classgroup cl og which are realisable by rings of integers of tame normal extensions ofkwith galois groupg we extend this to certain extensions which in general need be neither tame nor normal by replacingogwith a hopf order a and introducing the strongly a tame orders as an analogue of tame rings of integers we also describe the classes realised by principal homogeneous spaces over the dual of a

compressed nonnegative matrix factorization is fast and accurate nonnegative matrix factorization nmf has an established reputation as a useful data analysis technique in numerous applications however its usage in practical situations is undergoing challenges in recent years the fundamental factor to this is the increasingly growing size of the datasets available and needed in the information sciences to address this in this work we propose to use structured random compression that is random projections that exploit the data structure for two nmf variants classical and separable in separable nmf snmf the left factors are a subset of the columns of the input matrix we present suitable formulations for each problem dealing with different representative algorithms within each one we show that the resulting compressed techniques are faster than their uncompressed variants vastly reduce memory demands and do not encompass any significant deterioration in performance the proposed structured random projections for snmf allow to deal with arbitrarily shaped large matrices beyond the standard limit of tall and skinny matrices granting access to very efficient computations in this general setting we accompany the algorithmic presentation with theoretical foundations and numerous and diverse examples showing the suitability of the proposed approaches our implementations are publicly available

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

realizable hopf orders in kcp3 abstract let k be a finite extension of the p adic rationals q p with ring of integers r let c p 3 denote the cyclic group of order p 3 in this paper we construct a new collection of hopf orders in the group ring kc p 3 we determine which of these hopf orders are realizable as galois groups

lower bounds on nonnegative rank via nonnegative nuclear norms the nonnegative rank of an entrywise nonnegative matrix a in mathbb r m times n a r m n is the smallest integer r r such that a a can be written as a uv a uv where u in mathbb r m times r u r m r and v in mathbb r r times n v r r n are both nonnegative the nonnegative rank arises in different areas such as combinatorial optimization and communication complexity computing this quantity is np hard in general and it is thus important to find efficient bounding techniques especially in the context of the aforementioned applications in this paper we propose a new lower bound on the nonnegative rank which unlike most existing lower bounds does not solely rely on the matrix sparsity pattern and applies to nonnegative matrices with arbitrary support the idea involves computing a certain nuclear norm with nonnegativity constraints which allows to lower bound the nonnegative rank in the same way the standard nuclear norm gives lower bounds on the standard rank our lower bound is expressed as the solution of a copositive programming problem and can be relaxed to obtain polynomial time computable lower bounds using semidefinite programming we compare our lower bound with existing ones and we show examples of matrices where our lower bound performs better than currently known ones

ellipsoidal rounding for nonnegative matrix factorization under noisy separability we present a numerical algorithm for nonnegative matrix factorization nmf problems under noisy separability an nmf problem under separability can be stated as one of finding all vertices of the convex hull of data points the research interest of this paper is to find the vectors as close to the vertices as possible in a situation in which noise is added to the data points our algorithm is designed to capture the shape of the convex hull of data points by using its enclosing ellipsoid we show that the algorithm has correctness and robustness properties from theoretical and practical perspectives correctness here means that if the data points do not contain any noise the algorithm can find the vertices of their convex hull robustness means that if the data points contain noise the algorithm can find the near vertices finally we apply the algorithm to document clustering and report the experimental results

self scaled bounds for atomic cone ranks applications to nonnegative rank and cp rank the nonnegative rank of a matrix a is the smallest integer r such that a can be written as the sum of r rank one nonnegative matrices the nonnegative rank has received a lot of attention recently due to its application in optimization probability and communication complexity in this paper we study a class of atomic rank functions defined on a convex cone which generalize several notions of positive ranks such as nonnegative rank or cp rank for completely positive matrices the main contribution of the paper is a new method to obtain lower bounds for such ranks additionally the bounds we propose can be computed by semidefinite programming using sum of squares relaxations the idea of the lower bound relies on an atomic norm approach where the atoms are self scaled according to the vector or matrix in the case of nonnegative rank of interest this results in a lower bound that is invariant under scaling and that enjoys other interesting structural properties for the case of the nonnegative rank we show that our bound has an appealing connection with existing combinatorial bounds and other norm based bounds for example we show that our lower bound is a non combinatorial version of the fractional rectangle cover number while the sum of squares relaxation is closely related to the lovasz bar vartheta number of the rectangle graph of the matrix we also prove that the lower bound is always greater than or equal to the hyperplane separation bound and other similar norm based bounds we also discuss the case of the tensor nonnegative rank as well as the cp rank and compare our bound with existing results

tameness and local normal bases for objects of finite hopf algebras let r be a commutative ring s an r algebra h a hopf ralgebra both finitely generated and projective as r modules and suppose s is an h object so that h homr h r acts on s via a measuring let i be the space of left integrals of h we say s has normal basis if s h as h modules and s has local normal bases if sp hp as hp modules for all prime ideals p of r when r is a perfect field h is commutative and cocommutative and certain obvious necessary conditions on s hold then s has normal basis if and only if is r sh if r is a domain with quotient field k h is cocommutative and l s or k has normal basis as h 0 k module then s has local normal bases if and only if is r sh suppose k is a number field with ring of integers r l is a finite galois extension of k with galois group g and s is the integral closure of r in l then g acts as a group of automorphisms of s relative galois module theory seeks to understand s as an rg module via this action the most basic question is to inquire whether s is locally isomorphic to rg that is for all primes p of r sp has a local normal basis as a free rp module this question was answered by emmy noether who showed that s has a local normal basis at every prime p of r if and only if l k is tamely ramified here tamely ramified means that at each prime ideal p of r the ramification index of any prime p of s lying over p is relatively prime to the residue field characteristic it is well known that this latter condition is equivalent to the surjectivity of the trace map tr s r tr s ezacg s the purpose of this paper is to formulate and prove an extension of noether s theorem to objects of finite cocommutative hopf algebras assume now only that r is a commutative ring h is a hopf r algebra which is a finitely generated projective r module and h homr h r the dual hopf algebra a commutative r algebra s is an h object if s is a right h comodule via a map a s s 0 h which is an r algebra homomorphism if s is an h object then a induces a measuring a h s s in the sense of sweedler 21 galois module theory in this setting is the study of s as an h module via the measuring a the trivial example is s h itself with a a the comultiplication on h so we say that the h object has normal basis if s h as h modules and s has local normal bases if for all primes p of r sp hp as h modules since rgf rg as rg modules for g any finite group this notion of normal basis when specialized to h rg is equivalent to the classical notion received by the editors may 7 1985 and in revised form january 7 1986 1980 mathematics subject classification 1985 revision primary 13b05 secondary 16a24 1986 american mathematical society 0002 9947 86 1 00 25 per page

duality for hopf orders in this paper we use duality to construct new classes of hopf orders in the group algebra kc p3 where k is a finite extension of q p and c p3 denotes the cyclic group of order p 3 included in this collection is a subcollection of hopf orders which are realizable as galois groups

a geometric approach to archetypal analysis and nonnegative matrix factorization archetypal analysis and non negative matrix factorization nmf are staples in a statisticians toolbox for dimension reduction and exploratory data analysis we describe a geometric approach to both nmf and archetypal analysis by interpreting both problems as finding extreme points of the data cloud we also develop and analyze an efficient approach to finding extreme points in high dimensions for modern massive datasets that are too large to fit on a single machine and must be stored in a distributed setting our approach makes only a small number of passes over the data in fact it is possible to obtain the nmf or perform archetypal analysis with just two passes over the data

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

isogenies of polynomial formal groups leth hv i j be a greither order inkcp2and let spec hv i j spec b spec cbe a resolution of spec hv i j in the flat topology which appears as p2 gm gm gm gmover the fieldk under certain conditions the formal completions ofbandcgive rise to two dimensional polynomial formal groups f and g respectively the given resolution will then translate to an isogeny of polynomial formal groups f g with kernel spec hv i j