isometric dilations for infinite sequences of noncommuting operators this paper develops a dilation theory for t n l an infinite sequence of noncommuting operators on a hilbert space when the matrix t1 t2 is a contraction a wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for tn i are obtained some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given this results are used to extend the sz nagy foia lifting theorem to this noncommutative setting this paper is a continuation of 5 and develops a dilation theory for an infinite sequence tn n of noncommuting operators on a hilbert space when en 1 tn tn i i is the identity on s many of the results and techniques in dilation theory for one operator 8 and also for two operators 3 4 are extended to this setting first we extend wold decomposition 8 4 to the case of an infinite sequence vn i of isometries with orthogonal final spaces in 2 we obtain a minimal isometric dilation for tn 1 by extending the schaffer construction in 6 4 using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation finally we give some sufficient conditions on a sequence t7 n to be simultaneously quasi similar to a sequence rn n i of isometries on a hilbert space x with en rnrn n ix in 3 we use the above mentioned theorems to obtain the sz nagy foia5 lifting theorem 7 8 1 4 in our setting in a subsequent paper we will use the results of this paper for studying the characteristic function associated to a sequence tn lo 1 with ez 1 t t i throughout this paper a stands for the set 1 2 k k e n or the set n 1 2 received by the editors september 1 1987 1980 mathematics subject classification 1985 revision primary 47a20 secondary 47a45

quantization of poisson lie groups and applications letg be a connected poisson lie group we discuss aspects of the question of drinfel d can g be quantized and give some answers wheng is semisimple a case where the answer isyes we introduce quantizable poisson subalgebras ofc g related to harmonic analysis ong they are a generalization of f r t models of quantum groups and provide new examples of quantized poisson algebras

deformations of lie group and lie algebra representations a complete study of deformations of lie group and lie algebra representations including differentiability and integrability results is given adapted results are given in the semisimple case a notion of induced deformations is introduced various examples are given including deformations of indecomposable representations

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

separation of non negative mixture of non negative sources using a bayesian approach and mcmc sampling this paper addresses blind source separation in the case where both the source signals and the mixing coefficients are non negative the problem is referred to as non negative source separation and the main application concerns the analysis of spectrometric data sets the separation is performed in a bayesian framework by encoding non negativity through the assignment of gamma priors on the distributions of both the source signals and the mixing coefficients a markov chain monte carlo mcmc sampling procedure is proposed to simulate the resulting joint posterior density from which marginal posterior mean estimates of the source signals and mixing coefficients are obtained results obtained with synthetic and experimental spectra are used to discuss the problem of non negative source separation and to illustrate the effectiveness of the proposed method

blind separation of speech mixtures via time frequency masking binary time frequency masks are powerful tools for the separation of sources from a single mixture perfect demixing via binary time frequency masks is possible provided the time frequency representations of the sources do not overlap a condition we call w disjoint orthogonality we introduce here the concept of approximate w disjoint orthogonality and present experimental results demonstrating the level of approximate w disjoint orthogonality of speech in mixtures of various orders the results demonstrate that there exist ideal binary time frequency masks that can separate several speech signals from one mixture while determining these masks blindly from just one mixture is an open problem we show that we can approximate the ideal masks in the case where two anechoic mixtures are provided motivated by the maximum likelihood mixing parameter estimators we define a power weighted two dimensional 2 d histogram constructed from the ratio of the time frequency representations of the mixtures that is shown to have one peak for each source with peak location corresponding to the relative attenuation and delay mixing parameters the histogram is used to create time frequency masks that partition one of the mixtures into the original sources experimental results on speech mixtures verify the technique example demixing results can be found online at http alum mit edu www rickard bss html

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

blind separation of instantaneous mixtures of nonstationary sources most source separation algorithms are based on a model of stationary sources however it is a simple matter to take advantage of possible nonstationarities of the sources to achieve separation this paper develops novel approaches in this direction based on the principles of maximum likelihood and minimum mutual information these principles are exploited by efficient algorithms in both the off line case via a new joint diagonalization procedure and in the on line case via a newton like procedure some experiments showing the good performance of our algorithms and evidencing an interesting feature of our methods are presented their ability to achieve a kind of super efficiency the paper concludes with a discussion contrasting separating methods for non gaussian and nonstationary models and emphasizing that as a matter of fact what makes the algorithms work is strictly speaking not the nonstationarity itself but rather the property that each realization of the source signals has a time varying envelope

symmetrizable quantum affine superalgebras and their representations aspects of the algebraic structure and representation theory of the quantum affine superalgebras with symmetrizable cartan matrices are studied the irreducible integrable highest weight representations are classified and shown to be deformations of their classical counterparts it is also shown that jimbo type quantum affine superalgebras can be obtained by deforming universal enveloping algebras of ordinary i e nongraded affine algebras supplemented by certain parity operators

fast and robust fixed point algorithms for independent component analysis independent component analysis ica is a statistical method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible we use a combination of two different approaches for linear ica comon s information theoretic approach and the projection pursuit approach using maximum entropy approximations of differential entropy we introduce a family of new contrast functions for ica these contrast functions enable both the estimation of the whole decomposition by minimizing mutual information and estimation of individual independent components as projection pursuit directions the statistical properties of the estimators based on such contrast functions are analyzed under the assumption of the linear mixture model and it is shown how to choose contrast functions that are robust and or of minimum variance finally we introduce simple fixed point algorithms for practical optimization of the contrast functions

deformation quantization and nambu mechanics starting from deformation quantization star products the quantization problem of nambu mechanics is investigated after considering some impossibilities and pushing some analogies with field quantization a solution to the quantization problem is presented in the novel approach of zariski quantization of fields observables functions in this case polynomials this quantization is based on the factorization over of polynomials in several real variables we quantize the infinite dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is abelian associative and distributive this procedure is then adapted to derivatives needed for the nambu brackets which ensures the validity of the fundamental identity of nambu mechanics also at the quantum level our construction is in fact more general than the particular case considered here it can be utilized for quite general defining identities and for much more general star products

the hidden group structure of quantum groups strong duality rigidity and preferred deformations a notion of well behaved hopf algebra is introduced reflexivity for strong duality between hopf algebras of drinfeld type and their duals algebras of coefficients of compact semi simple groups is proved a hidden classical group structure is clearly indicated for all generic models of quantum groups moyal product like deformations are naturally found for all frt models on coefficients andc functions strong rigidity hbi2 0 under deformations in the category of bialgebras is proved and consequences are deduced

a simple method for constructing the inhomogeneous quantum group iglq n and its universal enveloping algebra uq igl n we propose a simple and concise method to construct the inhomogeneous quantum group iglq n and its universal enveloping algebra uq igl n our technique is based on embedding an n dimensional quantum space in an n 1 dimensional one as the set xn 1 1 this is possible only if one considers the multiparametric quantum space whose parameters are fixed in a specific way the quantum group iglq n is then the subset of glq n 1 which leaves the xn 1 1 subset invariant for the deformed universal enveloping algebra uq igl n we will show that it can also be embedded in uq gl n 1 provided one uses the multiparametric deformation of u gl n 1 with a specific choice of its parameters

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

a noncommutative hopf structure on c sl 2 c as a quantum lorentz group we introduce a classical frechet hopf algebra a containing the quantum deformation of the enveloping lorentz algebra of ogievestky and wess et al as a dense hopf subalgebra we also show that a is isomorphic to a quasitriangular twisted topological tensor product hopf algebra by duality we recover on the topological dual space of a the non commutative algebra structure introduced by podles and woronowicz finally this non commutative product can be extended to c sl 2 c which then becomes a topological hopf algebra for its natural frechet topology

algorithms for nonnegative matrix factorization with the β divergence this letter describes algorithms for nonnegative matrix factorization nmf with the divergence nmf the divergence is a family of cost functions parameterized by a single shape parameter that takes the euclidean distance the kullback leibler divergence and the itakura saito divergence as special cases 2 1 0 respectively the proposed algorithms are based on a surrogate auxiliary function a local majorization of the criterion function we first describe a majorization minimization algorithm that leads to multiplicative updates which differ from standard heuristic multiplicative updates by a dependent power exponent the monotonicity of the heuristic algorithm can however be proven for 0 1 using the proposed auxiliary function then we introduce the concept of the majorization equalization me algorithm which produces updates that move along constant level sets of the auxiliary function and lead to larger steps than mm simulations on synthetic and real data illustrate the faster convergence of the me approach the letter also describes how the proposed algorithms can be adapted to two common variants of nmf penalized nmf when a penalty function of the factors is added to the criterion function and convex nmf when the dictionary is assumed to belong to a known subspace

linear quadratic blind source separation using nmf to unmix urban hyperspectral images in this work we propose algorithms to perform blind source separation bss for the linear quadratic mixing model the linear quadratic model is less studied in the literature than the linear one in this paper we propose original methods that are based on non negative matrix factorization nmf this class of methods is well suited to many applications where the data are non negative we are here particularly interested in spectral unmixing extracting reflectance spectra of materials present in pixels and associated abundance fractions for urban hyperspectral images the originality of our work is that we developed extensions of nmf which is initially suited to the linear model for the linear quadratic model the proposed algorithms are tested with simulated hyperspectral images using real reflectance spectra and the obtained results are very satisfactory

quantum groups and deformation quantization explicit approaches and implicit aspects deformation quantization which gives a development of quantum mechanics independent of the operator algebra formulation and quantum groups which arose from the inverse scattering method and a study of yang baxter equations share a common idea abstracted earlier in algebraic deformation theory that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations i e quantizations in deformation quantization the algebraic object is the algebra of observables functions on symplectic phase space whose infinitesimal deformation is the poisson bracket and global deformation a star product in quantum groups it is a hopf algebra generally either of functions on a lie group or often its dual in the topological vector space sense as we briefly explain a completed universal enveloping algebra of a lie algebra with for infinitesimal a matrix satisfying the modified classical yang baxter equation mcybe frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract one success here comes from universal deformation formulas udfs expressions built from a lie algebra which deform any algebra on which the lie algebra operates as derivations the most famous of these is the moyal product a special case of a class in which the lie algebra is abelian another comes from recognition that the belavin drinfel d solutions to the mcybe are in fact infinitesimal deformations for which in the case of the special linear groups it is possible to give explicit formulas for the corresponding quantum yang baxter equations this review paper discusses necessarily in brief these and related topics including twisting as a form of udf and finding formulas for preferred deformations of hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation

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