corrected confidence bands for functional data using principal components functional principal components fpc analysis is widely used to decompose and express functional observations curve estimates implicitly condition on basis functions and other quantities derived from fpc decompositions however these objects are unknown in practice in this article we propose a method for obtaining correct curve estimates by accounting for uncertainty in fpc decompositions additionally pointwise and simultaneous confidence intervals that account for both model and decomposition based variability are constructed standard mixed model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition iterated expectation and variance formulas combine model based conditional estimates across the distribution of decompositions a bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities our method compares favorably to competing approaches in simulation studies that include both densely and sparsely observed functions we apply our method to sparse observations of cd4 cell counts and to dense white matter tract profiles code for the analyses and simulations is publicly available and our method is implemented in the r package refund on cran

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

stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear schrodinger equations abstract linear stability of both sign definite positive and sign indefinite solitary waves near pitchfork bifurcations is analyzed for the generalized nonlinear schrodinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions bifurcations of linear stability eigenvalues associated with pitchfork bifurcations are analytically calculated based on these eigenvalue bifurcation formulae linear stability of solitary waves near pitchfork bifurcations is then determined it is shown that the base solution branch switches stability at the bifurcation point in addition the two bifurcated solution branches and the base branch have the opposite same stability when their power slopes have the same opposite sign furthermore the stability of these solution branches can be determined almost exclusively from their power diagram especially for positive solitary waves these stability results are also compared with the hamiltonian krein index theory and they are shown to be consistent with each other lastly various numerical examples are presented and the numerical results confirm the analytical predictions both qualitatively and quantitatively

positive recurrent of stochastic coral reefs model abstract in this work we discuss permanence and ergodicity of a stochastic coral reefs model one of the distinctive features of our results is that our results enables characterization of the support of a unique invariant probability measure by lie groups and geometric control theory it is proved that the densities either converges to an invariant density or converges weakly to a singular measure

finding imaging patterns of structural covariance via non negative matrix factorization in this paper we investigate the use of non negative matrix factorization nnmf for the analysis of structural neuroimaging data the goal is to identify the brain regions that co vary across individuals in a consistent way hence potentially being part of underlying brain networks or otherwise influenced by underlying common mechanisms such as genetics and pathologies nnmf offers a directly data driven way of extracting relatively localized co varying structural regions thereby transcending limitations of principal component analysis pca independent component analysis ica and other related methods that tend to produce dispersed components of positive and negative loadings in particular leveraging upon the well known ability of nnmf to produce parts based representations of image data we derive decompositions that partition the brain into regions that vary in consistent ways across individuals importantly these decompositions achieve dimensionality reduction via highly interpretable ways and generalize well to new data as shown via split sample experiments we empirically validate nnmf in two data sets i a diffusion tensor dt mouse brain development study and ii a structural magnetic resonance smr study of human brain aging we demonstrate the ability of nnmf to produce sparse parts based representations of the data at various resolutions these representations seem to follow what we know about the underlying functional organization of the brain and also capture some pathological processes moreover we show that these low dimensional representations favorably compare to descriptions obtained with more commonly used matrix factorization methods like pca and ica

dynamics of kolmogorov systems of competitive type under the telegraph noise abstract this paper studies the dynamics of kolmogorov systems of competitive type under the telegraph noise the telegraph noise switches at random two kolmogorov competition type deterministic models the aim of this work is to describe the omega limit set of the system and investigates properties of stationary density

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

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

large cone nonnegative matrix factorization nonnegative matrix factorization nmf has been greatly popularized by its parts based interpretation and the effective multiplicative updating rule for searching local solutions in this paper we study the problem of how to obtain an attractive local solution for nmf which not only fits the given training data well but also generalizes well on the unseen test data based on the geometric interpretation of nmf we introduce two large cone penalties for nmf and propose large cone nmf lcnmf algorithms compared with nmf lcnmf will obtain bases comprising a larger simplicial cone and therefore has three advantages 1 the empirical reconstruction error of lcnmf could mostly be smaller 2 the generalization ability of the proposed algorithm is much more powerful and 3 the obtained bases of lcnmf have a low overlapping property which enables the bases to be sparse and makes the proposed algorithms very robust experiments on synthetic and real world data sets confirm the efficiency of lcnmf

threshold dynamics and ergodicity of an sirs epidemic model with markovian switching abstract this work studies the threshold dynamics and ergodicity of a stochastic sirs epidemic model with the disease transmission rate driven by a semi markov process the semi markov process used in this paper for describing a randomly changing environment is a very large extension of the most common markov regime switching process we define a basic reproduction number for the semi markov regime switching environment and show that its position with respect to 1 determines the extinction or persistence of the disease in the case of disease persistence we give mild sufficient conditions for ensuring the existence and absolute continuity of the invariant probability measure under the same conditions we also prove the global attractivity of the limit set of the system and the convergence in total variation norm of the transition probability to the invariant measure compared with the existing results in the markov regime switching environment the results generalized require almost no additional conditions

gap solitons and their linear stability in one dimensional periodic media an analytical theory utilizing exponential asymptotics is presented for one dimensional gap solitons that bifurcate from edges of bloch bands in the presence of a general periodic potential it is shown that two soliton families bifurcate out from every bloch band edge under self focusing or self defocusing nonlinearity and an asymptotic expression for the eigenvalues associated with the linear stability of these solitons is derived the locations of these solitons relative to the underlying potential are determined from a certain recurrence relation that contains information beyond all orders of the usual perturbation expansion in powers of the soliton amplitude moreover this same recurrence relation decides which of the two soliton families is unstable the analytical predictions for the stability eigenvalues are in excellent agreement with numerical results

dynamics of a multigroup sis epidemic model with standard incidence rates and markovian switching abstract in this paper we investigate a multigroup sis epidemic model with standard incidence rates and markovian switching first of all we establish sufficient conditions for extinction of the diseases then we obtain sufficient conditions for persistence in the mean of the diseases in addition in the case of persistence we obtain sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic lyapunov function with regime switching

on competitive lotka volterra model in random environments focusing on competitive lotka volterra model in random environments this paper uses regime switching diffusions to model the dynamics of the population sizes of n different species in an ecosystem subject to the random changes of the external environment it is demonstrated that the growth rates of the population sizes of the species are bounded above moreover certain long run average limits of the solution are examined from several angles a partial stochastic principle of competitive exclusion is also derived finally simple examples are used to demonstrate our findings

a bregman matrix and the gradient of mutual information for vector poisson and gaussian channels a generalization of bregman divergence is developed and utilized to unify vector poisson and gaussian channel models from the perspective of the gradient of mutual information the gradient is with respect to the measurement matrix in a compressive sensing setting and mutual information is considered for signal recovery and classification existing gradient of mutual information results for scalar poisson models are recovered as special cases as are known results for the vector gaussian model the bregman divergence generalization yields a bregman matrix and this matrix induces numerous matrix valued metrics the metrics associated with the bregman matrix are detailed as are its other properties the bregman matrix is also utilized to connect the relative entropy and mismatched minimum mean squared error two applications are considered 1 compressive sensing with a poisson measurement model and 2 compressive topic modeling for analysis of a document corpora word count data in both of these settings we use the developed theory to optimize the compressive measurement matrix for signal recovery and classification

existence of stationary distributions for kolmogorov systems of competitive type under telegraph noise abstract this work focuses on population dynamics of two species described by kolmogorov systems of competitive type under telegraph noise that is formulated as a continuous time markov chain with two states our main effort is on establishing the existence of an invariant or a stationary probability measure in addition the convergence in total variation of the instantaneous measure to the stationary measure is demonstrated under suitable conditions moreover the limit set of a model in which each species is dominant in a state of the telegraph noise is examined in detail

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

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

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

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

nonnegative decomposition of functional count data we present a novel decomposition of nonnegative functional count data that draws on concepts from nonnegative matrix factorization our decomposition which we refer to as narfd nonnegative and regularized function decomposition enables the study of patterns in variation across subjects in a highly interpretable manner prototypic modes of variation are estimated directly on the observed scale of the data are local and are transparently added together to reconstruct observed functions this contrasts with generalized functional principal component analysis an alternative approach that estimates functional principal components on a transformed scale produces components that typically vary across the entire functional domain and reconstructs observations using complex patterns of cancellation and multiplication of functional principal components narfd is implemented using an alternating minimization algorithm and we evaluate our approach in simulations we apply narfd to an accelerometer dataset comprising observations of physical activity for healthy older americans

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

registration for exponential family functional data we consider the problem of aligning curves from exponential family distributions the approach is based on the combination of alignment and functional principal components analysis and is facilitated by recent extensions of fpca to non gaussian settings our work is motivated by the study of physical activity using accelerometers wearable devices that provide around the clock monitoring of activity and produce non gaussian measurements we apply the proposed methods to activity counts using a poisson distribution and to a binary active vs inactive indicator using a binomial distribution after alignment the trajectories show clear peaks of activity in the morning and afternoon with a dip in the middle of the day

global analysis of the predator prey system with beddington deangelis functional response abstract in this paper predator prey systems with beddington deangelis functional response are considered by using divergency criterion global stability is established provided the system possesses a positive locally asymptotically stable equilibrium

uniqueness of limit cycles of the predator prey system with beddington deangelis functional response abstract the goal of this paper is to establish the uniqueness of limit cycles of the predator prey systems with beddington deangelis functional response through a change of variables the predator prey system can be transformed into a better studied gause type predator prey system as a result the uniqueness of limit cycles can be solved

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

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

vortex families near a spectral edge in the gross pitaevskii equation with a two dimensional periodic potential we examine numerically vortex families near band edges of the bloch wave spectrum for the gross pitaevskii equation with two dimensional periodic potentials and for the discrete nonlinear schrodinger equation we show that besides vortex families that terminate at a small distance from the band edges via fold bifurcations there exist vortex families that are continued all the way to the band edges

the threshold infection level for wolbachia invasion in random environments abstract dengue fever and zika are mosquito borne diseases threatening human health a novel strategy for mosquito borne disease control uses the bacterium wolbachia to block virus transmission it requires releasing wolbachia infected mosquitoes to exceed a threshold level since an accurate forecast for temperature and rainfall the major environmental conditions regulating the mosquito dynamics is often not available over a long time period it is important to explore how the threshold releasing level changes in random environments in this work we estimate the threshold level in a stochastic system of differential equations where the reproduction rates of mosquitoes change randomly we prove that the threshold level is surprisingly defined by a deterministic curve that does not fluctuate with environmental conditions the major difficulty in the proof is to construct various auxiliary curves to limit the dynamic behaviors of the whole family of innumerable solutions satisfying a given initial condition

generalized functional linear models we propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function a linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function and the expected value of the response is related to this linear predictor via a link function if in addition a variance function is specified this leads to a functional estimating equation which corresponds to maximizing a functional quasi likelihood this general approach includes the special cases of the functional linear model as well as functional poisson regression and functional binomial regression the latter leads to procedures for classification and discrimination of stochastic processes and functional data we also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data using a semiparametric quasi likelihood procedure an essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated karhunen loeve expansion we develop asymptotic inference for the proposed class of generalized regression models in the proposed asymptotic approach the truncation parameter increases with sample size and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics we establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable l 2 metric and is defined through a generalized covariance operator as a consequence we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model the proposed estimation inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study we find that the practical selection of the number of components works well with the aic criterion and this finding is supported by theoretical considerations we include an application to the classification of medflies regarding their remaining longevity status based on the observed initial egg laying curve for each of 534 female medflies

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

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

modelling sparse generalized longitudinal observations with latent gaussian processes summary in longitudinal data analysis one frequently encounters non gaussian data that are repeatedly collected for a sample of individuals over time the repeated observations could be binomial poisson or of another discrete type or could be continuous the timings of the repeated measurements are often sparse and irregular we introduce a latent gaussian process model for such data establishing a connection to functional data analysis the functional methods proposed are non parametric and computationally straightforward as they do not involve a likelihood we develop functional principal components analysis for this situation and demonstrate the prediction of individual trajectories from sparse observations this method can handle missing data and leads to predictions of the functional principal component scores which serve as random effects in this model these scores can then be used for further statistical analysis such as inference regression discriminant analysis or clustering we illustrate these non parametric methods with longitudinal data on primary biliary cirrhosis and show in simulations that they are competitive in comparisons with generalized estimating equations and generalized linear mixed models

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

stability of discrete solitons in nonlinear schrodinger lattices we consider the discrete solitons bifurcating from the anti continuum limit of the discrete nonlinear schrodinger nls lattice the discrete soliton in the anti continuum limit represents an arbitrary finite superposition of in phase or anti phase excited nodes separated by an arbitrary sequence of empty nodes by using stability analysis we prove that the discrete solitons are all unstable near the anti continuum limit except for the solitons which consist of alternating anti phase excited nodes we classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons

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

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

continuous markov semigroups and stability of transport equations a new theorem for asymptotic stability of markov semigroups is proved this result is applied to transport equations connected with diffusion and jumping processes and randomly perturbed dynamical systems

nonnegative matrix factorization with the itakura saito divergence with application to music analysis this letter presents theoretical algorithmic and experimental results about nonnegative matrix factorization nmf with the itakura saito is divergence we describe how is nmf is underlaid by a well defined statistical model of superimposed gaussian components and is equivalent to maximum likelihood estimation of variance parameters this setting can accommodate regularization constraints on the factors through bayesian priors in particular inverse gamma and gamma markov chain priors are considered in this work estimation can be carried out using a space alternating generalized expectation maximization sage algorithm this leads to a novel type of nmf algorithm whose convergence to a stationary point of the is cost function is guaranteed we also discuss the links between the is divergence and other cost functions used in nmf in particular the euclidean distance and the generalized kullback leibler kl divergence as such we describe how is nmf can also be performed using a gradient multiplicative algorithm a standard algorithm structure in nmf whose convergence is observed in practice though not proven finally we report a furnished experimental comparative study of euclidean nmf kl nmf and is nmf algorithms applied to the power spectrogram of a short piano sequence recorded in real conditions with various initializations and model orders then we show how is nmf can successfully be employed for denoising and upmix mono to stereo conversion of an original piece of early jazz music these experiments indicate that is nmf correctly captures the semantics of audio and is better suited to the representation of music signals than nmf with the usual euclidean and kl costs

bound states of two dimensional solitons in the discrete nonlinear schrodinger equation the existence and stability of bound states bss of solitary excitations in the two dimensional discrete nonlinear schrodinger lattice are considered stable bss of two or three solitons are categorized according to their vorticity s s 0 corresponds to ordinary pulses while s 1 corresponds to discrete vortex solitons interactions of s 0 solitons are found to have clear particle like characteristics which can be very accurately predicted in the framework of the perturbation theory that we develop triangular bound states of three solitons are also constructed and particle like characteristics in their behaviour are followed in numerical experiments finally stable bound states of two vortex solitons and s 0 1 stable complexes are constructed for the latter state we find a weak dependence of the interaction generated stability eigenvalue on the distance between the two solitons

on the asymptotics of penalized splines summavry we study the asymptotic behaviour of penalized spline estimators in the univariate case we use b splines and a penalty is placed on mth order differences of the coefficients the number of knots is assumed to converge to infinity as the sample size increases we show that penalized splines behave similarly to nadaraya watson kernel estimators with equivalent kernels depending upon m the equivalent kernels we obtain for penalized splines are the same as those found by silverman for smoothing splines the asymptotic distribution of the penalized spline estimator is gaussian and we give simple expressions for the asymptotic mean and variance provided that it is fast enough the rate at which the number of knots converges to infinity does not affect the asymptotic distribution the optimal rate of convergence of the penalty parameter is given penalized splines are not design adaptive

no stability switching at saddle node bifurcations of solitary waves in generalized nonlinear schrodinger equations saddle node bifurcations arise frequently in solitary waves of diverse physical systems previously it was believed that solitary waves always undergo stability switching at saddle node bifurcations just as in finite dimensional dynamical systems here we show that this is not true for a large class of generalized nonlinear schrodinger equations with real or complex potentials we prove that stability of solitary waves does not switch at saddle node bifurcations this analytical result is confirmed by numerical examples where both soliton branches are stable at saddle node bifurcations

minimum volume constrained nonnegative matrix factorization enhanced ability of learning parts nonnegative matrix factorization nmf with minimum volume constraint mvc is exploited in this paper our results show that mvc can actually improve the sparseness of the results of nmf this sparseness is l0 norm oriented and can give desirable results even in very weak sparseness situations thereby leading to the significantly enhanced ability of learning parts of nmf the close relation between nmf sparse nmf and the mvc nmf is discussed first then two algorithms are proposed to solve the mvc nmf model one is called quadratic programming mvc nmf qp mvc nmf which is based on quadratic programming and the other is called negative glow mvc nmf ng mvc nmf because it uses multiplicative updates incorporating natural gradient ingeniously the qp mvc nmf algorithm is quite efficient for small scale problems and the ng mvc nmf algorithm is more suitable for large scale problems simulations show the efficiency and validity of the proposed methods in applications of blind source separation and human face images analysis

non negative blind source separation algorithm based on minimum aperture simplicial cone we address the problem of blind source separation bss when the hidden sources are nonnegative n bss in this case the scatter plot of the mixed data is contained within the simplicial cone generated by the columns of the mixing matrix the proposed method termed scsa uns for simplicial cone shrinking algorithm for unmixing non negative sources aims at estimating the mixing matrix and the sources by fitting a minimum aperture simplicial cone masc to the cloud of mixed data points scsa uns is evaluated on both independent and correlated synthetic data and compared to other n bss methods simulations are also performed on real liquid chromatography mass spectrum lc ms data for the metabolomic analysis of a chemical sample and on real dynamic positron emission tomography pet images in order to study the pharmacokinetics of the 18f fdg fluorodeoxyglucose tracer in the brain

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

finding minimal convex nested polygons abstract we consider the problem of finding a polygon nested between two given convex polygons that has a minimal number of vertices our main result is an o n log k algorithm for solving the problem where n is the total number of vertices of the given polygons and k is the number of vertices of a minimal nested polygon we also present an o n sub optimal algorithm and a simple o nk optimal algorithm

from non local gap solitary waves to bound states in periodic media solitary waves in one dimensional periodic media are discussed by employing the nonlinear schrodinger equation with a spatially periodic potential as a model this equation admits two families of gap solitons that bifurcate from the edges of bloch bands in the linear wave spectrum these fundamental solitons may be positioned only at specific locations relative to the potential otherwise they become non local owing to the presence of growing tails of exponentially small amplitude with respect to the wave peak amplitude here by matching the tails of such non local solitary waves high order locally confined gap solitons or bound states are constructed details are worked out for bound states comprising two non local solitary waves in the presence of a sinusoidal potential a countable set of bound state families characterized by the separation distance of the two solitary waves is found and each family features three distinct solution branches that bifurcate near bloch band edges at small but finite amplitude power curves associated with these solution branches are computed asymptotically for large solitary wave separation and the theoretical predictions are consistent with numerical results

low dimensional polytope approximation and its applications to nonnegative matrix factorization in this study nonnegative matrix factorization is recast as the problem of approximating a polytope on the probability simplex by another polytope with fewer facets working on the probability simplex has the advantage that data are limited to a compact set with a known boundary making it easier to trace the approximation procedure in particular the supporting hyperplane that separates a point from a disjoint polytope a fact asserted by the hahn banach theorem can be calculated in finitely many steps this approach leads to a convenient way of computing the proximity map which in contrast to most existing algorithms where only an approximate map is used finds the unique and global minimum per iteration this paper sets up a theoretical framework outlines a numerical algorithm and suggests an effective implementation testing results strongly evidence that this approach obtains a better low rank nonnegative matrix approximation in fewer steps than conventional methods

underdetermined sparse blind source separation of nonnegative and partially overlapped data we study the solvability of sparse blind separation of n nonnegative sources from m linear mixtures in the underdetermined regime m n the geometric properties of the mixture matrix and the sparseness structure of the source matrix are closely related to the identification of the mixing matrix we first illustrate and establish necessary and sufficient conditions for the unique separation for the case of m mixtures and m 1 sources and develop a novel algorithm based on data geometry source sparseness and ell 1 minimization then we extend the results to any order m times n 3 leq m n based on the degree of degeneracy of the columns of the mixing matrix numerical results substantiate the proposed solvability conditions and show satisfactory performance of our approach

fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models summary recent work by reiss and ogden provides a theoretical basis for sometimes preferring restricted maximum likelihood reml to generalized cross validation gcv for smoothing parameter selection in semiparametric regression however existing reml or marginal likelihood ml based methods for semiparametric generalized linear models glms use iterative reml or ml estimation of the smoothing parameters of working linear approximations to the glm such indirect schemes need not converge and fail to do so in a non negligible proportion of practical analyses by contrast very reliable prediction error criteria smoothing parameter selection methods are available based on direct optimization of gcv or related criteria for the glm itself since such methods directly optimize properly defined functions of the smoothing parameters they have much more reliable convergence properties the paper develops the first such method for reml or ml estimation of smoothing parameters a laplace approximation is used to obtain an approximate reml or ml for any glm which is suitable for efficient direct optimization this reml or ml criterion requires that newton raphson iteration rather than fisher scoring be used for glm fitting and a computationally stable approach to this is proposed the reml or ml criterion itself is optimized by a newton method with the derivatives required obtained by a mixture of implicit differentiation and direct methods the method will cope with numerical rank deficiency in the fitted model and in fact provides a slight improvement in numerical robustness on the earlier method of wood for prediction error criteria based smoothness selection simulation results suggest that the new reml and ml methods offer some improvement in mean square error performance relative to gcv or akaike s information criterion in most cases without the small number of severe undersmoothing failures to which akaike s information criterion and gcv are prone this is achieved at the same computational cost as gcv or akaike s information criterion the new approach also eliminates the convergence failures of previous reml or ml based approaches for penalized glms and usually has lower computational cost than these alternatives example applications are presented in adaptive smoothing scalar on function regression and generalized additive model selection

coexistence and exclusion of stochastic competitive lotka volterra models abstract this work derives sufficient conditions for the coexistence and exclusion of a stochastic competitive lotka volterra model the conditions obtained are close to necessary in addition convergence in distribution of positive solutions of the model is also established a number of numerical examples are given to illustrate our results

symmetry breaking of solitons in two dimensional complex potentials symmetry breaking is reported for continuous families of solitons in the nonlinear schrodinger equation with a two dimensional complex potential this symmetry breaking is forbidden in generic complex potentials however for a special class of partially parity time symmetric potentials it is allowed at the bifurcation point two branches of asymmetric solitons bifurcate out from the base branch of symmetry unbroken solitons stability of these solitons near the bifurcation point are also studied and two novel properties for the bifurcated asymmetric solitons are revealed one is that at the bifurcation point zero and simple imaginary linear stability eigenvalues of asymmetric solitons can move directly into the complex plane and create oscillatory instability the other is that the two bifurcated asymmetric solitons even though having identical powers and being related to each other by spatial mirror reflection can possess different types of unstable eigenvalues and thus exhibit nonreciprocal nonlinear evolutions under random noise perturbations

generalized multilevel function on scalar regression and principal component analysis this manuscript considers regression models for generalized multilevel functional responses functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects this data structure is increasingly common across scientific domains and is exemplified by our motivating example in which binary curves indicating physical activity or inactivity are observed for nearly 600 subjects over 5 days we use a generalized linear model to incorporate scalar covariates into the mean structure and decompose subject specific and subject day specific deviations using multilevel functional principal components analysis thus functional fixed effects are estimated while accounting for within function and within subject correlations and major directions of variability within and between subjects are identified fixed effect coefficient functions and principal component basis functions are estimated using penalized splines model parameters are estimated in a bayesian framework using stan a programming language that implements a hamiltonian monte carlo sampler simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets in both cross sectional and multilevel scenarios code is publicly available in the application we identify effects of age and bmi on the time specific change in probability of being active over a 24 hour period in addition the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects

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

nonnegative least correlated component analysis for separation of dependent sources by volume maximization although significant efforts have been made in developing nonnegative blind source separation techniques accurate separation of positive yet dependent sources remains a challenging task in this paper a joint correlation function of multiple signals is proposed to reveal and confirm that the observations after nonnegative mixing would have higher joint correlation than the original unknown sources accordingly a new nonnegative least correlated component analysis nlca method is proposed to design the unmixing matrix by minimizing the joint correlation function among the estimated nonnegative sources in addition to a closed form solution for unmixing two mixtures of two sources the general algorithm of nlca for the multisource case is developed based on an iterative volume maximization ivm principle and linear programming the source identifiability and required conditions are discussed and proven the proposed nlca algorithm denoted by nlca ivm is evaluated with both simulation data and real biomedical data to demonstrate its superior performance over several existing benchmark methods

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

asymptotic decomposition of substochastic operators and semigroups abstract theorems on asymptotic decomposition of substochastic operators and semigroups are given general results are applied to show asymptotic periodicity asymptotic stability and sweeping of substochastic operators and semigroups