splitting scheme for the stability of strong shock profile abstract in this paper we use the laplace transform and dirichlet neumann map to give a systematical scheme to study the small wave perturbation of general shock profile with general amplitude here we use certain non classical shock waves for a rotationally invariant system of viscous conservation laws to demonstrate this scheme we derive an explicit solution and show that it converges pointwise to another over compressive profile exponentially when the perturbations of the initial data to a given over compressive shock profile are sufficiently small

spectral stability of small amplitude viscous shock waves in several space dimensions a planar viscous shock profile of a hyperbolic parabolic system of conservation laws is a steady solution in a moving coordinate frame the asymptotic stability of viscous profiles and the related vanishing viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions it is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile the evans function method provides a geometric dynamical systems framework to study the eigenvalue problem in this approach eigenvalues correspond to zeros of an essentially analytic function mathcal e rho lambda rho omega which detects nontrivial intersections of the so called stable and unstable spaces that is spaces of solutions that decay on one or the other side of the shock wave respectively in a series of pioneering papers kevin zumbrun and collaborators have established in various contexts that spectral stability that is the non vanishing of mathcal e rho lambda rho omega and the non vanishing of the lopatinski kreiss majda function imply nonlinear stability of viscous shock profiles in several space dimensions in this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions this is done by exploiting the slow fast nature of the small amplitude limit which was used in a previous paper by the authors to prove spectral stability of small amplitude shock waves in one space dimension geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior three qualitatively different regimes are distinguished that relate the small strength epsilon of the shock wave to appropriate ranges of values of the spectral parameters various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality

a unified convergence analysis of block successive minimization methods for nonsmooth optimization the block coordinate descent bcd method is widely used for minimizing a continuous function f of several block variables at each iteration of this method a single block of variables is optimized while the remaining variables are held fixed to ensure the convergence of the bcd method the subproblem of each block variable needs to be solved to its unique global optimal unfortunately this requirement is often too restrictive for many practical scenarios in this paper we study an alternative inexact bcd approach which updates the variable blocks by successively minimizing a sequence of approximations of f which are either locally tight upper bounds of f or strictly convex local approximations of f the main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods especially for the cases where the objective functions are either nondifferentiable or nonconvex our results unify and extend the existing convergence results

minimization of csi feedback dimension for interference alignment in mimo interference multicast networks it is well known that interference alignment ia can achieve substantial theoretical gains in multiple input multiple output mimo networks however the conventional works usually assume the full channel state information csi is available at the transmitter side which would create a overwhelming csi feedback overhead for practical frequency division duplexing fdd wireless systems to implement ia in practice it is highly desirable to reduce the amount of required csi feedback overhead in this paper we consider ia in mimo interference multicast networks under partial csi feedback and we attempt to minimize the csi feedback cost subject to ia feasibility constraints with a given degree of freedom dof requirements first we propose a csi feedback profile to embrace two important csi feedback reduction strategies and we use the metric of csi feedback dimension to quantify the associated csi feedback cost we then formulate the ia conditions under partial csi feedback in mimo interference multicast networks and we derive new ia feasibility conditions under the proposed partial csi feedback framework based on these results we consider the csi feedback dimension minimization subject to the ia feasibility constraints with a given dof requirements in mimo interference multicast networks which is formulated as a combinatorial optimization problem based on the specific problem structure we derive an asymptotically optimal solution for a category of network topologies and we further obtain closed form tradeoff results between dofs and the csi feedback cost for mimo interference multicast networks

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

complex valued matrix differentiation techniques and key results a systematic theory is introduced for finding the derivatives of complex valued matrix functions with respect to a complex valued matrix variable and the complex conjugate of this variable in the framework introduced the differential of the complex valued matrix function is used to identify the derivatives of this function matrix differentiation results are derived and summarized in tables which can be exploited in a wide range of signal processing related situations

multidimensional viscous shocks ii the small viscosity limit in this paper we prove the existence of curved multidimensional viscous shocks and also justify the small viscosity limit starting with a curved multidimensional inviscid shock solution to a system of hyperbolic conservation laws we show that the shock can be obtained as a small viscosity limit of solutions to an associated parabolic problem viscous shocks the two main hypotheses are a natural evans function assumption on the viscous profile together with a restriction on how much the shock can deviate from flatness the main tools are a conjugation lemma that removes xn dependence from the linearization of the parabolic problem about the viscous profile new degenerate kreiss type symmetrizers used to prove an l2 estimate for the linearized problem and a finite regularity calculus of semiclassical and mixed type classical semiclassical pseudodifferential operators 2003 wiley periodicals inc

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

unified development of multiplicative algorithms for linear and quadratic nonnegative matrix factorization multiplicative updates have been widely used in approximative nonnegative matrix factorization nmf optimization because they are convenient to deploy their convergence proof is usually based on the minimization of an auxiliary upper bounding function the construction of which however remains specific and only available for limited types of dissimilarity measures here we make significant progress in developing convergent multiplicative algorithms for nmf first we propose a general approach to derive the auxiliary function for a wide variety of nmf problems as long as the approximation objective can be expressed as a finite sum of monomials with real exponents multiplicative algorithms with theoretical guarantee of monotonically decreasing objective function sequence can thus be obtained the solutions of nmf based on most commonly used dissimilarity measures such as and divergence as well as many other more comprehensive divergences can be derived by the new unified principle second our method is extended to a nonseparable case that includes e g divergence and renyi divergence third we develop multiplicative algorithms for nmf using second order approximative factorizations in which each factorizing matrix may appear twice preliminary numerical experiments demonstrate that the multiplicative algorithms developed using the proposed procedure can achieve satisfactory karush kuhn tucker optimality we also demonstrate nmf problems where algorithms by the conventional method fail to guarantee descent at each iteration but those by our principle are immune to such violation

stability of discrete shocks for difference approximations to systems of conservation laws the asymptotic stability of weak discrete stationary shocks for systems of conservation laws in one space dimension is proved the difference approximation should be conservative dissipative and kth order accurate in space with odd k the problem is considered in a finite interval vert x vert le ell with appropriate boundary conditions where ell is large compared with the width of the shock layer varepsilon 1 vert u r u l vert 1 k the proof is based on the assumption that the corresponding continuous shocks for the scalar problem u t uu x i partial x k 1 u are stable the latter is known to be true for k 1 and k 3

a geometric singular perturbation approach for planar stationary shock waves abstract the non linear non equilibrium nature of shock waves in gas dynamics is investigated for the planar case along each streamline the euler equations with non equilibrium pressure are reduced to a set of ordinary differential equations defining a slow fast system and geometric singular perturbation theory is applied the proposed theory shows that an orbit on the slow manifold corresponds to the smooth part of the solution to the euler equation where non equilibrium effects are negligible and a relaxation motion from the unsteady to the steady branch of the slow manifold corresponds to a shock wave where the flow relaxes from non equilibrium to equilibrium recognizing the shock wave as a fast motion is found to be especially useful for shock wave detection when post processing experimental measured or numerical calculated flow fields various existing shock detection methods can be derived from the proposed theory in a rigorous mathematical manner the proposed theory provides a new shock detection method based on its non linear non equilibrium nature and may also serve as the theoretical foundation for many popular shock wave detection techniques

interference alignment with partial csi feedback in mimo cellular networks interference alignment ia is a linear precoding strategy that can achieve optimal capacity scaling at high snr in interference networks however most existing ia designs require full channel state information csi at the transmitters which could lead to significant csi signaling overhead there are two techniques namely csi quantization and csi feedback filtering that reduce csi feedback overhead in this paper we consider ia processing with csi feedback filtering in mimo cellular networks we introduce a novel metric namely the feedback dimension to quantify the first order csi feedback cost associated with the csi feedback filtering the csi feedback filtering poses several important challenges in ia processing first there is a hidden partial csi knowledge constraint in ia precoder design which cannot be handled using conventional ia design methodology furthermore existing results on the feasibility conditions of ia cannot be applied due to the partial csi knowledge finally it is very challenging to find out how much csi feedback is actually needed to support ia processing we shall address the above challenges and propose a new ia feasibility condition under partial csit knowledge in mimo cellular networks based on this we consider the csi feedback dimension minimization subject to the degrees of freedom requirements and further propose an asymptotically optimal solution and derive closed form trade off results between the csi feedback cost and ia performance in mimo cellular networks

analysis of multiple antenna systems with finite rate channel information feedback over spatially correlated fading channels this paper employs a high resolution quantization framework to study the effects of finite rate quantization of the channel state information csi on the performance of miso systems over correlated fading channels the contributions of this paper are twofold first as an application of the general distortion analysis tight lower bounds on the capacity loss of correlated miso systems due to the finite rate channel quantization are provided closed form expressions for the capacity loss in high signal to noise ratio snr and low snr regimes are also provided and their analysis reveals that the capacity loss of correlated miso channels is related to that of i i d fading channels by a simple multiplicative factor which is given by the ratio of the geometric mean to the arithmetic mean of the eigenvalues of the channel covariance matrix second this paper extends the general asymptotic distortion analysis to the important practical problem of suboptimal quantizers resulting from mismatches in the distortion functions source statistics and quantization criteria as a specific application two types of mismatched miso csi quantizers are investigated quantizers whose codebooks are designed with minimum mean square error mmse criterion but the distortion measure is the ergodic capacity loss i e mismatched design criterion and quantizers with codebook designed with a mismatched channel covariance matrix i e mismatched statistics bounds on the channel capacity loss of the mismatched codebooks are provided and compared to that of the optimal quantizers finally numerical and simulation results are presented and they confirm the tightness of theoretical distortion bounds

analysis of multiple antenna systems with finite rate feedback using high resolution quantization theory this paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple antenna systems with finite rate feedback inspired by the results of classical high resolution quantization theory the problem of finite rate quantized communication system is formulated as a general fixed rate vector quantization problem with side information available at the encoder or the quantizer but unavailable at the decoder the framework of the quantization problem is sufficiently general to include quantization schemes with general non mean squared distortion functions and constrained source vectors asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the bennett s integral specifically tight lower and upper bounds of the average asymptotic distortion are proposed sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed rate quantization problems the proposed general methodology provides a powerful analytical tool to study a wide range of finite rate feedback systems to illustrate the utility of the framework we consider the analysis of a finite rate feedback multiple input single output miso beamforming system over independent and identically distributed i i d rayleigh flat fading channels numerical and simulation results are presented that further confirm the accuracy of the analytical results

quantization bounds on grassmann manifolds and applications to mimo communications the grassmann manifold gn p l is the set of all p dimensional planes through the origin in the n dimensional euclidean space ln where l is either r or c this paper considers the quantization problem in which a source in gn p l is quantized through a code in gn q l with p and q not necessarily the same the analysis is based on the volume of a metric ball in gn p l with center in gn q l and our chief result is a closed form expression for the volume of a metric ball of radius at most one this volume formula holds for arbitrary n p q and l while previous results pertained only to some special cases based on this volume formula several bounds are derived for the rate distortion tradeoff assuming that the quantization rate is sufficiently high the lower and upper bounds on the distortion rate function are asymptotically identical and therefore precisely quantify the asymptotic rate distortion tradeoff we also show that random codes are asymptotically optimal in the sense that they achieve the minimum possible distortion in probability as n and the code rate approach infinity linearly finally as an application of the derived results to communication theory we quantify the effect of beamforming matrix selection in multiple antenna communication systems with finite rate channel state feedback

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

stability of viscous scalar shock fronts in several dimensions we prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions the proof uses the integrated equation and an effective equation for the motion of the front itself we derive energy estimates that balance terms from the integrated equation with terms from the front motion equation in this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions these multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts lu and in shock waves with phase changes the proof here is a rigorous version of arguments that are used to derive effective equations such as the kuramoto sivashinsky equation to describe the behavior of fronts the one dimensional stability for scalar conservation laws was proven by ii in and oleinik io using the integrated equation see below and a max imum principle another proof based on weighted norms and spectral theory for the linearized problem was given by sattinger s the multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by kawashima and mat sumura km and goodman go and completed by liu li these proofs use l2 energy estimates for the integrated equation we consider equations of the form ut f u x g u y uxx uyy where f u is a strictly convex function of u

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

stability analysis of multiplicative update algorithms and application to nonnegative matrix factorization multiplicative update algorithms have proved to be a great success in solving optimization problems with nonnegativity constraints such as the famous nonnegative matrix factorization nmf and its many variants however despite several years of research on the topic the understanding of their convergence properties is still to be improved in this paper we show that lyapunov s stability theory provides a very enlightening viewpoint on the problem we prove the exponential or asymptotic stability of the solutions to general optimization problems with nonnegative constraints including the particular case of supervised nmf and finally study the more difficult case of unsupervised nmf the theoretical results presented in this paper are confirmed by numerical simulations involving both supervised and unsupervised nmf and the convergence speed of nmf multiplicative updates is investigated

stability of large ekman boundary layers in rotating fluids the aim of this paper is to investigate the stability of ekman boundary layers for rotating fluids when the ekman number and the rossby number go to zero more precisely we prove that spectral stability implies linear and nonlinear stabilities of approximate solutions in particular we replace the smallness condition obtained with energy methods in 5 by a weaker spectral condition which is sharp

combining beamforming and orthogonal space time block coding multiple transmit and receive antennas can be used in wireless systems to achieve high data rate communication efficient space time codes have been developed that utilize a large portion of the available capacity these codes are designed under the assumption that the transmitter has no knowledge about the channel in this work on the other hand we consider the case when the transmitter has partial but not perfect knowledge about the channel and how to improve a predetermined code so that this fact is taken into account a performance criterion is derived for a frequency nonselective fading channel and then utilized to optimize a linear transformation of the predetermined code the resulting optimization problem turns out to be convex and can thus be efficiently solved using standard methods in addition a particularly efficient solution method is developed for the special case of independently fading channel coefficients the proposed transmission scheme combines the benefits of conventional beamforming with those given by orthogonal space time block coding simulation results for a narrow band system with multiple transmit antennas and one or more receive antennas demonstrate significant gains over conventional methods in a scenario with nonperfect channel knowledge

automatic relevance determination in nonnegative matrix factorization with the beta divergence this paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization nmf with the divergence the divergence is a family of cost functions that includes the squared euclidean distance kullback leibler kl and itakura saito is divergences as special cases learning the model order is important as it is necessary to strike the right balance between data fidelity and overfitting we propose a bayesian model based on automatic relevance determination ard in which the columns of the dictionary matrix and the rows of the activation matrix are tied together through a common scale parameter in their prior a family of majorization minimization mm algorithms is proposed for maximum a posteriori map estimation a subset of scale parameters is driven to a small lower bound in the course of inference with the effect of pruning the corresponding spurious components we demonstrate the efficacy and robustness of our algorithms by performing extensive experiments on synthetic data the swimmer dataset a music decomposition example and a stock price prediction task

hyperbolic boundary value problems for symmetric systems with variable multiplicities abstract we extend the kreiss majda theory of stability of hyperbolic initial boundary value and shock problems to a class of systems notably including the equations of magnetohydrodynamics mhd for which majda s block structure condition does not hold namely simultaneously symmetrizable systems with characteristics of variable multiplicity satisfying at points of variable multiplicity either a totally nonglancing or a nonglancing and linearly splitting condition at the same time we give a simple characterization of the block structure condition as geometric regularity of characteristics defined as analyticity of associated eigenprojections the totally nonglancing or nonglancing and linearly splitting conditions are generically satisfied in the simplest case of crossings of two characteristics and likewise for our main physical examples of mhd or maxwell equations for a crystal together with previous analyses of spectral stability carried out by gardner kruskal and blokhin trakhinin this yields immediately a number of new results of nonlinear inviscid stability of shock waves in mhd in the cases of parallel or transverse magnetic field and recovers the sole previous nonlinear result obtained by blokhin trakhinin by direct dissipative integral methods of stability in the zero magnetic field limit we also discuss extensions to the viscous case

cubic regularization of newton method and its global performance in this paper we provide theoretical analysis for a cubic regularization of newton method as applied to unconstrained minimization problem for this scheme we prove general local convergence results however the main contribution of the paper is related to global worst case complexity bounds for different problem classes including some nonconvex cases it is shown that the search direction can be computed by standard linear algebra technique

degrees of freedom of the k user m times n mimo interference channel we provide inner bound and outer bound for the total number of degrees of freedom of the k user multiple input multiple output mimo gaussian interference channel with m antennas at each transmitter and n antennas at each receiver if the channel coefficients are time varying and drawn from a continuous distribution the bounds are tight when the ratio max m n min m n r is equal to an integer for this case we show that the total number of degrees of freedom is equal to min m n k if k r and min m n r r 1 k if k r achievability is based on interference alignment we also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some mimo interference channels with constant channel coefficients

stability of systems of viscous conservation laws we consider systems of conservation laws and give conditions for nonlinear stability of viscous shock profiles the analysis applies to classical shocks of arbitrary strength 1998 john wiley sons inc

asymptotic stability of relaxation shock profiles for hyperbolic conservation laws abstract this paper studies the asymptotic stability of traveling relaxation shock profiles for hyperbolic systems of conservation laws under a stability condition of subcharacteristic type the large time relaxation dynamics on the level of shocks is shown to be determined by the equilibrium conservation laws the proof is due to the energy principle using the weighted norms the interaction of waves from various modes is treated by imposing suitable weight matrix

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

boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems abstract in this paper we study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero the boundary is noncharacteristic for the hyperbolic system we in particular describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions a counterexample is given when this condition is not satisfied and the solution blows up as the viscosity goes to 0

existence and stability of multidimensional shock fronts in the vanishing viscosity limit in this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of majda on existence and stability of multidimensional shock waves more precisely given a curved lax shock solution u0 to a hyperbolic system of conservation laws we construct nearby viscous shock solutions ue to a parabolic viscous perturbation of the hyperbolic system which converge to u0 as viscosity e 0 and satisfy an appropriate conormal version of majda s stability estimate

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

viscous boundary value problems for symmetric systems with variable multiplicities abstract extending investigations of metivier and zumbrun in the hyperbolic case we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity specifically the construction of symmetrizers in the low frequency regime where variable multiplicity plays a role at the same time we extend the boundary layer theory to real or partially parabolic viscosities neumann or mixed type parabolic boundary conditions and systems with nonconservative form in addition proving a more fundamental version of the zumbrun serre rousset theorem valid for variable multiplicities characterizing the limiting hyperbolic system and boundary conditions as a nonsingular limit of a reduced viscous system the new effects of viscosity are seen to be surprisingly subtle in particular viscous coupling of crossing hyperbolic modes may induce a destabilizing effect we illustrate the theory with applications to magnetohydrodynamics

existence of relaxation shock profiles for hyperbolic conservation laws in this paper we prove that like viscosity relaxation can also smooth weak shocks for strict hyperbolic conservation laws as equilibium systems of hyperbolic relaxation systems under some natural structural conditions these conditions were derived previously by yong from stability considerations and hence are entirely analogous to the majda pego criterion of the viscous case the proof involves a new parametrization of hugoniot curves and a delicate analysis of the algebraic structure of relaxation systems as well as construction of an appropriate center manifold

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

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

spectral stability of small shock waves this paper is the first in a series of articles that study the eigenvalue problem for small amplitude shock waves in systems of conservation laws with viscosity or relaxation the papers show in various contexts that in the zero amplitude limit appropriately scaled versions of the associated evans bundles converge to suspensions of evans bundles for fixed shock waves in related reduced systems with lower dimensional state space in this article the new geometric framework establishing this connection is introduced in the simplest context that of shock waves associated with a simple genuinely nonlinear mode in systems with identity viscosity in one space dimension

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

design and analysis of transmit beamforming based on limited rate feedback this paper deals with design and performance analysis of transmit beamformers for multiple input multiple output mimo systems based on bandwidth limited information that is fed back from the receiver to the transmitter by casting the design of transmit beamforming based on limited rate feedback as an equivalent sphere vector quantization svq problem multiantenna beamformed transmissions through independent and identically distributed i i d rayleigh fading channels are first considered the rate distortion function of the vector source is upper bounded and the operational rate distortion performance achieved by the generalized lloyd s algorithm is lower bounded although different in nature the two bounds yield asymptotically equivalent performance analysis results the average signal to noise ratio snr performance is also quantified finally beamformer codebook designs are studied for correlated rayleigh fading channels and a low complexity codebook design that achieves near optimal performance is derived

multidimensional viscous shocks i degenerate symmetrizers and long time stability received by the editors september 18 2002 2000 mathematics subject classification primary 35l60 secondary 35b35 35b45 35k55 35l65 76l05

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

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

transmit beamforming in multiple antenna systems with finite rate feedback a vq based approach this paper investigates quantization methods for feeding back the channel information through a low rate feedback channel in the context of multiple input single output miso systems we propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization vq design algorithm the criterion is based on maximizing the mean squared weighted inner product mswip between the optimum and the quantized beamforming vector the performance of systems with quantized beamforming is analyzed for the independent fading case this requires finding the density of the squared inner product between the optimum and the quantized beamforming vector which is obtained by considering a simple approximation of the quantization cell the approximate density function is used to lower bound the capacity loss due to quantization the outage probability and the bit error probability the resulting expressions provide insight into the dependence of the performance of transmit beamforming miso systems on the number of transmit antennas and feedback rate computer simulations support the analytical results and indicate that the lower bounds are quite tight

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

stability of viscous shocks on finite intervals consider the cauchy problem for a system of viscous conservation laws with a solution consisting of a thin viscous shock layer connecting smooth regions we expect the time dependent behavior of such a solution to involve two processes one process consists of the large scale evolution of the solution this process is well modeled by the corresponding inviscid equations the other process is the adjustment in shape and position of the shock layer to the large scale solution the time scale of the second process is much faster than the first 1 v compared to 1 the second process can be divided into two parts adjustment of the shape and of the position during this adjustment the end states are essentially constant in order to answer the question of stability we have developed a technique where the two processes can be separated to isolate the fast process we consider the region in the vicinity of the shock layer the equations are augmented with special boundary conditions that reflect the slow change of the end states we show that for the isolated fast process the perturbations decay exponentially in time

stability of small amplitude boundary layers for mixed hyperbolic parabolic systems we consider an initial boundary value problem for a symmetrizable mixed hyperbolic parabolic system of conservation laws with a small viscosity e u e t f u e x e b u e u e x x when the boundary is noncharacteristic for both the viscous and the inviscid system and the boundary condition dissipative we show that u e converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small this generalizes previous results obtained for b invertible and the linear study of serre and zumbrun obtained for a pure dirichlet s boundary condition

stability of multidimensional viscous shocks for symmetric systems with variable multiplicities we establish long time stability of multi dimensional viscous shocks of a general class of symmetric hyperbolic parabolic systems with variable multiplicities notably including the equations of compressible magnetohydrodynamics mhd in dimensions d 2 this extends the existing result established by k zumbrun for systems with characteristics of constant multiplicity to the ones with variable multiplicity yielding the first such a stability result for fast mhd shocks at the same time we are able to drop a technical assumption on structure of the so called glancing set that was necessarily used in previous analyses the key idea to the improvements is to introduce a new simple argument for obtaining a l l resolvent bound in low frequency regimes by employing the recent construction of degenerate kreiss symmetrizers by o gues g metivier m williams and k zumbrun thus at the low frequency resolvent bound level our analysis gives an alternative to the earlier pointwise green s function approach of k zumbrun high frequency solution operator bounds have been previously established entirely by nonlinear energy estimates

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

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

stability of one dimensional boundary layers by using green s functions the aim of this paper is to investigate the stability of one dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and more precisely to prove that spectral stability implies linear and nonlinear stability of approximate solutions in particular we replace the smallness condition obtained by the energy method 10 13 by a weaker spectral condition 2001 john wiley sons inc

nonclassical multidimensional viscous and inviscid shocks extending our earlier work on lax type shocks of systems of conservation laws we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general lax or undercompressive type shock waves of nonconservative hyperbolic systems with parabolic regularization the hyperbolic equations may be of variable multiplicity and the parabolic regularization may be of real or partially parabolic type we prove an existence result for inviscid nonconservative shocks that extends to multidimensional shocks a one dimensional result of x lin proved by quite different methods in addition we construct families of smooth viscous shocks converging to a given inviscid shock as viscosity goes to zero thereby justifying the small viscosity limit for multidimensional nonconservative shocks in our previous work on shocks we made use of conservative form especially in parts of the low frequency analysis thus most of the new analysis of this paper is concentrated in this area by adopting the more general nonconservative viewpoint we are able to shed new light on both the viscous and inviscid theory for example we can now provide a clearer geometric motivation for the low frequency analysis in the viscous case also we show that one may in the treatment of inviscid stability of nonclassical and or nonconservative shocks remove an apparently restrictive technical assumption made by mokrane and coulombel in their work on respectively shocktype nonconservative boundary problems and conservative undercompressive shocks another advantage of the nonconservative perspective is that lax and undercompressive shocks can be treated by exactly the same analysis

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

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

incremental majorization minimization optimization with application to large scale machine learning majorization minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function these upper bounds are tight at the current estimate and each iteration monotonically drives the objective function downhill such a simple principle is widely applicable and has been very popular in various scientific fields especially in signal processing and statistics we propose an incremental majorization minimization scheme for minimizing a large sum of continuous functions a problem of utmost importance in machine learning we present convergence guarantees for nonconvex and convex optimization when the upper bounds approximate the objective up to a smooth error we call such upper bounds first order surrogate functions more precisely we study asymptotic stationary point guarantees for nonconvex problems and for convex ones we provide convergence rates for the expected objective function value we apply our scheme to composite optimization and obtain a new incremental proximal gradient algorithm with linear convergence rate for strongly convex functions our experiments show that our method is competitive with the state of the art for solving machine learning problems such as logistic regression when the number of training samples is large enough and we demonstrate its usefulness for sparse estimation with nonconvex penalties

l1nonlinear stability of traveling waves for a hyperbolic system with relaxation where x t r r the unknowns u v belong to r the function f f u is in c r and 0 is a fixed constant relaxation systems often arise in many physical situations let us recall for example gases not in thermodynamic equilibrium vk kinetic theory ce cip pi chromatography raa river flows traffic flows and more general waves wh more recently the study of a special class of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws jx an the 2 2 relaxation hyperbolic systems of conservation laws were first analyzed by liu li2 who justified some nonlinear stability criteria for diffusion waves expansion waves and traveling waves article no 0180

csi feedback reduction for mimo interference alignment interference alignment ia is a linear precoding strategy that can achieve optimal capacity scaling at high snr in interference networks most of the existing ia designs require full channel state information csi at the transmitters which induces a huge csi signaling cost hence it is desirable to improve the feedback efficiency for ia and in this paper we propose a novel ia scheme with a significantly reduced csi feedback to quantify the csi feedback cost we introduce a novel metric namely the feedback dimension this metric serves as a first order measurement of csi feedback overhead due to the partial csi feedback constraint conventional ia schemes can not be applied and hence we develop a novel ia precoder decorrelator design and establish new ia feasibility conditions via dynamic feedback profile design the proposed ia scheme can also achieve a flexible tradeoff between the degree of freedom dof requirements for data streams the antenna resources and the csi feedback cost we show by analysis and simulations that the proposed scheme achieves substantial reductions of csi feedback overhead under the same dof requirement in mimo interference networks

zero dissipation limit of solutions with shocks for systems of hyperbolic conservation laws we consider the convergence of solutions of conservation laws with viscosity to solutions having shocks of hyperbolic conservation laws without viscosity as the viscosity tends to zero our analysis reveals a rich structure of nonlinear wave interactions due to the presence of shocks and initial layers these interactions generate four different wave patterns initial layers shock layers diffusion waves and coupling waves we study the propagation and interactions of the four wave patterns by a detailed pointwise analysis

limited feedback design for interference alignment on mimo interference networks with heterogeneous path loss and spatial correlations interference alignment is degree of freedom optimal on k user mimo interference channels and many previous works have studied the transceiver designs however these works predominantly focus on networks with perfect channel state information at the transmitters and symmetrical interference topology in this paper we consider a limited feedback system with heterogeneous path loss and spatial correlations and investigate how the dynamics of the interference topology can be exploited to improve the feedback efficiency we propose a novel spatial codebook design and perform dynamic quantization via bit allocations to adapt to the asymmetry of the interference topology we bound the system throughput under the proposed dynamic scheme in terms of the transmit snr feedback bits and the interference topology parameters it is shown that when the number of feedback bits scales with snr as c s cdot log hbox snr cal o 1 the sum degrees of freedom of the network are preserved moreover the value of scaling coefficient c s can be significantly reduced in networks with asymmetric interference topology

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