generalized principal component analysis projection of saturated model parameters principal component analysis pca is very useful for a wide variety of data analysis tasks but its implicit connection to the gaussian distribution can be undesirable for discrete data such as bi

interactions of two dimensionally localized pulses of the regularized long wave equation abstract interactions between positive and or negative cylindrical bell shaped pulses are investigated numerically for a two dimensional regularized long wave equation overtaking collisions of two same signed pulses are found to be fairly elastic head on collisions between positive and negative pulses are inelastic and their overall features are analogous to those in the one dimentional case some properties of collision processes are discussed in terms of the conversed quantities

stability and interactions of solitons in two component active systems we demonstrate that solitary pulses in linearly coupled nonlinear schr odinger equations with gain in one mode and losses in another one which is a model of an asymmetric erbium doped nonlinear optical coupler exist and are stable as was recently predicted analytically next we consider interactions between the pulses the in phase pulses attract each other and merge into a single one numerical and analytical consideration of the repulsive interaction between ensuremath pi out of phase pulses reveals the existence of their robust pseudobound state when a final separation between them takes an almost constant minimum value as a function of the initial separation mathit t mathrm in in a certain interval of mathit t mathrm in in the case of the phase difference ensuremath pi 2 the interaction is also repulsive textcopyright 1996 the american physical society

bifurcations of travelling wave solutions for 2 1 dimensional boussinesq type equation variants of the two dimensional boussinesq type equations with positive and negative exponents are studied the bifurcation theory of dynamic systems is fruitfully used to carry out the analysis the dynamical behavior of different physical structure solitary patterns solitons kink breaking and periodic wave solutions is obtained the quantitative change in the physical structure of the solutions is shown to depend on the systemic parameters under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given

numerical approximations of algebraic riccati equations for abstract systems modelled by analytic semigroups and applications this paper provides a numerical approximation theory of algebraic riccati operator equations with unbounded coefficient operators a and b such as arise in the study of optimal quadratic cost problems over the time interval 0 oo for the abstract dynamics j ay bu here a is the generator of a strongly continuous analytic semigroup and b is an unbounded operator with any degree of unboundedness less than that of a convergence results are provided for the riccati operators as well as for all the other relevant quantities which enter into the dynamic optimization problem the present numerical theory is the counterpart of a known continuous theory several examples of partial differential equations with boundary point control where all the required assumptions are verified illustrate the theory they include parabolic equations with l2 dirichlet control as well as plate equations with a strong degree of damping and point control

matrix completion and low rank svd via fast alternating least squares the matrix completion problem has attracted a lot of attention largely as a result of the celebrated net flix competition two popular approaches for solving the problem are nuclear norm regularized matrix approximation candes and tao 2009 mazumder et al 2010 and maximum margin matrix factorization srebro et al 2005 these two procedures are in some cases solving equivalent problems but with quite different algorithms in this article we bring the two approaches together leading to an efficient algorithm for large matrix factorization and completion that outperforms both of these we develop a software package softimpute in r for implementing our approaches and a distributed version for very large matrices using the spark cluster programming environment

convergence rates for the approximations of the solutions to algebraic riccati equations with unbounded coefficients case of analytic semigroups approximations schemes for the solutions of the algebraic riccati equations will be considered we shall concentrate on the case when the input operator is unbounded and the dynamics of the system is described by an analytic semigroup the main goal of the paper is to establish the optimal rates of convergence of the underlying approximations by optimal we mean such approximations which would reconstruct the optimal regularity of the original solutions as well as the best approximation properties of the finite dimensional subspaces it turns out that if one aims to obtain the optimal rates in the case of unbounded input operators the choice of the approximations to the generator as well as to the control operator is very critical while the convergence results hold with any consistent approximations the optimal rates require a careful selection of the approximating schemes our theoretical results will be illustrated by several examples of boundary point control problems where the optimal rates of convergence are achieved with the appropriate numerical algorithms

spectral regularization algorithms for learning large incomplete matrices we use convex relaxation techniques to provide a sequence of regularized low rank solutions for large scale matrix completion problems using the nuclear norm as a regularizer we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm our algorithm soft impute iteratively replaces the missing elements with those obtained from a soft thresholded svd with warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter the computationally intensive part of our algorithm is in computing a low rank svd of a dense matrix exploiting the problem structure we show that the task can be performed with a complexity of order linear in the matrix dimensions our semidefinite programming algorithm is readily scalable to large matrices for example soft impute takes a few hours to compute low rank approximations of a 106 x 106 incomplete matrix with 107 observed entries and fits a rank 95 approximation to the full netflix training set in 3 3 hours our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state of the art techniques

main effects and interactions in mixed and incomplete data frames a mixed data frame mdf is a table collecting categorical numerical and count observations the use of mdf is widespread in statistics and the applications are numerous from abundance data in ec

poisson matrix recovery and completion we extend the theory of low rank matrix recovery and completion to the case when poisson observations for a linear combination or a subset of the entries of a matrix are available which arises in various applications with count data we consider the usual matrix recovery formulation through maximum likelihood with proper constraints on the matrix m of size d 1 by d 2 and establish theoretical upper and lower bounds on the recovery error our bounds for matrix completion are nearly optimal up to a factor on the order of cal o log d 1 d 2 these bounds are obtained by combining techniques for recovering sparse vectors with compressed measurements in poisson noise those for analyzing low rank matrices as well as those for one bit matrix completion davenport 1 bit matrix completion information and inference information and inference vol 3 no 3 pp 189 223 sep 2014 although these two problems are different in nature the adaptation requires new techniques exploiting properties of the poisson likelihood function and tackling the difficulties posed by the locally sub gaussian characteristic of the poisson distribution our results highlight a few important distinctions of the poisson case compared to the prior work including having to impose a minimum signal to noise requirement on each observed entry and a gap in the upper and lower bounds we also develop a set of efficient iterative algorithms and demonstrate their good performance on synthetic examples and real data

simple structure in component analysis techniques for mixtures of qualitative and quantitative variables several methods have been developed for the analysis of a mixture of qualitative and quantitative variables and one called pcamix includes ordinary principal component analysis pca and multiple correspondence analysis mca as special cases the present paper proposes several techniques for simple structure rotation of a pcamix solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for pca in addition a recently developed technique for the analysis of mixtures of qualitative and quantitative variables called indomix is shown to construct component scores without rotational freedom maximizing the quartimax criterion over all possible sets of component scores a numerical example is used to illustrate the implication that when used for qualitative variables indomix provides axes that discriminate between the observation units better than do those generated from mca

stable two dimensional solitary pulses in linearly coupled dissipative kadomtsev petviashvili equations we present a two dimensional 2d generalization of the stabilized kuramoto sivashinsky system based on the kadomtsev petviashvili kp equation including dissipation of the generic newell whitehead segel nws type and gain the system directly applies to the description of gravity capillary waves on the surface of a liquid layer flowing down an inclined plane with a surfactant diffusing along the layer s surface actually the model is quite general offering a simple way to stabilize nonlinear media combining the weakly 2d dispersion of the kp type with gain and nws dissipation other applications are internal waves in multilayer fluids flowing down an inclined plane double front flames in gaseous mixtures etc parallel to this weakly 2d model we also introduce and study a semiphenomenological one whose dissipative terms are isotropic rather than of the nws type in order to check if qualitative results are sensitive to the exact form of the lossy terms the models include an additional linear equation of the advection diffusion type linearly coupled to the main kp nws equation the extra equation provides for stability of the zero background in the system thus opening a way for the existence of stable localized pulses we focus on the most interesting case when the dispersive part of the system is of the kp i type which corresponds e g to capillary waves and makes the existence of completely localized 2d pulses possible treating the losses and gain as small perturbations and making use of the balance equation for the field momentum we find that the equilibrium between the gain and losses may select two steady state solitons from their continuous family existing in the absence of the dissipative terms the latter family is found in an exact analytical form and is numerically demonstrated to be stable the selected soliton with the larger amplitude is expected to be stable direct simulations completely corroborate the analytical predictions for both the physical and phenomenological models

new compact and noncompact solutions for two variants of a modified camassa holm equation in this paper exact solutions of two variants of a modified camassa holm equation are determined by using direct ansatze as a result compactons solutions solitons with the absence of infinite tails solitons nonlinear localized waves of infinite support solitary patterns solutions having infinite slopes or cusps and plane periodic solutions are established it is also found that the qualitative change in the physical structure of solutions depends mainly on whether the exponent of the wave function u x t and the expression 2k c are positive or negative

cylindrical quasi solitons of the zakharov kuznetsov equation abstract evolutions and interactions of two dimensional solitary waves of the zakharov kuznetsov equation are investigated numerically formations of cylindrical bell shaped pulses are observed in the initial value problems a single bell shaped pulse propagates stably without any deformation like a soliton two similar pulses exchange their amplitudes without merging and two dissimilar ones undergo overtaking collision after a collision of two pulses the strong pulse becomes somewhat stronger and the weak one becomes weaker with radiation of ripples so that the collision process is slightly inelastic generated ripples are very small for center to center collision of similar pulses the cylindrically symmetric pulses of the zakharov kuznetov equation are thus found to behave approximately soliton like some properties of collision process are interpreted in terms of the conservation laws

nonlinear dispersive special type of the zakharov kuznetsov equation zk n n with compact and noncompact structures in this paper we study nonlinear dispersive special type of the zakharov kuznetsov equation with positive and negative exponents the approach depends mainly on the sine cosine algorithm compactons solitary patterns solitons and periodic solutions are formally derived

nonlinear variants of the improved boussinesq equation with compact and noncompact structures variants of the improved boussinesq equation with positive and negative exponents are investigated it is formally shown that these nonlinear variants give rise to compact and non compact physical structures where compactons solitons solitary patterns and periodic solutions are developed the presented sine cosine ansatz is reliable

composite self concordant minimization we propose a variable metric framework for minimizing the sum of a self concordant function and a possibly non smooth convex function endowed with an easily computable proximal operator we theoretically establish the convergence of our framework without relying on the usual lipschitz gradient assumption on the smooth part an important highlight of our work is a new set of analytic step size selection and correction procedures based on the structure of the problem we describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numerically on both synthetic and real data

galerkin approximations of abstract parabolic boundary value problems with rough boundary data l p theory etude des approximations de galerkin d un probleme aux valeurs limites parabolique abstrait avec des donnees aux limites grossieres determination des vitesses de convergence optimales dans les normes lp

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

robust matrix decomposition with sparse corruptions suppose a given observation matrix can be decomposed as the sum of a low rank matrix and a sparse matrix and the goal is to recover these individual components from the observed sum such additive decompositions have applications in a variety of numerical problems including system identification latent variable graphical modeling and principal components analysis we study conditions under which recovering such a decomposition is possible via a combination of l1 norm and trace norm minimization we are specifically interested in the question of how many sparse corruptions are allowed so that convex programming can still achieve accurate recovery and we obtain stronger recovery guarantees than previous studies moreover we do not assume that the spatial pattern of corruptions is random which stands in contrast to related analyses under such assumptions via matrix completion

variants of the two dimensional boussinesq equation with compactons solitons and periodic solutions variants of the two dimensional boussinesq equation with positive and negative exponents are studied the sine cosine ansatz is fruitfully used to carry out the analysis exact solutions of different physical structures compactons solitary patterns solitons and periodic solutions are obtained the quantitative change in the physical structure of the solutions is shown to depend mainly on the exponent of the wave function u x t and on the ratio a b of the derivatives of u x t

robust matrix completion this paper considers the problem of estimation of a low rank matrix when most of its entries are not observed and some of the observed entries are corrupted the observations are noisy realizations of a sum of a low rank matrix which we wish to estimate and a second matrix having a complementary sparse structure such as elementwise sparsity or columnwise sparsity we analyze a class of estimators obtained as solutions of a constrained convex optimization problem combining the nuclear norm penalty and a convex relaxation penalty for the sparse constraint our assumptions allow for simultaneous presence of random and deterministic patterns in the sampling scheme we establish rates of convergence for the low rank component from partial and corrupted observations in the presence of noise and we show that these rates are minimax optimal up to logarithmic factors

the camassa holm kp equations with compact and noncompact travelling wave solutions abstract in this paper two variants of camassa holm kp equations are investigated compactons solitons with the absence of infinite tails solitons nonlinear localized waves of infinite support solitary patterns having infinite slopes or cusps and plane periodic solutions are formally derived the work highlights the qualitative change in the physical structures of the obtained solutions

matrix completion with covariate information this article investigates the problem of matrix completion from the corrupted data when the additional covariates are available despite being seldomly considered in the matrix completion literatu

a conservative spectral method for several two dimensional nonlinear wave equations a conservative spectral method is proposed to solve several two dimensional nonlinear wave equations the conventional fast fourier transform is used to approximate the spatial derivatives and a three level difference scheme with a free parameter is to advance the solution in time our time discretization is semi implicit in the sense that the linear terms are treated implicitly while the nonlinear terms are evaluated only by previous time levels and thus treated explicitly however the cost of the algorithm is no greater than that of a fully explicit method because the linear boundary value problem that must be solved at each time step is almost trivial in a spectral spatial discretization a linear stability analysis shows that the method leads to a less restrictive stability condition than the corresponding explicit one the method is conservative and the ratio of the numerical dispersion to the physical dispersion is of the order o t2 applications of our method to the kadomtsev petviashvili and the zakharov kuznetsov equations exhibit excellent results

a max norm constrained minimization approach to 1 bit matrix completion we consider in this paper the problem of noisy 1 bit matrix completion under a general non uniform sampling distribution using the max norm as a convex relaxation for the rank a max norm constrained maximum likelihood estimate is introduced and studied the rate of convergence for the estimate is obtained information theoretical methods are used to establish a minimax lower bound under the general sampling model the minimax upper and lower bounds together yield the optimal rate of convergence for the frobenius norm loss computational algorithms and numerical performance are also discussed

cylindrical solitary pulses in a two dimensional stabilized kuramoto sivashinsky system abstract by linearly coupling a generalized zakharov kuznetsov equation alias the two dimensional 2d benney equation to an extra linear dissipative equation a 2d extension of a recently proposed stabilized kuramoto sivashinsky system is developed the model applies to the description of surface waves on 2d liquid layers in various physical settings the extra equation provides for the stability of the zero state in the system thus paving a way to the existence of stable 2d localized solitary pulses sps a perturbation theory based on a family of cylindrical solitons existing in the conservative counterpart of the system is developed by treating dissipation and gain in the model as small perturbations it is shown that the system may select two steady state solitons from the continuous family provided by the conservative counterpart of which the one with larger amplitude is expected to be stable numerical simulations support the analytical predictions quite well additionally it is found that a shallow quasi one dimensional 1d trough is attached to the stable sp if the integration domain is not very large and an explanation to this feature is proposed stable double humped bound states of two pulses are found too

algebraic riccati equation arising in boundary control problems an algebraic riccati equation is studied with application to the optimal control of deterministic and stochastic parabolic systems with boundary control the control function can act on the boundary through dirichlet or neumann conditions

generalized boussinesq type of equations with compactons solitons and periodic solutions generalized boussinesq type of equations with positive and negative exponents are examined the analysis depends mainly on the sine cosine ansatz it is formally shown that these nonlinear models give rise to compactons solitary patterns solitons and periodic solutions depending on the exponents and the coefficients of the derivatives of u x t the presented scheme reveals quite a number of remarkable features that will be helpful for identical problems

dirichlet boundary control problems for parabolic equations with quadratic cost analyticity and riccati s feedback synthesis riccati type feedback synthesis of optimal controls for dirichlet boundary parabolic equations is considered the functional cost penalizes the l2 energy over 0 t of state and control action u and also final state y t at t t this latter fact makes the functional cost discontinuous on the space of admissible controls l2 0t after overcoming some technical difficulties related to the above mentioned discontinuity we prove that the optimal control uo can be written in the desired feedback form uo t cp t yo t for all 0 t t

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

convergence rates for the feedback operators arising in the linear quadratic regulator problem governed by parabolic equations the linear quadratic control problem for systems governed by evolution equations of parabolic type is considered the optimal feedback control uses a riccati operator which is approximated by a galerkin scheme optimal convergence rates except for a logarithmic factor are derived for the approximation of the riccati operator for both the finite and infinite time interval this is done in an abstract hilbert space framework assuming a certain rate of convergence for the uncontrolled system in the numerical part the galerkin method is compared with the collocation method with special emphasis on the feedback gains