a characterization of proper minimal points as solutions of sublinear optimization problems abstract proper minimal points are of special interest in multiple objective optimization problems in this paper proper minimal points of sets in locally convex spaces and proper minimal solutions of constrained vector problems in topological linear spaces are characterized as solutions of scalar optimization problems in the latter case no differentiability assumption is made

blind deconvolution using convex programming we consider the problem of recovering two unknown vectors w and x of length l from their circular convolution we make the structural assumption that the two vectors are members of known subspaces one with dimension n and the other with dimension k although the observed convolution is nonlinear in both w and x it is linear in the rank 1 matrix formed by their outer product wx this observation allows us to recast the deconvolution problem as low rank matrix recovery problem from linear measurements whose natural convex relaxation is a nuclear norm minimization program we prove the effectiveness of this relaxation by showing that for generic signals the program can deconvolve w and x exactly when the maximum of n and k is almost on the order of l that is we show that if x is drawn from a random subspace of dimension n and w is a vector in a subspace of dimension k whose basis vectors are spread out in the frequency domain then nuclear norm minimization recovers wx without error we discuss this result in the context of blind channel estimation in communications if we have a message of length n which we code using a random l x n coding matrix and the encoded message travels through an unknown linear time invariant channel of maximum length k then the receiver can recover both the channel response and the message when l n k to within constant and log factors

fast preconditioned conjugate gradient algorithms for wiener hopf integral equations in this paper the authors study circulant approximations of finite sections of a wiener hopf integral equation on the half line such circulant operators are defined by periodic kernel functions they approximate finite sections of the wiener hopf operator within a sum of a small operator and an operator with fixed finite rank constructions are given of two such circulant operators and their use as preconditioners for the conjugate gradients algorithm is explained numerical examples are included to illustrate the results

convex optimization approaches for blind sensor calibration using sparsity we investigate a compressive sensing framework in which the sensors introduce a distortion to the measurements in the form of unknown gains we focus on blind calibration using measures performed on multiple unknown but sparse signals and formulate the joint recovery of the gains and the sparse signals as a convex optimization problem we divide this problem in 3 subproblems with different conditions on the gains specifially i gains with different amplitude and the same phase ii gains with the same amplitude and different phase and iii gains with different amplitude and phase in order to solve the first case we propose an extension to the basis pursuit optimization which can estimate the unknown gains along with the unknown sparse signals for the second case we formulate a quadratic approach that eliminates the unknown phase shifts and retrieves the unknown sparse signals an alternative form of this approach is also formulated to reduce complexity and memory requirements and provide scalability with respect to the number of input signals finally for the third case we propose a formulation that combines the earlier two approaches to solve the problem the performance of the proposed algorithms is investigated extensively through numerical simulations which demonstrates that simultaneous signal recovery and calibration is possible with convex methods when sufficiently many unknown but sparse calibrating signals are provided

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

necessary and sufficient conditions for oscillations of higher order delay differential equations we obtain necessary and sufficient conditions under which ail solutions of certain n th order delay differential equations oscillate when n is odd and necessary and sufficient conditions under which all bounded solutions oscillate when n is even the conditions derived involve the characteristic equation of the delay differential equation

factorized banded inverse preconditioners for matrices with toeplitz structure in this paper we study factorized banded inverse preconditioners for matrices with toeplitz structure we show that if a toeplitz matrix t has certain off diagonal decay property then the factorized banded inverse preconditioner approximates t 1 accurately and the spectra of these preconditioned matrices are clustered around 1 in nonlinear image restoration applications toeplitz related systems of the form i t d t are required to solve where d is a positive nonconstant diagonal matrix we construct inverse preconditioners for such matrices numerical results show that the performance of our proposed preconditioners are superior to that of circulant preconditioners a two dimensional nonlinear image restoration example is also presented to demonstrate the effectiveness of the proposed preconditioner

on the oscillation of certain functional differential equations via comparison methods abstract several new criteria for the oscillation of the functional differential equations of the form d dt 1 a n 1 t d dt 1 a n 2 t d dt 1 a 1 t d dt x t q t f x g t 0 are established in this paper

positive proper efficient points and related cone results in vector optimization theory positive proper efficient points are defined as solutions of appropriate linear scalar optimization problems a geometric characterization of positive proper efficient points is given as well as conditions under which the set of positive proper efficient points is dense in the set of all efficient points it is shown that these results are applicable in the normed vector lattices c a b l p and l p for 1 leqq p leqq infty and that previous related results which required the ordering cone to have a compact or weak compact base are not applicable in many normed vector lattices including c a b l p and l p for 1 leqq p leqq infty

identifiability in bilinear inverse problems with applications to subspace or sparsity constrained blind gain and phase calibration bilinear inverse problems bips the resolution of two vectors given their image under a bilinear mapping arise in many applications without further constraints bips are usually ill posed in practice the properties of natural signals are exploited to solve bips for example subspace constraints or sparsity constraints are imposed to reduce the search space these approaches have shown some success in practice however there are few results on uniqueness in bips for most bips the fundamental question of under what condition the problem admits a unique solution is yet to be answered for example blind gain and phase calibration bgpc is a structured bip which arises in many applications including inverse rendering in computational relighting albedo estimation with unknown lighting blind phase and gain calibration in sensor array processing and multichannel blind deconvolution mbd it is interesting to study the uniqueness of such problems in this paper we define identifiability of a bip up to a group of transformations we derive necessary and sufficient conditions for such identifiability i e the conditions under which the solutions can be uniquely determined up to the transformation group these conditions take the form of dividing the identifiability of the pair of unknown variables into the individual identifiability of each variable although verifying these individual conditions requires problem specific procedures this framework is universally applicable to all bips applying these results to bgpc we derive sufficient conditions for unique recovery under several scenarios including subspace joint sparsity and sparsity models for bgpc with joint sparsity or sparsity constraints we develop a procedure to compute the transformation groups corresponding to inherent ambiguities we also give necessary conditions in the form of tight lower bounds on sample complexities and demonstrate the tightness of these bounds by numerical experiments the results for bgpc not only demonstrate the application of the proposed general framework for identifiability analysis but are also of interest in their own right

truncated power method for sparse eigenvalue problems this paper considers the sparse eigenvalue problem which is to extract dominant largest sparse eigenvectors with at most k non zero components we propose a simple yet effective solution called truncated power method that can approximately solve the underlying nonconvex optimization problem a strong sparse recovery result is proved for the truncated power method and this theory is our key motivation for developing the new algorithm the proposed method is tested on applications such as sparse principal component analysis and the densest k subgraph problem extensive experiments on several synthetic and real world data sets demonstrate the competitive empirical performance of our method

stability of localized operators let lp 1 p be the space of all p summable sequences and ca be the convolution operator associated with a summable sequence a it is known that the lp stability of the convolution operator ca for different 1 p are equivalent to each other i e if ca has lp stability for some 1 p then ca has lq stability for all 1 q in the study of spline approximation wavelet analysis time frequency analysis and sampling there are many localized operators of non convolution type whose stability is one of the basic assumptions in this paper we consider the stability of those localized operators including infinite matrices in the sjostrand class synthesis operators with generating functions enveloped by shifts of a function in the wiener amalgam space and integral operators with kernels having certain regularity and decay at infinity we show that the lp stability or lp stability of those three classes of localized operators are equivalent to each other and we also prove that the left inverse of those localized operators are well localized

monotone convex sequences and cholesky decomposition of symmetric toeplitz matrices this paper studies off diagonal decay in symmetric toeplitz matrices it is shown that if the generating sequence of the matrix is monotone positive and convex then the monotonicity and positivity are maintained through triangular decomposition the work is motivated by recent results on explicit bounds for inverses of triangular matrices

fast and guaranteed blind multichannel deconvolution under a bilinear system model we consider the multichannel blind deconvolution problem where we observe the output of multiple channels that are all excited with the same unknown input from these observations we wish to estimate the impulse responses of each of the channels we show that this problem is well posed if the channels follow a bilinear model where the ensemble of channel responses is modeled as lying in a low dimensional subspace but with each channel modulated by an independent gain under this model we show how the channel estimates can be found by minimizing a quadratic function over a non convex set we analyze two methods for solving this non convex program and provide performance guarantees for each the first is a method of alternating eigenvectors that breaks the program down into a series of eigenvalue problems the second is a truncated power iteration which can roughly be interpreted as a method for finding the largest eigenvector of a symmetric matrix with the additional constraint that it adheres to our bilinear model as with most non convex optimization algorithms the performance of both of these algorithms is highly dependent on having a good starting point we show how such a starting point can be constructed from the channel measurements our performance guarantees are non asymptotic and provide a sufficient condition on the number of samples observed per channel in order to guarantee channel estimates of certain accuracy our analysis uses a model with a generic subspace that is drawn at random and we show the performance bounds hold with high probability mathematically the key estimates are derived by quantifying how well the eigenvectors of certain random matrices approximate the eigenvectors of their mean we also present a series of numerical results demonstrating that the empirical performance is consistent with the presented theory

wiener s lemma for infinite matrices in this paper we introduce a class of infinite matrices related to the beurling algebra of periodic functions and we show that it is an inverse closed subalgebra of mathcal b ell q w the algebra of all bounded linear operators on the weighted sequence space ell q w for any 1 q and any discrete muckenhoupt aq weight w

oscillation of functional trinomial differential equations with positive and negative term abstract in the paper we establish new technique for investigation of properties of trinomial differential equations with positive and negative term b t a t x t p t f x t q t h x t 0 we offer new criteria for asymptotic properties of nonoscillatory solutions for the studied equations we support our results with illustrative examples

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

understanding blind deconvolution algorithms blind deconvolution is the recovery of a sharp version of a blurred image when the blur kernel is unknown recent algorithms have afforded dramatic progress yet many aspects of the problem remain challenging and hard to understand the goal of this paper is to analyze and evaluate recent blind deconvolution algorithms both theoretically and experimentally we explain the previously reported failure of the naive map approach by demonstrating that it mostly favors no blur explanations we show that using reasonable image priors a naive simulations map estimation of both latent image and blur kernel is guaranteed to fail even with infinitely large images sampled from the prior on the other hand we show that since the kernel size is often smaller than the image size a map estimation of the kernel alone is well constrained and is guaranteed to succeed to recover the true blur the plethora of recent deconvolution techniques makes an experimental evaluation on ground truth data important as a first step toward this experimental evaluation we have collected blur data with ground truth and compared recent algorithms under equal settings additionally our data demonstrate that the shift invariant blur assumption made by most algorithms is often violated

oscillation of third order trinomial delay differential equations the purpose of this paper is to study oscillation and asymptotic behavior of solutions to a third order linear delay differential equation y t p t y t q t y tau t 0 new comparison th

iterative image restoration using approximate inverse preconditioning removing a linear shift invariant blur from a signal or image can be accomplished by inverse or wiener filtering or by an iterative least squares deblurring procedure because of the ill posed characteristics of the deconvolution problem in the presence of noise filtering methods often yield poor results on the other hand iterative methods often suffer from slow convergence at high spatial frequencies this paper concerns solving deconvolution problems for atmospherically blurred images by the preconditioned conjugate gradient algorithm where a new approximate inverse preconditioner is used to increase the rate of convergence theoretical results are established to show that fast convergence can be expected and test results are reported for a ground based astronomical imaging problem

oscillation criteria for certain nth order differential equations with deviating arguments abstract oscillation criteria for nth order differential equations with deviating arguments of the form x n 1 t 1 x n 1 t f t x g t 0 n even are established where g c t0 r f c t0 r r and 0 is a constant

on oscillation of third order noncanonical delay differential equations abstract in the paper we give new criteria for the oscillation of third order delay differential equations with noncanonical operators our results essentially improve and simplify the ones in dzurina et al 2018 the results are illustrated by a euler type delay differential equation

remarks on the paper by sun and meng appl math comput 174 2006 in the paper the second order half linear delay differential equation r t u t 1 u t p t u t 1 u t 0 1 is studied under the condition r 1 t d t the oscillation criterion proved by sun and meng 2006 theorem 2 2 is carefully examined and improved by simplifying and sharpening the estimates used in the original proof and removing unnecessary assumptions among others using elementary arguments we show that an arbitrary positive nondecreasing function from this criterion can be safely and with no loss of generality replaced by a constant function an extension to other related equations such as neutral equations is also provided

multichannel sparse blind deconvolution on the sphere multichannel blind deconvolution is the problem of recovering an unknown signal f and multiple unknown channels x i from their circular convolution y i x i circledast f i 1 2 dots n we consider the case where the x i s are sparse and convolution with f is invertible our nonconvex optimization formulation solves for a filter h on the unit sphere that produces sparse output y i circledast h under some technical assumptions we show that all local minima of the objective function correspond to the inverse filter of f up to an inherent sign and shift ambiguity and all saddle points have strictly negative curvatures this geometric structure allows successful recovery of f and x i using a simple manifold gradient descent mgd algorithm the same approach is also applicable to blind gain and phase calibration with a fourier sensing matrix our algorithm and analysis require fewer assumptions than previous algorithms for the same problem our theoretical findings are complemented by numerical experiments which demonstrate superior performance of the proposed approach over the previous methods empirically our algorithm has low computation cost converging in a small number of iterations and low memory footprint solving only for the inverse filter of f

degrees of efficiency and degrees of minimality in this work we characterize different types of solutions of a vector optimization problem by means of a scalarization procedure usually different scalarizing functions are used in order to obtain the various solutions of the vector problem here we consider different kinds of solutions of the same scalarized problem our results allow us to establish a parallelism between the solutions of the scalarized problem and the various efficient frontiers stronger solution concepts of the scalar problem correspond to more restrictive notions of efficiency besides the usual notions of weakly efficient and efficient points which are characterized as global and strict global solutions of the scalarized problem we also consider some restricted notions of efficiency such as strict and proper efficiency which are characterized as tikhonov well posed minima and sharp minima for the scalarized problem

multiple shooting with dichotomically stable formulae for linear boundary value problems abstract this paper describes an improvement of england and mattheij s code mutssym for solving linear boundary value problems for ordinary differential equations with may or not give rise to sharp boundary layers the method is based on multiple shooting with a decoupling strategy allowing the calculation of stable solutions according to the increasing or decreasing fundamental modes the integration of the associated initial value problems is performed using a 4 th order symmetric implicit runge kutta method with the dichotomic stability property if the problem is well conditioned the method calculates discrete decaying growing modes controlled by initial terminal conditions corresponding to similar continuous modes a special step size control strategy permits efficient calculation of the numerical solution throughout the interval 1999 elsevier science ltd all rights reserved

a least squares approach to blind channel identification conventional blind channel identification algorithms are based on channel outputs and knowledge of the probabilistic model of channel input in some practical applications however the input statistical model may not be known or there may not be sufficient data to obtain accurate enough estimates of certain statistics in this paper we consider the system input to be an unknown deterministic signal and study the problem of blind identification of multichannel fir systems without requiring the knowledge of the input statistical model a new blind identification algorithm based solely on the system outputs is proposed necessary and sufficient identifiability conditions in terms of the multichannel systems and the deterministic input signal are also presented

fir perfect signal reconstruction from multiple convolutions minimum deconvolver orders given p co prime finite impulse response fir filters h sub i spl isin r m sub h it well known that there exist q sub i spl isin r m sub q such that spl sigma sub i h sub i q sub i spl delta importantly this enables signal recovery from its convolutions with p spl ges 2 co prime fir filters by fir filtering we show that such q sub i exist almost surely if and only if m sub q spl ges m sub h 1 p 1 where x is the smallest integer greater than or equal to x the results also provide conditions for full rank of certain key matrices arising in the blind multichannel deconvolution problem

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

oscillation criteria of a certain class of third order nonlinear delay differential equations with damping abstract in this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form r 2 t r 1 t y p t y q t f y g t 0 by using a generalized riccati transformation and integral averaging technique we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero in particular several examples are given to illustrate the importance of our results

multichannel blind deconvolution of seismic signals a new algorithm for simultaneous wavelet estimation and deconvolution of seismic reflection signals is given to remove the inherent ambiguity in this blind deconvolution problem we introduce relevant a priori information our major assumption is sparseness of the reflectivity which corresponds to a layered earth model this allows nonminimum phase wavelets to be recovered reliably and closely spaced reflectors to be resolved to combine a priori knowledge and data we use a bayesian framework and derive a maximum a posteriori estimate computing this estimate is a difficult optimization problem solved by a suboptimal iterative procedure the procedure alternates steps of wavelet estimation and reflectivity estimation the first step only requires a simple least squares fit while the second step is solved by the iterated window maximization algorithm proposed by kaaresen this enables better efficiency and optimality than established alternatives the resulting optimization method can easily handle multichannel models with only a moderate increase of the computational load lateral continuity of the reflectors is achieved by modeling local dependencies between neighboring traces major improvements in both wavelet and reflectivity estimates are obtained by taking the wavelet to be invariant across several traces the practicality of the algorithm is demonstrated on synthetic and real seismic data an application to multivariate well log segmentation is also given

complete dictionary recovery over the sphere ii recovery by riemannian trust region method we consider the problem of recovering a complete i e square and invertible matrix a 0 from y in mathbb r n times p with y boldsymbol a 0 x 0 provided x 0 is sufficiently sparse this recovery problem is central to theoretical understanding of dictionary learning which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning we give the first efficient algorithm that provably recovers a 0 when x 0 has o left n right nonzeros per column under suitable probability model for x 0 our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint and hence is naturally phrased in the language of manifold optimization in a companion paper we have showed that with high probability our nonconvex formulation has no spurious local minimizers and around any saddle point the objective function has a negative directional curvature in this paper we take advantage of the particular geometric structure and describe a riemannian trust region algorithm that provably converges to a local minimizer with from arbitrary initializations such minimizers give excellent approximations to the rows of x 0 the rows are then recovered by a linear programming rounding and deflation

krylov subspace algorithms and circulant embedding method for efficient wideband single carrier equalization wider bandwidth allows higher data rate by transmitting narrower pulses however doing so would also increase the discrete channel memory length for single carrier communication systems this results in higher computational burden at the receiver we are concerned with single carrier nonblock transmission schemes with receiver oversampling as they can provide higher spectral efficiency than block transmission schemes in the presence of large delay spreads we first propose a simple finite impulse response fir equalizer that is based on the circulant embedding ce method and analyze its performance by investigating the relationship between solutions of various finite dimensional models and the original infinite dimensional problem we show that under proper conditions the ce fir equalizer converges exponentially fast to the iir equalizer we then focus on the conjugate gradient cg algorithm as an efficient means for equalization that is specifically well suited for dealing with large delay spread channels we discuss the importance of stopping the iterations for the cg algorithm at the right time in the presence of noise and present several reliable low cost stopping criteria it turns out that the cg algorithm equipped with appropriate stopping criteria can outperform mmse equalizers since both the ce and the cg methods can be efficiently implemented via fast fourier transforms equalization complexity is only in the order of n log n for n data symbols several numerical experiments demonstrate the performance of the proposed methods

a p q version of bourgain s theorem let 1 p q satisfy 1 p 1 q 1 we construct an orthonormal basis b n for l 2 r such that p b n and q bn are both uniformly bounded in n here f inf r x a f x 2 dx 1 2 this generalizes a theorem of bourgain and is closely related to recent results on the balian low theorem

toeplitz preconditioners for hermitian toeplitz systems abstract we propose a new type of preconditioners for hermitian positive definite toeplitz systems a n x b where a n are assumed to be generated by functions f that are positive and 2 periodic our approach is to precondition a n by the toeplitz matrix a n generated by 1 f we prove that the resulting preconditioned matrix a n a n will have clustered spectrum when a n cannot be formed efficiently we use quadrature rules and convolution products to construct nearby approximations to a n we show that the resulting approximations are toeplitz matrices which can be written as sums of circulant matrices as a side result we prove that any toeplitz matrix can be written as a sum of circulant matrices we then show that our toeplitz preconditioners t n are generalizations of circulant preconditioners and the way they are constructed is similar to the approach used in the additive schwarz method for elliptic problems we finally prove that the preconditioned systems t n a n will have clustered spectra around 1

oscillation of third order differential equations with noncanonical operators abstract new oscillation criteria for third order delay differential equations with noncanonical operators are presented contrary to existing results oscillation of the studied equation is attained via only two conditions our criteria not only improve extend and significantly simplify existing ones but the newly proposed approach could hopefully serve as a reference in the less developed theory of noncanonical equations of higher order the importance of the results obtained is illustrated via euler type equations

four short stories about toeplitz matrix calculations the stories told in this paper are dealing with the solution of finite infinite and biinfinite toeplitz type systems a crucial role plays the off diagonal decay behavior of toeplitz matrices and their inverses classical results of gelfand et al on commutative banach algebras yield a general characterization of this decay behavior we then derive estimates for the approximate solution of bi infinite toeplitz systems by the finite section method showing that the approximation rate depends only on the decay of the entries of the toeplitz matrix and its condition number furthermore we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems finally some quantitative results on the construction of preconditioners via circulant embedding are derived which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems

oscillation of certain functional differential equations sufficient conditions are presented for all bounded solutions of the linear system of delay differential equations 1 m 1 frac d m y i t dt m sum limits j 1 n q ij y j t h jj 0 text m geqslant 1 i 1 2 n to be oscillatory where qije hjj 0 i j 1 2 n also we study the oscillatory behavior of all bounded solutions of the linear system of neutral differential equations 1 m 1 frac d m dt m y i t cy i t g sum limits j 1 n q ij y j t h 0 where c g and h are real constants and i 1 2 n

symmetry and inverse closedness of matrix algebras and functional calculus for infinite matrices we investigate the symbolic calculus for a large class of matrix algebras that are defined by the off diagonal decay of infinite matrices applications are given to the symmetry of some highly non commutative banach algebras to the analysis of twisted convolution and to the theory of localized frames

the oscillation of higher order differential equations with deviating arguments in this paper some new criteria for the oscillation of higher order functional differential equations of the form lnx t f t x g t 0 n is even are established some of our results are obtained via comparing it with second order ordinary linear and first order delay differential equations the oscillation of this equation when n is odd is also considered then we shall use the obtained results to study the oscillatory behavior of the neutral functional differential equations of the form ln x t c t x t f t x g t 0 n is even and the damped functional differential equations of the type lnx t h t x g t ddtx h t 0 n is even the obtained results extend improve and correlate a number of existing criteria

on the existence and stability of approximate solutions of perturbed vector equilibrium problems abstract in this paper we consider several concepts of approximate minima of a set in normed vector spaces and we provide some results concerning the stability of these minima under perturbation of the underlying set with a sequence of sets converging in the sense of painleve kuratowski to the initial set then we introduce the concept of approximate solution for equilibrium problem governed by set valued maps and we study the stability of these solutions the particular case of linear continuous operators is considered as well

some observations on variance matrices abstract a class of special matrices is introduced it is a symmetric matrix depending on the order and a parameter which can also be a square matrix itself the connected minors of the matrix can be completely described there are also connections with totally positive matrices

evam an eigenvector based algorithm for multichannel blind deconvolution of input colored signals a new algorithm is proposed for the deconvolution of an unknown possibly colored gaussian or nonstationary signal that is observed through two or more unknown channels described by rational system transfer functions more specifically not only the root pole and zero locations but also the orders of the channel transfer functions are unknown it is assumed that the channel orders may be overestimated the proposed algorithm estimates the orders and root locations of the channel transfer functions therefore it can also be used in multichannel system identification problems the input signal is allowed to be nonstationary and the channel transfer functions may be a nonminimum phase as well as noncausal hence the proposed algorithm is particularly suitable for applications such as dereverberation of speech signals recorded through multiple microphones several experimental results indicate improvement compared to the existing methods in the literature

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

approximate duality for vector quasi equilibrium problems and applications abstract in this paper we introduce new versions of dual problems of a vector quasi equilibrium problem with set valued maps and we give an duality result between approximate solutions of the primal and dual problems as the first application of the main result we obtain an duality for a vector quasi equilibrium problem whose solutions are understood in the sense of proper efficiency the second application is devoted to an duality for a vector optimization problem with set valued maps

new criteria for blind deconvolution of nonminimum phase systems channels a necessary and sufficient condition for blind deconvolution without observing the input of nonminimum phase linear time invariant systems channels is derived based on this condition several optimization criteria are proposed and their solution is shown to correspond to the desired response these criteria involve the computation only of second and fourth order moments implying a simple tap update procedure the proposed methods are universal in the sense that they do not impose any restrictions on the probability distribution of the unobserved input sequence it is shown that in several important cases e g when the additive noise is gaussian the proposed criteria are essentially unaffected

complete dictionary recovery over the sphere i overview and the geometric picture we consider the problem of recovering a complete i e square and invertible matrix a 0 from y in mathbb r n times p with y a 0 x 0 provided x 0 is sufficiently sparse this recovery problem is central to theoretical understanding of dictionary learning which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning we give the first efficient algorithm that provably recovers a 0 when x 0 has o left n right nonzeros per column under suitable probability model for x 0 in contrast prior results based on efficient algorithms either only guarantee recovery when x 0 has o sqrt n zeros per column or require multiple rounds of semidefinite programming relaxation to work when x 0 has o n nonzeros per column our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint in this paper we provide a geometric characterization of the objective landscape in particular we show that the problem is highly structured with high probability 1 there are no spurious local minimizers and 2 around all saddle points the objective has a negative directional curvature this distinctive structure makes the problem amenable to efficient optimization algorithms in a companion paper we design a second order trust region algorithm over the sphere that provably converges to a local minimizer from arbitrary initializations despite the presence of saddle points

gradient descent with random initialization fast global convergence for nonconvex phase retrieval this paper considers the problem of solving systems of quadratic equations namely recovering an object of interest varvec x natural in mathbb r n from m quadratic equations samples y i varvec a i top varvec x natural 2 1 le i le m this problem also dubbed as phase retrieval spans multiple domains including physical sciences and machine learning we investigate the efficacy of gradient descent or wirtinger flow designed for the nonconvex least squares problem we prove that under gaussian designs gradient descent when randomly initialized yields an epsilon accurate solution in o big log n log 1 epsilon big iterations given nearly minimal samples thus achieving near optimal computational and sample complexities at once this provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval without the need of i carefully designed initialization ii sample splitting or iii sophisticated saddle point escaping schemes all of these are achieved by exploiting the statistical models in analyzing optimization algorithms via a leave one out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data

on the oscillation of certain third order nonlinear functional differential equations abstract we offer some sufficient conditions for the oscillation of all solutions of third order nonlinear functional differential equations of the form d d t a t d 2 d t 2 x t q t f x g t 0 and d d t a t d 2 d t 2 x t q t f x g t p t h x t when a 1 s d s the case when a 1 s d s is also included

left inverses of matrices with polynomial decay abstract it is known that the algebra of schur operators on l 2 namely operators bounded on both l 1 and l is not inverse closed when l 2 l 2 x where x is a metric space one can consider elements of the schur algebra with certain decay at infinity for instance if x has the doubling property then q sun has proved that the weighted schur algebra a x for a strictly polynomial weight is inverse closed in this paper we prove a sharp result on left invertibility of the these operators namely if an operator a a x satisfies a f p f p for some 1 p then it admits a left inverse in a x the main difficulty here is to obtain the above inequality in l 2 the author was both motivated and inspired by a previous work of aldroubi baskarov and krishtal 2008 1 where similar results were obtained through different methods for x z d under additional conditions on the decay