dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions we establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions also called linear threshold or semilinear transfer functions as hahnloser sarpeshkar mahowald douglas and seung 2000 showed networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit to obtain this behavior the network must be multistable and nondivergent and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification the first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron the second condition is restricted to symmetric networks but can also take into account the stabilizing effect of nonlocal inhibitory interactions we demonstrate the application of the conditions on a simple example and the orientation selectivity model of ben yishai lev bar or and sompolinsky 1995 we show that the conditions can be used to identify in their model regions of maximal orientation selective amplification and symmetry breaking

pseudo skeleton approximations with better accuracy estimates abstract we propose a priori accuracy estimates for low rank matrix approximations that use just a small number of the rows and columns this number is greater than the approximation rank unlike the existing methods of pseudo skeleton approximation but the estimates are more accurate than previously known ones this paper generalizes the results of 12 13

chebyshev hankel matrices and the splitting approach for centrosymmetric toeplitz plus hankel matrices abstract centrosymmetric toeplitz plus hankel matrices are investigated on the basis of their splitting property which is their similarity to the direct sum of two special toeplitz plus hankel matrices these matrices can be considered as hankel matrices moment matrices in bases of chebyshev polynomials and are called chebyshev hankel matrices chebyshev hankel matrices have similar properties like hankel matrices this concerns inversion formulas and fast algorithms a superfast algorithm for solving chebyshev hankel and centrosymmetric toeplitz plus hankel systems is presented that is based on real trigonometric transforms the main tool of investigation is the interpretation of chebyshev hankel matrices as matrices of restricted multiplication operators with respect to chebyshev bases

impedance profile inversion via the first transport equation abstract the subject of this paper is the inverse reflection problem for a stratified elastic half space that is a linear elastic medium whose elastic properties depend only on depth from a planar free surface is stimulated at t 0 by a plane wave impulsive source the motion of a typical surface element is recorded for 0 t 2t it is shown that this surface trace determines the acoustic impedance of the medium as a function of travel time to travel time depth t moreover we give a precise characterization of those functions which may appear as surface traces and show uniqueness existence and continuous dependence of the logarithm of the impedance as a function of the surface trace in the sobolev h1 topology

fixed point attractor analysis for a class of neurodynamics nearly all models in neural networks start from the assumption that the input output characteristic is a sigmoidal function on parameter space we present a systematic and feasible method for analyzing the whole spectrum of attractors all saturated all but one saturated all but twosaturated and so on of a neurodynamical system with a saturated sigmoidal function as its input output characteristic we present an argument that claims under a mild condition that only all saturated or all but one saturated attractors are observable for the neurodynamics for any given all saturated configuration all but one saturated configuration the article shows how to construct an exact parameter region r such that if and only if the parameters fall within r then is an attractor a fixed point of the dynamics the parameter region for an all saturated fixed point attractor is independent of the specific choice of a saturated sigmoidal function whereas for an all but one saturated fixed point it

differential and integral methods for multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this procedure operates by recursively reconstructing the radon transform of the potential from the jump in the radon transform of the scattered field at the wave front this reconstructed potential is then used to propagate the wave front and scattered field differentially further into the support of the potential the connections between this differential procedure and integral equation procedures are then illustrated by the derivations of two well known exact integral equation procedures using the radon transform and a generalized radon transform these procedures as well as the layer stripping procedure are then reduced to the familiar born approximation result for this problem by neglecting multiple scattering events this illustrates the central role of the radon transform in both exact and approximate inversion procedures

generalization errors of the simple perceptron to find an exact form for the generalization error of a learning machine is an open problem even in the simplest case simple perceptron learning we introduce a new approach to tackle the problem the generalization error of the simple perceptron is expressed as a linear combination of extreme values of inputs with the help of extreme value theory in statistics we then obtain an exact form of the generalization error of the simple perceptron in the case of the worst learning generalization errors of the higher order perceptron taking the form of an inverse power law in the number of examples are also considered

scattering by a potential using hyperbolic methods etude du probleme de diffusion inverse developpement d un algorithme correspondant pour deux variables

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

a kalman filter solution of the inverse scattering problem with a rational reflection coefficient abstract this paper presents a new inverse scattering method for reconstructing the reflectivity function of symmetric two component wave equations or the potential of a schrodinger equation when the reflection coefficient is rational this method relies on the so called chandrasekhar equations which implement the kalman filter associated to a stationary state space model these equations are derived by using first a general layer stripping principle to obtain some differential equations for reconstructing a general scattering medium and by specializing these recursions to the case when the probing waves have a state space model

goupillaud inverse problem with arbitrary input a new impulse inverse formula and it s generalization including arbitrary input are proposed the new generalized inverse formula has the advantage of not being necessary to specify the first digital value of the input numerical instability of the inverse formula for arbitrary input is investigated carefully an analysis of the perturbed stability and lyapunov exponent are carried out as the result of this analysis it is shown that the estimated error is sensitive to the experimental error and previously estimated error as a sequential algorithm simulation using experimental data is also shown in the paper

tt cross approximation for multidimensional arrays abstract as is well known a rank r matrix can be recovered from a cross of r linearly independent columns and rows and an arbitrary matrix can be interpolated on the cross entries other entries by this cross or pseudo skeleton approximation are given with errors depending on the closeness of the matrix to a rank r matrix and as well on the choice of cross in this paper we extend this construction to d dimensional arrays tensors and suggest a new interpolation formula in which a d dimensional array is interpolated on the entries of some tt cross tensor train cross the total number of entries and the complexity of our interpolation algorithm depend on d linearly so the approach does not suffer from the curse of dimensionality we also propose a tt cross method for computation of d dimensional integrals and apply it to some examples with dimensionality in the range from d 100 up to d 4000 and the relative accuracy of order 10 10 in all constructions we capitalize on the new tensor decomposition in the form of tensor trains tt decomposition

the one dimensional inverse problem of reflection seismology an impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth the ensuing particle velocity at the surface is assumed to be measured the problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one dimensional inverse problem of reflection seismology the problem is set up mathematically as an inverse problem for a first order 2 times 2 hyperbolic system the role of propagation of singularities is explained and the relation between the solution of the inverse problem and the cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem it is found that recursive numerical techniques such as the levinson algorithm and certain other fast methods for cholesky factorization are intimately related to explicit finite difference schemes for solving hyperbolic systems the order of approximation between the

capacity of the hopfield model for a given if neurons deviating from the memorized patterns are allowed we constructively show that if and only if all stored patterns are fixed points of the hopfield model if neurons are allowed with then where is the distribution function of the normal distribution the result obtained by amit and co workers only formally coincides with the latter case which indicates that the replica trick approach to the capacity of the hopfield model is only valid in the case

an analysis on neural dynamics with saturated sigmoidal functions we propose a unified approach to study the relation between the set of saturated attractors and the set of system parameters of the hopfield model linsker s model and the dynamic link network dln which use saturated sigmoidal functions in its dynamics of the state or weight the key point for this approach is to rigorously derive a necessary and sufficient condition to test whether a given saturated state in the hopfield model or weight vector in linsker s model and the dln is stable or not for any given set of system parameters and used this to determine the complete regime in the parameter space over which the given state or weight is stable our approach allows us to give an exact characterization between the parameters and the capacity in the hopfield model to generalize our previous results on linsker s network and the dln to have a better understanding of the underlying mechanism among these models the method reported here could be adopted to analyze a variety of models in the field of the neural networks

fast matrix factorizations via discrete transmission lines abstract fast algorithms associated with the names of schur and levinson are known for the triangular factorization of symmetric positive definite toeplitz matrices and their inverses in this paper we show that these algorithms can be derived from simple arguments involving causality symmetry and energy conservation in discrete lossless transmission lines the results not only provide a nice interpretation of the classical schur and levinson algorithms and a certain toeplitz inversion formula of gohberg and semencul but they also show immediately that the same fast algorithms apply not only to toeplitz matrices but to all matrices with so called displacement inertia 1 1 the results have been helpful in suggesting new digital filter structures and in the study of nonstationary second order processes

lyapunov functions for neural nets with nondifferentiable input output characteristics i construct lyapunov functions for asynchronous dynamics and synch ronous dynamics of neural networks with nondifferentiable input output characteristics

source wavelet and its angular spectrum from plane wave seismograms in a typical seismic experiment land or marine the generated source wavelet is either unknown or known only approximately in order to solve many geophysical problems an accurate wavelet estimation is crucial this paper demonstrates a fast numerical algorithm which allows not only estimating the source wavelet but also its angular spectrum radiation pattern from plane wave decomposed seismograms surprisingly literature on this subject is virtually missing although a nonuniform angular spectrum of the generated wavelet can substantially affect the recorded data the technique presented here is based on the downward continuation dc and reverse time extrapolation rte of the recorded data both pressure and its vertical derivative the proposed method is iterative the rate of convergence does not depend on phase characteristics of the generated wavelet we demonstrate that both the wavelet signature and its angular spectrum can be estimated accurately from plane wave seismograms we also illu

rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle abstract the techniques of topological dynamics and differential dynamical systems are used to study polynomials orthogonal with respect to a measure supported on the unit circle it is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process in particular the techniques mentioned above are used to prove a gap labelling result

on neurodynamics with limiter function and linsker s developmental model the limiter function is used in many learning and retrieval models as the constraint controlling the magnitude of the weight or state vectors in this paper we developed a new method to relate the set of saturated fixed points to the set of system parameters of the models that use the limiter function and then as a case study applied this method to linsker s hebbian learning network we derived a necessary and sufficient condition to test whether a given saturated weight or state vector is stable or not for any given set of system parameters and used this condition to determine the whole regime in the parameter space over which the given state is stable this approach allows us to investigate the relative stability of the major receptive fields reported in linsker s simulations and to demonstrate the crucial role played by the synaptic density functions

acoustic impulse responses for nonuniform media abstract the acoustic impulse responses of a family of nonuniform lossless one dimensional media are determined the nonuniform portions of the media are confined to layers of either finite or infinite thicknesses the acoustic impedances of the nonuniform layers as functions of travel time are assumed known the impulse responses are determined by evaluating the fourier transforms of their reflection coefficients using infinite series expansions and contour integrations

fast algorithms for estimation and signal processing an inverse scattering framework the inverse scattering problem for a lossless transmission line has been used as a framework for deriving the levinson and schur fast algorithms for toeplitz matrices here the same framework is used to derive and attach novel simple scattering interpretations to the lattice algorithm and the split algorithms this demonstrates that the scattering and interpretation is more fundamental to fast signal processing algorithms than previously noted

inverse scattering for discrete transmission line models this paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform discrete transmission line from its response to a given forcing function this problem is the prototype for a wealth of one dimensional inverse scattering problems arising in various fields a unified straightforward approach is presented providing all known inversion procedures and some novel ones the derivations readily follow from causality of signal propagation on the transmission line it is shown that a direct exploitation of the signal propagation model leads to recursive computationally efficient and reliable inversion algorithms in particular the relationships between layer peeling schur type difference equation and layer adjoining levinson type integral equation algorithms are clearly displayed

an iterative method for the inversion of the two dimensional wave equation with a potential a numerical method for a nonlinear inversion problem for the 2d wave equation with a potential is discussed in order to avoid the ill posedness we substitute a coupled system of one way wave equations for the original wave equation an iterative algorithm is constructed to improve the accuracy of the inversion numerical experiments are performed on several examples to examine the effectiveness of this method

layer stripping solutions of multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this method operates by recursively reconstructing the potential from the jump in the scattered field at the wave front and then using the reconstructed potential to propagate the wave front and the scattered field further into the inhomogeneous region it is thus a generalization of algorithms that have been developed for one dimensional inverse scattering problems although the procedure has not yet been numerically tested the corresponding one dimensional algorithms have performed well on synthetic data the procedure is applied to a two dimensional inverse seismic problem connections between simplifications of this method and born approximation inverse scattering methods are also noted

reconstruction of a spherically symmetric speed of sound consider the inverse acoustic scattering problem for a spherically symmetric inhomogeneity of compact support that arises among other places in nondestructive testing define the corresponding homogeneous and inhomogeneous interior transmission problems see e g d colton and p monk quart j mech math 41 1988 pp 97 125 here the authors study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem it is shown in mclaughlin and polyakov j differential equations to appear that these eigenvalues are the zeros of an average of the scattering amplitude and a uniqueness theorem for the inverse acoustic scattering problem is presented where these eigenvalues are the given data in the present paper an algorithm for finding the solution of the inverse acoustic scattering problem from this subset of transmission eigenvalues is developed and implemented the method given here completely determines the sound

fast algorithms for toeplitz and hankel matrices abstract the paper gives a self contained survey of fast algorithms for solving linear systems of equations with toeplitz or hankel coefficient matrices it is written in the style of a textbook algorithms of levinson type and schur type are discussed their connections with triangular factorizations pade recursions and lanczos methods are demonstrated in the case in which the matrices possess additional symmetry properties split algorithms are designed and their relations to butterfly factorizations are developed

preconditioning linear least squares problems by identifying a basis matrix we study the solution of the linear least squares problem min x b a x 2 2 where the matrix a in bbb r m times n m ge n has rank n and is large and sparse we assume that a is available as a matrix not an operator the preconditioning of this problem is difficult because the matrix a does not have the properties of differential problems that make standard preconditioners effective incomplete cholesky techniques applied to the normal equations do not produce a well conditioned problem we attempt to bypass the ill conditioning by finding an n times n nonsingular submatrix b of a that reduces the euclidean norm of ab 1 we use b to precondition a symmetric quasi definite linear system whose condition number is then independent of the condition number of a and has the same solution as the original least squares problem we illustrate the performance of our approach on some standard test problems and show it is competitive with other approaches

inverse problem in nonstationary multidimensional medium the problem of a scalar wave propagation from the point impulsive source in the layer of a nonstationary multidimensional medium is considered the boundary problem for the wave equation is reformulated in the problem with the initial condition using the invariant imbedding method the integral differential inverse procedures of the various orders were obtained from the imbedding equations using the singularities method the order of inverse procedure is defined by the degree of a polynomial in the analytical representation of the medium characteristic near the layer boundary it was shown that the coefficients of the polynomial are calculated with the help of the differential characteristics of the point impulsive source in the inhomogeneous medium the cause and character of the multidimensional inverse problem overdefiniteness are considered the application of the proposed procedure for a statistical problem is discussed

matrix representations of split bezoutians abstract inverses of symmetric or skewsymmetric toeplitz matrices as well as of centrosymmetric or centro skewsymmetric toeplitz plus hankel matrices can be represented as sums of two split bezoutians which are highly structured matrices since all of their rows and columns are symmetric or skewsymmetric vectors thus it is desirable to find matrix representations for split bezoutians b this is the main aim of the present paper recursion formulas for the entries of b are presented bases of very simple split bezoutians or of sparse matrices are constructed and b is represented as a corresponding linear combination moreover matrix representations of gohberg semencul type are established

numerical methods for reflection inverse problems convergence and nonimpulsive sources we study numerical methods for a one dimensional reflection inverse problem a normally incident impulsive plane wave is sent into a stratified elastic half space from an adjoining homogeneous elastic half space and the reflected wave the reflectance is measured the characteristic impedance of the medium is to be recovered as a function of travel time this inverse problem occurs in several applications including reflection seismology and determining vocal tract shape for speech synthesis we show that if the step reflectance or ramp reflectance the first or second time integral of the reflectance is sampled appropriately and the problem is discretized accordingly then the solution of the discrete inverse problem converges uniformly to the impedance profile and the convergence is second order in the mesh width the proofs involve analyzing a discretization of an integral equation obtained by burridge which is a variant of the marchenko equation the discrete inverse problem can be solved by integ

estimation of coefficients and boundary parameters in hyperbolic systems we consider semi discrete galerkin approximation schemes in connection with inverse problems for the estimation of spatially varying coefficients and boundary condition parameters in second order hyperbolic systems typical of those arising in 1 d surface seismic problems spline based algorithms are proposed for which theoretical convergence results along with a representative sample of numerical findings are glven

rectangular maximum volume submatrices and their applications abstract we introduce a definition of the volume of a general rectangular matrix which is equivalent to an absolute value of the determinant for square matrices we generalize results of square maximum volume submatrices to the rectangular case show a connection of the rectangular volume with an optimal experimental design and provide estimates for a growth of coefficients and an approximation error in spectral and chebyshev norms three promising applications of such submatrices are presented recommender systems finding maximal elements in low rank matrices and preconditioning of overdetermined linear systems the code is available online

inversion of p sv seismic data in a layered elastic material density shear velocity and compressional velocity can be found at any depth from broadband surface measurements at two distinct nonzero precritical values of plane wave incidence angle layer stripping inversion uses three component surface velocity measurements generated by a polarized surface source to determine subsurface properties incrementally the surface velocity measurements initialize a first order nonlinear matrix riccati equation derived from the elastic wave equation which takes advantage of an attractive fixed point condition in the complex frequency plane to extract subsurface mechanical impedances subsurface density and velocities are recovered from the inverted impedances at two or more plane wave incidence angles general properties of the matrix riccati equation in the complex frequency plane aid in incorporating bandwidth constraints inversion of synthetic plane wave data from a piece wise continuous model illustrates inversion effects when only

analytical methods for recovering coefficients in differential equations from spectral data in this paper analytical methods for recovering coefficients from spectral data are discussed properties of these methods are given and mathematical models for which these methods might apply are presented

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

iterative trace deconvolution and noncausal transform for processing band limited data in 1983 carrion and patton showed that if recorded seismic data do not have low frequencies in their spectrum inversion for acoustic impedance is unstable this concept is commonly accepted here we infer acoustic impedance from band limited data without low frequencies using iterative trace deconvolution with the so called noncausal projection which moves initial data samples corresponding to small eigenvalues to negative times we show that this procedure can solve ill posed problems by migrating large residuals uncertainties in the trend of the recovered impedance to the bottom of the model an important property of the proposed method is that it converges to the true solution independently of the chosen initial model another advantage of the proposed algorithm is that unlike conventional dynamic deconvolution characteristic integration schemes it increases neither errors nor the condition number with depth the method is illustrated on synthetic and real data

convergence of numerical inversion methods for discontinuous impedance profiles we study numerical methods for the one dimensional inverse problem of reflection seismology an impulsive plane wave in pressure is applied to the surface of a piecewise smooth stratified elastic half space and the resulting particle velocity is measured at the surface the characteristic impedance of the medium is to be recovered as a function of travel time we consider two methods for the numerical solution of this inverse problem a discretization of the gopinath sondhi integral equation and a downward continuation method both methods are exact for so called goupillaud layered media both use second order difference schemes to reconstruct a solution of the system of partial differential equations governing motion in the medium a formally second order accurate equation based on the theory of geometrical acoustics is used to recover the impedance by analyzing the discretization of the gopinath sondhi integral equation we prove that if the impulse response is measured by sampling the step response or

subset selection for matrices abstract we consider the question of how to delete m k rows from a matrix x r m n so that the resulting matrix a r k n is as non singular as possible bounds for the singular values of a are derived which decrease only algebraically with m and n in addition a number of applications where subset selection is necessary are examined

asymptotic impact behavior of goupillaud type layered elastic media abstract in this paper we use the method of riemann invariants to develop a system of recurrence relations for the one dimensional elastic impact problem in which a semi infinite flyer collides with and adheres to a stationary goupillaud type layered elastic target plate of finite thickness that is bonded to a semi infinite half space we derive explicit expressions for the long time asymptotes of stress and particle velocity that are independent of the layered target s elastic properties this result is verified by numerical simulations of impact into layered targets with periodic or randomly distributed elastic properties and holds when the half space backing the target is replaced by either free or fixed boundary conditions our discrete recurrence relation solutions for impact into layered media generalize our recent analytical results concerning impact into homogeneous single layer targets as well as prior discrete recurrence relation solutions for stress wave propagation in goupillaud type layered elastic media