a stable normalized least mean fourth algorithm with improved transient and tracking behaviors the stability problems of the least mean fourth lmf algorithm put a limitation on its tracking capability the paper investigates the possibility of solving this problem via stabilization of the algorithm the analysis is done for a markov plant it is found that the available stable normalized lmf nlmf algorithm has a tracking limitation for high signal to noise ratio the paper presents a new stable nlmf algorithm that is free of this limitation mean square stability of the algorithm is proved expressions are derived for the minimum steady state mean square deviation msd and the corresponding convergence time the new algorithm outperforms the available stable nlmf algorithm in both the transient and steady states the new algorithm is also compared with the nlms algorithm when the adaptation parameter of each algorithm is set to the value that minimizes its steady state msd for large initial msd the algorithm outperforms the nlms algorithm even for gaussian noise for small initial msd the algorithm outperforms the nlms algorithm for sub gaussian noise while the situation is opposite for gaussian noise analytical results are supported by simulations

the bezout number for linear piecewise algebraic curves a piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function this paper discusses the bezout number the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite on regular triangulations we give an upper bound of the bezout number for linear piecewise algebraic curves bn 1 0 1 0 d on the triangulation with an odd interior vertex for the triangulations which satisfy a vertex coloring condition we compute the exact value of the bezout number bn 1 0 1 0 d

real root classification of parametric spline functions abstract the real root classification of a given parametric spline function is a collection of possible cases of its real root distribution on every interval together with the conditions of its coefficients must be satisfied for each case this paper presents an algorithm to deal with the real root classification of a given parametric spline function two examples are provided to illustrate the proposed algorithm is flexible

an upper bound of the bezout number for piecewise algebraic curves a piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function based on the discussion of the number of the zeros of homogeneous trigonometric splines with different smoothness and the common points of two piecewise algebraic curves over a star partition a better upper bound of bezout number of two piecewise algebraic curves over arbitrary triangulation is found moreover upper bounds of the bezout number b n m r n r for piecewise algebraic curves over several special partitions such as rectangular partition type 1 triangulation and type 2 triangulation are obtained

dimensions of spline spaces over t meshes a t mesh is basically a rectangular grid that allows t junctions in this paper we propose a method based on bezier nets to calculate the dimension of a spline function space over a t mesh when the order of the smoothness is less than half of the degree of the spline functions a dimension formula is derived which involves only the topological quantities of the t mesh the construction of basis functions is briefly discussed furthermore the dimension formulae for t meshes after mesh operations such as edge insertion and mesh merging are also obtained

an unconditionally convergent method for computing zeros of splines and polynomials we present a simple and ecient method for computing zeros of spline functions the method exploits the close relationship between a spline and its control polygon and is based on repeated knot insertion like newton s method it is quadratically convergent but the new method overcomes the principal problem with newton s method in that it always converges and no starting value needs to be supplied by the user

analysis of the momentum lms algorithm several modifications of the well known lms algorithm have been proposed for improved operation this work analyzes one such algorithm that corresponds to the standard lms algorithm with an additional update term parameterized by the scalar factor alpha where mod alpha mod

adaptive thinning algorithm of nonsingular triangulation this paper studies adaptive thinning strategies for the non singular triangulation of scattered data by c 1 rational spline function given a set of points in r 2 luo liu and chen have presented a triangulation algorithm which ensures the non singularity of s 2 1 and s 3 1 spaces in this paper we improve the algorithm to reduce the number of knots of the triangulation within a given tolerance while the non singularity of s 2 1 and s 3 1 spaces is ensured our strategies presented here depend on both the locations of the data points in the plane and the data values at these points we give the definition of discrete norm for c 1 rational spline function by using its coefficients then a weight is assigned to each knot which is a measure of the importance of knot in the representation of spline when the weight of the knot is less than the given tolerance its influence is regarded negligible then it can be removed it s a discrete method in the end of this paper several numerical examples are presented to show the feasibility and validity of our algorithm

mean square stability analysis of a normalized least mean fourth algorithm for a markov plant recently it has been shown that the stability of the least mean fourth lmf algorithm depends on the nonstationarity of the plant the present paper investigates the possibility of overcoming this problem by normalization of the weight vector update term a rigorous mean square stability analysis is provided for a recent normalized lmf algorithm which is normalized by a term that is second order in the estimation error and fourth order in the regressor the analysis is done for a markov plant with a stationary white input with even probability density function and a stationary zero mean white noise it is proved that the mean square deviation msd of the algorithm is bounded for all finite values of the input variance noise variance initial msd and mean square plant parameter increment analytical results are supported by simulations

convergence of the sign algorithm for adaptive filtering with correlated data convergence of a decreasing gain sign algorithm sa for adaptive filtering is analyzed the presence of the hard limiter in the algorithm makes a rigorous analysis difficult therefore there are few results available such results normally include restrictive assumptions such as the assumptions that successive observation vectors are independent and the new error signal of the adaptive filter has a time invariant probability density function the former assumption is not valid in the context of adaptive filtering since two successive observation vectors share most of their components while the latter assumption is a restriction on the adaptive weights whose evolution is a priori unknown in lieu of using these assumptions an almost sure convergence of the sa is proved under the assumption that the sequence of observation vectors is m dependent this assumption allows strong correlation between successive observations

cayley bacharach theorems and conjectures a theorem of pappus of alexandria proved in the fourth century a d began a long development in algebraic geometry in its changing expressions one can see reflected the changing concerns of the field from synthetic geometry to projective plane curves to riemann surfaces to the modern development of schemes and duality we survey this development historically and use it to motivate a brief treatment of a part of duality theory we then explain one of the modern developments arising from it a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them we give a proof of the conjectures in a new

solving parametric piecewise polynomial systems we deal with c r smooth continuity conditions for piecewise polynomial functions on d where d is an algebraic hypersurface partition of a domain w in r n piecewise polynomial functions of degree at most k on d that are continuously differentiable of order r form a spline space c k r we present a method for solving parametric systems of piecewise polynomial equations of the form z f 1 f n email protected email protected f 1 v x 0 f n v x 0 where f email protected c k w r w d and f w s email protected q v x for each n cell s i in d v u 1 u 2 u t is the set of parameters and x x 1 x 2 x n is the set of variables s 1 s 2 s m are all the n dimensional cells in d and email protected i 1 email protected i based on the discriminant variety method presented by lazard and rouillier we show that solving a parametric piecewise polynomial system z f 1 f n is reduced to the computation of discriminant variety of z the variety can then be used to solve the parametric piecewise polynomial system we also propose a general method to classify the parameters of z f 1 f n this method allows us to say that if there exist an open set of the parameters space where the system admits exactly a given number of distinct torsion free real zeros in every n cells in d

a complete algorithm for counting real solutions of polynomial systems of equations and inequalities abstract we present a complete and practical algorithm which can determine the number of distinct real solutions of a given polynomial system of equations and inequalities with integer coefficients mechanically based on this algorithm a program called nearsolve has been implemented in maple the algorithm and program have been successfully applied to many problems with various backgrounds and to automated discovering and proving for inequality type theorems

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

stochastic analysis of gradient adaptive identification of nonlinear systems with memory for gaussian data and noisy input and output measurements this paper investigates the statistical behavior of two gradient search adaptive algorithms for identifying an unknown nonlinear system comprised of a discrete time linear system h followed by a zero memory nonlinearity g spl middot the input and output of the unknown system are corrupted by additive independent noises gaussian models are used for all inputs two competing adaptation schemes are analyzed the first is a sequential adaptation scheme where the lms algorithm is first used to estimate the linear portion of the unknown system the lms algorithm is able to identify the linear portion of the unknown system to within a scale factor the weights are then frozen at the end of the first adaptation phase recursions are derived for the mean and fluctuation behavior of the lms algorithm which are in excellent agreement with monte carlo simulations when the nonlinearity is modeled by a scaled error function the second part of the sequential gradient identification scheme is shown to correctly learn the scale factor and the error function scale factor mean recursions for the scale factors show good agreement with monte carlo simulations for slow learning the stationary points of the gradient algorithm closely agree with the stationary points of the theoretical recursions the second adaptive scheme simultaneously learns both the linear and nonlinear portions of the unknown channel the mean recursions for the linear and nonlinear portions show good agreement with monte carlo simulations for slow learning the stationary points of the gradient algorithm also agree with the stationary points of the theoretical recursions

cayley bacharach theorem of piecewise algebraic curves the piecewise algebraic curve determined by a bivariate spline function is a generalization of the classical algebraic curve in this paper by using bezout s theorem and nother type theorem of piecewise algebraic curves the cayley bacharach theorem and hilbert function of c0 piecewise algebraic curves are presented

on the probability density function of the lms adaptive filter weights the joint probability density function of the weight vector in least mean square lms adaptation is studied for gaussian data models an exact expression is derived for the characteristic function of the weight vector at time n 1 conditioned on the weight vector at time n the conditional characteristic function is expanded in a taylor series and averaged over the unknown weight density to yield a first order partial differential difference equation in the unconditioned characteristic function of the weight vector the equation is approximately solved for small values of the adaption parameter and the weights are shown to be jointly gaussian with time varying mean vector and covariance matrix given as the solution to well known difference equations for the weight vector mean and covariance matrix the theoretical results are applied to analyzing the use of the weights in detection and time delay estimation simulations that support the theoretical results are also presented

lagrange interpolation by bivariate splines on cross cut partitions the piecewise algebraic curve is a generalization of the classical algebraic curve in this paper we give the nother type theorem of piecewise algebraic curves on cross cut partitions by using interpolation along a piecewise algebraic curve we also present a method to construct lagrange interpolation sets for bivariate spline spaces on cross cut partitions

stabilization of high order stochastic gradient adaptive filtering algorithms the paper is concerned with stabilizing the family of adaptive filtering algorithms based on minimizing the 2lth moment of the estimation error with l being an integer greater than 1 stabilization is attained via a proposed normalization of the algorithm mean square stability of the normalized algorithm is proved for a markov plant for all l 1 transient and steady state performances of the algorithm are analyzed for a time invariant plant expressions are derived for the steady state misadjustment and convergence time tradeoff between the transient and steady state performances is evaluated dependence of this tradeoff on the value of l is studied this tradeoff is compared with the tradeoff of the nlms algorithm the proposed algorithm is dramatically superior to the nlms algorithm for sub gaussian noise the superiority increases with l initial mean square deviation and signal to noise ratio analytical results are supported by simulations

a novel family of adaptive filtering algorithms based on the logarithmic cost we introduce a novel family of adaptive filtering algorithms based on a relative logarithmic cost inspired by the competitive methods from the online learning literature the competitive or regret based approaches stabilize or improve the convergence performance of adaptive algorithms through relative cost functions the new family elegantly and gradually adjusts the conventional cost functions in its optimization based on the error amount we introduce important members of this family of algorithms such as the least mean logarithmic square lmls and least logarithmic absolute difference llad algorithms however our approach and analysis are generic such that they cover other well known cost functions as described in the paper the lmls algorithm achieves comparable convergence performance with the least mean fourth lmf algorithm and enhances the stability performance significantly the llad and least mean square lms algorithms demonstrate similar convergence performance in impulse free noise environments while the llad algorithm is robust against impulsive interferences and outperforms the sign algorithm sa we analyze the transient steady state and tracking performance of the introduced algorithms and demonstrate the match of the theoretical analyses and simulation results we show the enhanced stability performance of the lmls algorithm and analyze the robustness of the llad algorithm against impulsive interferences finally we demonstrate the performance of our algorithms in different scenarios through numerical examples

stochastic gradient adaptation under general error criteria examines a family of adaptive filter algorithms of the form w sub k 1 w sub k spl mu f d sub k w sub k sup t x sub k x sub k in which f spl middot is a memoryless odd symmetric nonlinearity acting upon the error such algorithms are a generalization of the least mean square lms adaptive filtering algorithm for even symmetric error criteria for this algorithm family the authors derive general expressions for the mean and mean square convergence of the filter coefficients for both arbitrary stochastic input data and gaussian input data they then provide methods for optimizing the nonlinearity to minimize the algorithm misadjustment for a given convergence rate using the calculus of variations it is shown that the optimum nonlinearity to minimize misadjustment near convergence under slow adaptation conditions is independent of the statistics of the input data and can be expressed as p x p x where p x is the probability density function of the uncorrelated plant noise for faster adaptation under the white gaussian input and noise assumptions the nonlinearity is shown to be x 1 spl mu spl lambda x sup 2 spl sigma sub k sup 2 where spl lambda is the input signal power and spl sigma sub k sup 2 is the conditional error power thus the optimum stochastic gradient error criterion for gaussian noise is not mean square it is shown that the equations governing the convergence of the nonlinear algorithm are exactly those which describe the behavior of the optimum scalar data nonlinear adaptive algorithm for white gaussian input simulations verify the results for a host of noise interferences and indicate the improvement using non mean square error criteria

bounds on the number of solutions of polynomial systems and the betti numbers of real piecewise algebraic hypersurfaces abstract this paper based on bihan and sottile s method which reduces a polynomial system to its gale dual system and then bounds the number of solutions of this gale system proves that a real coefficient polynomial system with n equations and with n variables involving n k 1 monomials has fewer than 27 e 5 3 88 90 i 0 k 1 2 i n 1 1 positive solutions and 27 e 10 3 88 90 i 0 k 1 2 i n 1 1 non degenerate non zero real solutions this dramatically improves f bihan and f sottile s bounds of e 2 3 4 2 k 2 n k and e 4 3 4 2 k 2 n k respectively using the new upper bound for positive solutions we establish restrictions to the sum of the betti numbers of real piecewise algebraic hypersurfaces and real piecewise algebraic curves a new bound on the number of compact components of algebraic hypersurfaces in r n is also given

counting positive solutions for polynomial systems with real coefficients based on the subresultant polynomial chain and the method presented by hongguang fu and lu yang et al we propose a practical algorithm for computing the number of sign changes of the leading coefficients and the constant terms of a generalized sturm sequence gss with the principal minors sequence of the discrimination matrix of the polynomial and the number of sign changes of the gss a formula determining the number of positive solutions of a given bivariate polynomial systems with real coefficients with a finite number of complex solutions and its algorithm are presented using the techniques of the b net form of bivariate splines function this algorithm can be used for computing the number of real intersection points of two piecewise algebraic curves and its distributions in all cells when common points are finite a piecewise algebraic curve is defined by a bivariate spline function

the maximum number and its distribution of singular points for parametric piecewise algebraic curves abstract the piecewise algebraic curve as the zero set of a bivariate spline function is a generalization of the classical algebraic curve based on the previous method presented by lai et al 2009 we show that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial systems in this article we give a method to compute the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve this method also produces necessary and sufficient conditions of its parameters must be satisfied an illustrated example shows that our method is flexible

global stabilization of the least mean fourth algorithm the least mean fourth algorithm has several stability problems its stability depends on the variance and distribution type of the adaptive filter input the noise variance and the initialization of the filter weights the present correspondence provides a global solution to all these stability problems this is achieved by normalizing the weight vector update term by a term that is fourth order in the regressor and second order in the estimation error the former property stabilizes the algorithm against the variance and distribution type of the filter input while the latter stabilizes the algorithm against the noise variance and the weight initialization the obtained algorithm is stable for all values of the step size between 0 and 2 the stability of the algorithm is supported by simulations

the principal components of natural images revisited this paper investigates the principal components pcs of natural gray and color images a horizontal and vertical typology of pcs is found which leads to the identification of groups of basis functions for steerable bandpass filters using this system the contribution of spatio chromatic structure to the total variance can be quantified for selected spatial frequencies

adaptive algorithms with nonlinear data and error functions the tools of nonlinear system theory are used to examine several common nonlinear variants of the lms algorithm and derive a persistence of excitation criterion for local exponential stability the condition is tight when the inputs are periodic and a generic counterexample is demonstrated which gives local instability for a large class of such nonlinear versions of lms specifically those which utilize a nonlinear data function the presence of a nonlinear error function is found to be relatively benign in that it does not affect the stability of the error system rather it defines the cost function the algorithm tends to minimize specific examples include the dead zone modification the cubed data nonlinearity the cubed error nonlinearity the signed regressor algorithm and a single layer version of the backpropagation algorithm

homology of smooth splines generic triangulations and a conjecture of strang for a a triangulated d dimensional region in rd let sr a denote the vector space of all cr functions f on a that restricted to any simplex in a are given by polynomials of degree at most m we consider the problem of computing the dimension of such spaces we develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds a in the plane getting lower bounds on the dimension of sr a for all r for r 1 we prove a conjecture of strang concerning the generic dimension of the space of cl splines over a triangulated manifold in r2 finally we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region

the correspondence between multivariate spline ideals and piecewise algebraic varieties as a piecewise polynomial with a certain smoothness the spline plays an important role in computational geometry the algebraic variety is the most important subject in classical algebraic geometry as the zero set of multivariate splines the piecewise algebraic variety is a generalization of the algebraic variety in this paper the correspondence between piecewise algebraic varieties and spline ideals is discussed furthermore hilbert s nullstellensatz for the piecewise algebraic variety is also studied

multivariate weak spline function space in this paper b net method for studying multivariate weak spline is discussed and the dimension of w21 i1 1 is presented several b splines on type 1 triangulation are also obtained

polynomial systems with few real zeroes we study some systems of polynomials whose support lies in the convex hull of a circuit giving a sharp upper bound for their numbers of real solutions this upper bound is non trivial in that it is smaller than either the kouchnirenko or the khovanskii bounds for these systems when the support is exactly a circuit whose affine span is z n this bound is 2n 1 while the khovanskii bound is exponential in n 2 the bound 2n 1 can be attained only for non degenerate circuits our methods involve a mixture of combinatorics geometry and arithmetic

scattered data interpolation based upon bivariate recursive polynomials abstract in this paper firstly based on new recursive algorithms of non tensor product typed bivariate divided differences scattered data interpolation schemes are constructed in the cases of odd and even interpolating nodes respectively moreover the corresponding error estimation is worked out and equivalent formulae are obtained between bivariate high order non tensor product typed divided differences and high order partial derivatives furthermore the operation count for the addition subtractions multiplication and divisions approximates o n 2 in the computation of the interpolating polynomials presented while the operation count approximates o n 3 in the case of radial basis functions for sufficiently large n finally several numerical examples show that it is valid for the recursive interpolating polynomial schemes and these interpolating polynomials change as the order of the interpolating nodes although the node collection is the same

on the learning mechanism of adaptive filters this paper highlights both analytically and by simulations some interesting phenomena regarding the behavior of ensemble average learning curves of adaptive filters that may have gone unnoticed among other results the paper shows that even ensemble average learning curves of single tap lms filters actually exhibit two distinct rates of convergence one for the initial time instants and another faster one for later time instants in addition such curves tend to converge faster than predicted by mean square theory and can converge even when a mean square stability analysis predicts divergence these effects tend to be magnified by increasing the step size two of the conclusions that follow from this work are 1 the mean square stability alone may not be the most appropriate performance measure especially for larger step sizes a combination of mean square stability and almost sure a s stability seems to be more appropriate 2 care is needed while interpreting ensemble average curves for larger step sizes the curves can lead to erroneous conclusions unless a large number of experiments are averaged at times of the order of tens of thousands or higher

piecewise algebraic curve a piecewise algebraic curve is defined by a bivariate spline function using the techniques of the b net form of bivariate splines function discriminant sequence of polynomial cf yang lu et al sci china ser e 39 6 1996 628 and yang lu et al nonlinear algebraic equation system and automated theorem proving shanghai scientific and technological education publishing house shanghai 1996 and the number of sign changes in the sequence of coefficients of the highest degree terms of sturm sequence we determine the number of real intersection points of two piecewise algebraic curves whose common points are finite a lower bound of the number of real intersection points is given in terms of the method of rotation degree of vector field

avoiding divergence in the shalvi weinstein algorithm the most popular algorithms for blind equalization are the constant modulus algorithm cma and the shalvi weinstein algorithm swa it is well known that swa presents a higher convergence rate than cma at the expense of higher computational complexity if the forgetting factor is not sufficiently close to one if the initialization is distant from the optimal solution or if the signal to noise ratio is low swa can converge to undesirable local minima or even diverge in this paper we show that divergence can be caused by an inconsistency in the nonlinear estimate of the transmitted signal or when the algorithm is implemented in finite precision by the loss of positiveness of the estimate of the autocorrelation matrix or by a combination of both in order to avoid the first cause of divergence we propose a dual mode swa in the first mode of operation the new algorithm works as swa in the second mode it rejects inconsistent estimates of the transmitted signal assuming the persistence of excitation condition we present a deterministic stability analysis of the new algorithm to avoid the second cause of divergence we propose a dual mode lattice swa which is stable even in finite precision arithmetic and has a computational complexity that increases linearly with the number of adjustable equalizer coefficients the good performance of the proposed algorithms is confirmed through numerical simulations

transient analysis of adaptive affine combinations in this correspondence we provide a transient analysis of an affinely constrained mixture method that adaptively combines the outputs of adaptive filters running in parallel on the same task the affinely constrained mixture is adapted using a stochastic gradient update to minimize the square of the prediction error although we specifically carry out the transient analysis for a combination of two equal length adaptive filters trying to learn a linear model working on real valued data we also provide the final equations and the necessary extensions in order to generalize the transient analysis to mixtures combining more than two filters using newton based updates to train the mixture weights working on complex valued data or unconstrained mixtures the derivations are generic such that the constituent filters can be trained using unbiased updates including the least mean squares or recursive least squares updates this correspondence concludes with numerical examples and final remarks

structure and dimension of multivariate spline space of lower degree on arbitrary triangulation in this paper we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules on the base of analyzing the algebraic and geometric results about singularity of s21 ms we build the structure of triangulation and give some useful geometric conditions such that s 1 space is singular and we obtain an algebraic condition which is necessary and sufficient for the singularity of s 1 spaces as well as their dimension formulae moreover the structure matrix of spline spaces over any given partition is defined which has been used to discuss the structure of s31 and s21 spaces over arbitrary triangulation and to prove the nonsingularity of s31 spaces this partially settles a conjecture on the singularity of spline spaces in wang et al multivariate spline and its applications kluwer press dordrecht 2002 academic press beijing 1994 in chinese meanwhile the dimension formulae of s31 s21 spaces and the dimension formulae of s 1 1 spaces are also given in this paper

recent researches on multivariate spline and piecewise algebraic variety the multivariate splines as piecewise polynomials have become useful tools for dealing with computational geometry computer graphics computer aided geometrical design and image processing it is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials recently the surfaces are mainly represented by multivariate piecewise polynomials i e multivariate splines so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry moreover the piecewise algebraic variety will be also important in computational geometry computer graphics computer aided geometrical design and image processing the purpose of this paper is to introduce some recent researches on multivariate spline piecewise algebraic variety curve and their applications

minimum bit error rate design for space time equalization based multiuser detection a novel minimum bit error rate mber space time equalization ste based multiuser detector mud is proposed for multiple receive antenna assisted space division multiple access systems it is shown that the mber ste aided mud significantly outperforms the standard minimum mean square error design in terms of the achievable bit error rate ber adaptive implementations of the mber ste are considered and both the block data based and sample by sample adaptive mber algorithms are proposed the latter referred to as the least ber lber algorithm is compared with the most popular adaptive algorithm known as the least mean square lms algorithm it is shown that in case of binary phase shift keying the computational complexity of the lber ste is about half of that required by the classic lms ste simulation results demonstrate that the lber algorithm performs consistently better than the classic lm salgorithm both in terms of its convergence speed and steady state ber performance

stochastic stability analysis for the constant modulus algorithm we derive an easy to compute approximate bound for the range of step sizes for which the constant modulus algorithm cma will remain stable if initialized close to a minimum of the cm cost function our model highlights the influence of the signal constellation used in the transmission system for smaller variation in the modulus of the transmitted symbols the algorithm will be more robust and the steady state misadjustment will be smaller the theoretical results are validated through several simulations for long and short filters and channels

stochastic analysis of a stable normalized least mean fourth algorithm for adaptive noise canceling with a white gaussian reference the least mean fourth lmf algorithm has several stability problems its stability depends on the variance and distribution type of the adaptive filter input the noise variance and the initialization of the filter weights a global solution to these stability problems was presented recently for a normalized lmf nlmf algorithm here a stochastic analysis of the mean square deviation msd of the globally stable nlmf algorithm is provided the analysis is done in the context of adaptive noise canceling with a white gaussian reference input and gaussian binary and uniform desired signals the analytical model is shown to accurately predict the results of monte carlo simulations comparisons of the nlmf and nlms algorithms are then made for various parameter selections it is then shown under what conditions the nlmf algorithm is superior to nlms algorithm for adaptive noise canceling

the least mean fourth lmf adaptive algorithm and its family new steepest descent algorithms for adaptive filtering and have been devised which allow error minimization in the mean fourth and mean sixth etc sense during adaptation the weights undergo exponential relaxation toward their optimal solutions time constants have been derived and surprisingly they turn out to be proportional to the time constants that would have been obtained if the steepest descent least mean square lms algorithm of widrow and hoff had been used the new gradient algorithms are insignificantly more complicated to program and to compute than the lms algorithm their general form is w j 1 w j 2 mu k epsilon j 2k 1 x j where w j is the present weight vector w j 1 is the next weight vector epsilon j is the present error x j is the present input vector mu is a constant controlling stability and rate of convergence and 2k is the exponent of the error being minimized conditions have been derived for weight vector convergence of the mean and of the variance for the new gradient algorithms the behavior of the least mean fourth lmf algorithm is of special interest in comparing this algorithm to the lms algorithm when both are set to have exactly the same time constants for the weight relaxation process the lmf algorithm under some circumstances will have a substantially lower weight noise than the lms algorithm it is possible therefore that a minimum mean fourth error algorithm can do a better job of least squares estimation than a mean square error algorithm this intriguing concept has implications for all forms of adaptive algorithms whether they are based on steepest descent or otherwise

a mean square stability analysis of the least mean fourth adaptive algorithm this paper presents a new convergence analysis of the least mean fourth lmf adaptive algorithm in the mean square sense the analysis improves previous results in that it is valid for non gaussian noise distributions and explicitly shows the dependence of algorithm stability on the initial conditions of the weights analytical expressions are derived presenting the relationship between the step size the initial weight error vector and mean square stability the analysis assumes a white zero mean gaussian reference signal and an independent identically distributed i i d measurement noise with any even probability density function pdf it has been shown by nascimento and bermudez probability of divergence for the least mean fourth lmf algorithm ieee transactions on signal processing vol 54 no 4 pp 1376 1385 apr 2006 that the lmf algorithm is not mean square stable for reference signals whose pdfs have infinite support however the probability of divergence as a function of the step size value tends to rise abruptly only when it moves past a given threshold our analysis provides a simple and yet precise estimate of the region of quick rise in the probability of divergence hence the present analysis is useful for predicting algorithm instability in most practical applications

on structure of bivariate spline space of lower degree over arbitrary triangulation the structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property a new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces the upper bound of the dimension of spline spaces over triangulation given by l l schumaker is slightly improved via the new vertex coding method the structure of multivariate spline spaces s 2 1 and s 3 1 over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by luo and wang a kind of sufficient conditions on judging non singularity of the s 2 1 and s 3 1 spaces over arbitrary triangulation is given which only depends on the topological property of the triangulation from the sufficient conditions a triangulation strategy is presented at the end of the paper the strategy ensures that the constructed triangulation is non singular or generic for s 2 1 and s 3 1

the covering problem and spl mu dependent adaptive algorithms the paper presents a family of techniques called adaptive covering algorithms which solve a particular covering problem how to best cover a target shape using a set of simply parameterized elements the algorithms inspired by adaptive filtering techniques provide a computationally simple robust and efficient alternative to more traditional methods such as bayesian approaches convex hulls and multi layer perceptrons the paper develops a theoretical understanding of the adaptive covering algorithms by relating their behavior via weak convergence techniques to the evolution of a deterministic ordinary differential equation ode in the process the authors give new convergence results for a class of step size dependent recursive algorithms stability and instability of the ode can be interpreted in terms of the local stability instability of the algorithm in terms of the covering problem candidate coverings tend to improve as more data is gathered whenever the ode is stable several examples are given which demonstrate the ideas and which verify that the analysis accurately predicts the true behavior of the algorithms

the stability of lms new and weak conditions are given under which the lms algorithm is exponentially convergent with probability one in a stochastic setting these results show that lms works under very broad conditions a small gain input signal with finite fourth moments time varying persistence of excitation and weak assumptions on the correlation structure of the input signal previous results at this level of generality linked convergence to peak signal amplitude rather than average amplitude under the same conditions stochastic boundedness of a forced lms system is also established

comment on the unnecessary assumption of statistical independence between reference signal and filter weights in feedforward adaptive systems minkoff see ieee trans signal processing vol 45 p 2993 3004 1997 presented a formulation in which the time evolving weight iteration equation for random signals is derived without the necessity of invoking the usual unsatisfactory assumption that is customarily made namely that the weights w and the reference signal x the weight iteration equation are statistically independent minkoff neglected however to give a physical argument for it that is in this derivation it is not necessary for w to be independent of x but only of xx which does not contain the phase information of x the off diagonal terms of xx contain only phase differences which could be produced by an infinite number of different arbitrary xs

multivariate spline and algebraic geometry abstract the purpose of this survey is to emphasize the special relationship between multivariate spline and algebraic geometry we will not only point out the algebraic geometric method of multivariate spline but also the algebraic geometric background and essence of multivariate spline especially we have made an introduction to the so called piecewise algebraic curve piecewise algebraic variety and some of their properties

deformable kernels for early vision early vision algorithms often have a first stage of linear filtering that extracts from the image information at multiple scales of resolution and multiple orientations a common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs a technique is presented that allows 1 computing the best approximation of a given family using linear combinations of a small number of basis functions and 2 describing all finite dimensional families i e the families of filters for which a finite dimensional representation is possible with no error the technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations the relevant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated experimental results are presented that demonstrate the applicability of the technique to generating multiorientation multi scale 2d edge detection kernels the implementation issues are also discussed

almost sure convergence of a decreasing gain sign algorithm for adaptive filtering the convergence analysis of a sign algorithm with decreasing gain when governing the weights of an adaptive filter is presented the analysis is done in a noiseless case a rigorous proof of almost sure convergence to the optimal filter is attained under a weak ergodicity assumption that includes the case important in practice of correlated observations

dependence of the stability of the least mean fourth algorithm on target weights non stationarity the paper investigates a new stability problem of the least mean fourth lmf algorithm which is the dependence of the algorithm stability on the time variation of the target weights of the adaptive filter the analysis is done in the context of tracking a markov plant with a stationary white gaussian input it is found that the algorithm diverges if the mean square increment of the plant parameter vector exceeds a threshold value that depends on the step size input variance and noise moments the paper also derives a closed form of the steady state mean square deviation without the usual assumption of a strong noise comparison of the tracking capabilities of the lmf and lms algorithms is provided the comparison is done in terms of the minimum mean square deviation attained by each algorithm over the stability range of its step size gaussian uniform and binary distributions of the noise are considered conditions that make one algorithm outperform the other are determined analytical results are supported by simulations

probability of divergence for the least mean fourth algorithm in this paper it is shown that the least mean fourth lmf adaptive algorithm is not mean square stable when the regressor input is not strictly bounded as happens for example if the input has a gaussian distribution for input distributions with infinite support even for the gaussian distribution the lmf always has a nonzero probability of divergence no matter how small the step size is chosen this result is proven for a slight modification of the gaussian distribution in a one tap filter and corroborated with several simulations in addition an upper bound is given for the probability of divergence of lmf as a function of the filter length input power step size and noise variance for the case of gaussian regressors the results reported in this paper provide tools for designers to better understand the behavior of the lmf algorithm and decide on the convenience or not of its use for a given application

asymptotic analysis of stochastic gradient based adaptive filtering algorithms with general cost functions this paper presents an analysis of stochastic gradient based adaptive algorithms with general cost functions the analysis holds under mild assumptions on the inputs and the cost function the method of analysis is based on an asymptotic analysis of fixed stepsize adaptive algorithms and gives almost sure results regarding the behavior of the parameter estimates whereas previous stochastic analyses typically considered mean and mean square behavior the parameter estimates are shown to enter a small neighborhood about the optimum value and remain there for a finite length of time furthermore almost sure exponential bounds are given for the rate of convergence of the parameter estimates the asymptotic distribution of the parameter estimates is shown to be gaussian with mean equal to the optimum value and covariance matrix that depends on the input statistics specific adaptive algorithms that fall under the framework of this paper are signed error least mean square lms dual sign lms quantized state lms least mean fourth dead zone algorithms momentum algorithms and leaky lms

multivariate spline space over cross cut partition in this paper we mainly study the generator bases of the prime module determined by the global conformality condition of multivariate spline over star cross cut partition the results can help us to get the dimension and construct the spline bases for multivariate spline space over cross cut partition quasi cross cut partition and super cross cut partition

weak convergence and local stability properties of fixed step size recursive algorithms a recursive equation that subsumes several common adaptive filtering algorithms is analyzed for general stochastic inputs and disturbances by relating the motion of the parameter estimate errors to the behavior of an unforced deterministic ordinary differential equation ode the odes describing the motion of several common adaptive filters are examined in some simple settings including the least mean square lms algorithm and all three of its signed variants the signed regressor the signed error and the sign sign algorithms stability and instability results are presented in terms of the eigenvalues of a correlation like matrix this generalizes known results for lms signed regressor lms and signed error lms and gives new stability criteria for the sign sign algorithm the ability of the algorithms to track moving parameterizations can be analyzed in a similar manner by relating the time varying system to a forced ode the asymptotic distribution about the forced ode is an ornstein uhlenbeck process the properties of which can be described in a straightforward manner

the dimension and basis of spaces of multivariate splines abstract the purpose of this paper is to study the recent development of certain aspects of multivariate spline spaces more specifically we will survey the existence of bivariate spline the dimension and basis of the bivariate spline spaces s k with various partitions furthermore we will also introduce some results on the higher dimensional splines

isolating the real roots of the piecewise algebraic variety the piecewise algebraic variety as a set of the common zeros of multivariate splines is a kind of generalization of the classical algebraic variety in this paper we present an algorithm for isolating the zeros of the zero dimensional piecewise algebraic variety which is primarily based on the interval zeros of univariate interval polynomials numerical example illustrates that the proposed algorithm is flexible