growing radial basis neural networks merging supervised and unsupervised learning with network growth techniques this paper proposes a framework for constructing and training radial basis function rbf neural networks the proposed growing radial basis function grbf network begins with a small number of prototypes which determine the locations of radial basis functions in the process of training the grbf network gross by splitting one of the prototypes at each growing cycle two splitting criteria are proposed to determine which prototype to split in each growing cycle the proposed hybrid learning scheme provides a framework for incorporating existing algorithms in the training of grbf networks these include unsupervised algorithms for clustering and learning vector quantization as well as learning algorithms for training single layer linear neural networks a supervised learning scheme based on the minimization of the localized class conditional variance is also proposed and tested grbf neural networks are evaluated and tested on a variety of data sets with very satisfactory results

state reduction information and entropy in quantum measurement processes information obtained by a quantum measurement process performed on a physical system and the entropy change of the measured physical system are considered in detail it is shown that the condition for the amount of information obtained by the quantum measurement process to be represented by the shannon mutual entropy is that the intrinsic observable of the measured physical system commutes with the operational observable defined by the quantum measurement process when some measurement outcome is obtained the decrease of the shannon entropy of the measured system is compared with that of the von neumann entropy furthermore a condition is established under which the amount of information that can be established by the quantum measurement process becomes equal to the decrease of the shannon entropy of the measured physical system

the paley wiener theorem in rn with the clifford analysis setting abstract we prove the paley wiener theorem in the clifford algebra setting as an application we derive the corresponding result for conjugate harmonic functions

pattern recognition with ordered labels abstract we consider a multi class pattern recognition problem with linearly ordered labels and a loss function which is a sum of squared deviations of decisions from true classes it is proved that the optimal in the bayesian sense decision rule is the a posteriori mean which is then rounded to the nearest integer then the plug in type estimator of the optimal decision rule is proposed we also investigate its asymptotic properties finally we discuss briefly the structure of a neural net which approximates the optimal decision rule

an axiomatic approach to soft learning vector quantization and clustering this paper presents an axiomatic approach to soft learning vector quantization lvq and clustering based on reformulation the reformulation of the fuzzy c means fcm algorithm provides the basis for reformulating entropy constrained fuzzy clustering ecfc algorithms according to the proposed approach the development of specific algorithms reduces to the selection of a generator function linear generator functions lead to the fcm and fuzzy learning vector quantization algorithms while exponential generator functions lead to ecfc and entropy constrained learning vector quantization algorithms the reformulation of lvq and clustering algorithms also provides the basis for developing uncertainty measures that can identify feature vectors equidistant from all prototypes these measures are employed by a procedure developed to make soft lvq and clustering algorithms capable of identifying outliers in the data set this procedure is evaluated by testing the algorithms generated by linear and exponential generator functions on speech data

a new class of wavelet networks for nonlinear system identification a new class of wavelet networks wns is proposed for nonlinear system identification in the new networks the model structure for a high dimensional system is chosen to be a superimposition of a number of functions with fewer variables by expanding each function using truncated wavelet decompositions the multivariate nonlinear networks can be converted into linear in the parameter regressions which can be solved using least squares type methods an efficient model term selection approach based upon a forward orthogonal least squares ols algorithm and the error reduction ratio err is applied to solve the linear in the parameters problem in the present study the main advantage of the new wn is that it exploits the attractive features of multiscale wavelet decompositions and the capability of traditional neural networks by adopting the analysis of variance anova expansion wns can now handle nonlinear identification problems in high dimensions

construction of the least informative observable conserved by a given quantum instrument for a quantum measurement process described by a quantum instrument i and a system observable corresponding to a positive operator valued measure povm e i is said to conserve the information of e if the joint successive measurement of i followed by e is equivalent to a single measurement of e we show that for any quantum instrument i we can construct a povm conserved by i intuitively the construction gives the infinite joint successive measurement of i we also show that the constructed povm is the least informative observable among povms conserved by i i e the constructed povm can be realized by a classical post processing of any povm conserved by i as typical examples of quantum instruments we explicitly evaluate povms of infinite successive measurements for photon counting and quantum counter instruments

fueter s theorem and its generalizations in dunkl clifford analysis in this paper we give a construction of dunkl monogenic and dunkl harmonic functions starting from holomorphic functions in the plane this construction has the advantage of not needing dunkl s intertwining operator or dunkl spherical harmonics to this end we study vekua type systems and prove a version of fueter s theorem in the case of finite reflection groups important examples such as a dunkl monogenic gaussian distribution or a cauchy kernel will be given at the end

evolving space filling curves to distribute radial basis functions over an input space an evolutionary neural network training algorithm is proposed for radial basis function rbf networks the locations of basis function centers are not directly encoded in a genetic string but are governed by space filling curves whose parameters evolve genetically this encoding causes each group of codetermined basis functions to evolve to fit a region of the input space a network produced from this encoding is evaluated by training its output connections only networks produced by this evolutionary algorithm appear to have better generalization performance on the mackey glass time series than corresponding networks whose centers are determined by k means clustering

fourier multipliers on lipschitz curves we develop the theory of fourier multipliers acting on l p where is a lipschitz curve of the form x ig x with g and g the aim is to better understand convolution singular integrals b defined naturally on such curves by bu z p v z u d for almost all z

convolution singular integrals on lipschitz surfaces we prove the lp boundedness of convolution singular integral op erators on a lipschitz surface g x eo x e r 1 x e rn where g is a lipschitz function which satisfies ilvgli we have embedded r i in the clifford algebra r n with identity eo and are considering convolution with right monogenic functions 0 which satisfy 10 x i 0 so x xo x e r lxol where p a w provided there exists an l function q satisfying 0o r 0 r xern then the related convolution singular integral operator t u x lim o f yei x y n y u y dsy 0 en x u x e o jl is bounded on lp for 1 tronics macquarie university nsw 2109 australia e mail address chun li chun macadam mpce mq edu au e mail address alan mcintosh alan macadam mpce mq edu au stephen semmes department of mathematics rice university houston texas 77251 e mail address semmes rice edu this content downloaded from 157 55 39 143 on wed 15 jun 2016 04 40 33 utc all use subject to http about jstor org terms

on information gain by quantum measurements of continuous observables a generalization of shannon s amount of information into quantum measurements of continuous observables is introduced a necessary and sufficient condition for measuring processes to have a non negative amount of information is obtained this resolves groenewold s conjecture completely including the case of measurements of continuous observables as an application the approximate position measuring process considered by von neumann and later by davies is shown to have a non negative amount of information

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

the h functional calculus based on the s spectrum for quaternionic operators and for n tuples of noncommuting operators abstract in this paper we extend the h functional calculus to quaternionic operators and to n tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus called s functional calculus the s functional calculus has two versions one for quaternionic valued functions and one for clifford algebra valued functions and can be considered the riesz dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties the s functional calculus is based on the notion of s spectrum which in the case of quaternionic normal operators on a hilbert space is also the notion of spectrum that appears in the quaternionic spectral theorem the main purpose of this paper is to construct the h functional calculus based on the notion of s spectrum for both quaternionic operators and for n tuples of noncommuting operators we remark that the h functional calculus for n 1 tuples of operators applies in particular to the dirac operator

some conformally flat spin manifolds dirac operators and automorphic forms abstract in this paper we study clifford and harmonic analysis on some examples of conformal flat manifolds that have a spinor structure or more generally at least a pin structure the examples treated here are manifolds that can be parametrized by u where u is a subdomain of either s n or r n and is a kleinian group acting discontinuously on u the examples studied here include rp n and the hopf manifolds s 1 s n 1 also some hyperbolic manifolds will be treated special kinds of clifford analytic automorphic forms associated to the different choices of are used to construct explicit cauchy kernels cauchy integral formulas green s kernels and formulas together with hardy spaces and plemelj projection operators for l p spaces of hypersurfaces lying in these manifolds

quantum measuring processes of continuous observables the purpose of this paper is to provide a basis of theory of measurements of continuous observables we generalize von neumann s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables and show how every such measuring process determines the state change caused by the measurement we establish a one to one correspondence between completely positive instruments in the sense of davies and lewis and the state changes determined by the measuring processes we also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis a proof of the wigner araki yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework we also discuss the implication of these results for the recent results due to srinivas and due to mercer on measurements of continuous observables

canonical approximate quantum measurements in order to generalize the wigner formula for the joint probability distribution of successive canonical measurements of discrete observables to continuous observables a mathematical model is investigated of a canonical approximate measurement of an arbitrary observable which generalizes von neumann s model of measuring interaction and davies s covariant instrument for approximate position measurement calculations of two types of root mean square errors show that this model clears certain requirements for good approximate measurements a modified wigner formula for successive canonical approximate measurements of arbitrary observables is established it is also shown that the canonical approximate measurement of a discrete observable is reduced to the conventional measurement satisfying the von neumann luders collapse postulate by a minor modification in the process of measurement

a natural convolution of quaternion valued functions and its applications abstract we introduce a natural convolution of two suitable quaternion valued functions on r and list down its properties using this convolution first we get the convolution theorem for fourier transform on quaternion valued functions next we modify the existing definition of wavelet transform on square integrable quaternion valued functions in a natural manner so that parseval s identity is obtained without any additional conditions applying the parseval s identity we derive the inversion formula for the wavelet transform and we also prove the other properties like linearity continuity and injectivity finally we construct two boehmian space of quaternion valued functions and extend the wavelet transform as a continuous linear injection from one boehmian space into the other space

a class of fourier multipliers on starlike lipschitz surfaces abstract in this paper we consider a class of fourier multipliers whose symbols are controlled by a polynomial on starlike lipschitz surfaces and get the l 2 boundedness of these operators on sobolev spaces and their endpoint estimates

reformulated radial basis neural networks trained by gradient descent this paper presents an axiomatic approach for constructing radial basis function rbf neural networks this approach results in a broad variety of admissible rbf models including those employing gaussian rbfs the form of the rbfs is determined by a generator function new rbf models can be developed according to the proposed approach by selecting generator functions other than exponential ones which lead to gaussian rbfs this paper also proposes a supervised learning algorithm based on gradient descent for training reformulated rbf neural networks constructed using the proposed approach a sensitivity analysis of the proposed algorithm relates the properties of rbfs with the convergence of gradient descent learning experiments involving a variety of reformulated rbf networks generated by linear and exponential generator functions indicate that gradient descent learning is simple easily implementable and produces rbf networks that perform considerably better than conventional rbf models trained by existing algorithms

fourier analysis on starlike lipschitz surfaces a theory of a class of singular integrals on starlike lipschitz surfaces in rn is established the class of singular integrals forms an operator algebra identical to the class of bounded holomorphic fourier multipliers as well as to the cauchy dunford bounded holomorphic functional calculus of the spherical dirac operator the study proposes a new method inducing clifford holomorphic functions from holomorphic functions of one complex variable by means of which problems on the sphere are reduced to those on the unit circle

efficient relighting of rbf based illumination adjustable images an illumination adjustable image iai contains a large number of prerecorded images under various light directions relighting a scene under complicated lighting conditions can be achieved from the iai using the radial basis function rbf approach to represent an iai is proven to be more efficient than using the spherical harmonic approach however to represent high frequency lighting effects we need to use many rbfs hence the relighting speed could be very slow this brief investigates a partial reconstruction scheme for relighting an iai based on the locality of rbfs compared with the conventional rbf and spherical harmonics sh approaches the proposed scheme has a much faster relighting speed under the similar distortion performance

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

probabilistically reversible measurements probabilistically reversible quantum measurements are considered from the information theoretical point of view it is found that an upper bound on the probability of a reversing quantum measurement is given by the generalized bhattacharyya overlap of conditional probabilities of the measurement outcomes the relation between the upper bound of the probability and the information obtained by the quantum measurement is also discussed

on the construction and training of reformulated radial basis function neural networks presents a systematic approach for constructing reformulated radial basis function rbf neural networks which was developed to facilitate their training by supervised learning algorithms based on gradient descent this approach reduces the construction of radial basis function models to the selection of admissible generator functions the selection of generator functions relies on the concept of the blind spot which is introduced in the paper the paper also introduces a new family of reformulated radial basis function neural networks which are referred to as cosine radial basis functions cosine radial basis functions are constructed by linear generator functions of a special form and their use as similarity measures in radial basis function models is justified by their geometric interpretation a set of experiments on a variety of datasets indicate that cosine radial basis functions outperform considerably conventional radial basis function neural networks with gaussian radial basis functions cosine radial basis functions are also strong competitors to existing reformulated radial basis function models trained by gradient descent and feedforward neural networks with sigmoid hidden units

the inverse fueter mapping theorem for axially monogenic functions of degree k abstract in this paper we first prove an important formula for the fractional laplacian and then we use it to invert the fueter mapping theorem for axially monogenic functions of degree k in fact we prove that for every axially monogenic function of degree k f x a x 0 x x x b x 0 x p k x x r n 1 there exists a holomorphic intrinsic function f k in c such that f x k f k x k n 1 2 f k x p k x where n can be any positive integer k can be any non negative integer f k is the slice monogenic function induced by f k and p k x is an inner spherical monogenic polynomial of degree k using the maps k k 0 1 2 we obtain a decomposition of a monogenic function for any value of the dimension n

dynamics of learning near singularities in layered networks we explicitly analyze the trajectories of learning near singularities in hierarchical networks such as multilayer perceptrons and radial basis function networks which include permutation symmetry of hidden nodes and show their general properties such symmetry induces singularities in their parameter space where the fisher information matrix degenerates and odd learning behaviors especially the existence of plateaus in gradient descent learning arise due to the geometric structure of singularity we plot dynamic vector fields to demonstrate the universal trajectories of learning near singularities the singularity induces two types of plateaus the on singularity plateau and the near singularity plateau depending on the stability of the singularity and the initial parameters of learning the results presented in this letter are universally applicable to a wide class of hierarchical models detailed stability analysis of the dynamics of learning in radial basis function networks and multilayer perceptrons will be presented in separate work