coherency of su 1 1 barut girardello type and entanglement for spherical harmonics barut girardello coherent states corresponding to the l m and l m integer discrete irreducible representations of su 1 1 lie algebra are calculated by the spherical harmonics ylm their explicit compact forms and also to realize the resolution of the identity their corresponding positive definite measures on the complex plane are obtained in terms of the known functions it is also shown that coherent states of both positive and negative representations separately lead us to construct new representation bases for su 1 1 lie algebra then it is shown that the su 1 1 barut girardello coherent states corresponding to two particles containing the spatial parity symmetries of a bipartite quantum system can be entangled in ten different ways

rates of convergence for a class of global stochastic optimization algorithms inspired and motivated by the recent advances in simulated annealing algorithms this paper analyzes the convergence rates of a class of recursive algorithms for global optimization via monte carlo methods by using perturbed liapunov function methods stability results of the algorithms are established then the rates of convergence are ascertained by examining the asymptotic properties of suitably scaled estimation error sequences

cooling schedules for optimal annealing a monte carlo optimization technique called simulated annealing is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima the level of randomization is determined by a control parameter t called temperature which tends to zero according to a deterministic cooling schedule we give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states in the special case that the cooling schedule has parametric form tt c log1 t the condition for convergence is that c be greater than or equal to the depth suitably defined of the deepest local minimum which is not a global minimum state

every completely polynomially bounded operator is similar to a contraction it is proved that a bounded operator on a hilbert space is similar to a contraction if and only if it is completely polynomially bounded this gives a partial answer to problem 6 of halmos bull amer math soc 76 1970 877 933 the set of completely bounded maps between c algebras is studied to obtain some structure representation and extension theorems for this class of maps these allow a characterization of the completely bounded representations on a hilbert space of any subalgebra of a c algebra to be obtained the result in the title follows by applying this characterization to the disk algebra

distributed representation of geometrically correlated images with compressed linear measurements this paper addresses the problem of distributed coding of images whose correlation is driven by the motion of objects or the camera positioning it concentrates on the problem where images are encoded with compressed linear measurements we propose a geometry based correlation model that describes the common information in pairs of images we assume that the constitutive components of natural images can be captured by visual features that undergo local transformations e g translation in different images we first identify prominent visual features by computing a sparse approximation of a reference image with a dictionary of geometric basis functions we then pose a regularized optimization problem in order to estimate the corresponding features in correlated images that are given by quantized linear measurements the correlation model is thus given by the relative geometric transformations between corresponding features we then propose an efficient joint decoding algorithm that reconstructs the compressed images such that they are consistent with both the quantized measurements and the correlation model experimental results show that the proposed algorithm effectively estimates the correlation between images in multiview data sets in addition the proposed algorithm provides effective decoding performance that advantageously compares to independent coding solutions and state of the art distributed coding schemes based on disparity learning

global stochastic optimization with low dispersion point sets this study concerns a generic model free stochastic optimization problem requiring the minimization of a risk function defined on a given bounded domain in a euclidean space smoothness assumptions regarding the risk function are hypothesized and members of the underlying space of probabilities are presumed subject to a large deviation principle however the risk function may well be nonconvex and multimodal a general approach to finding the risk minimizer on the basis of decision observation pairs is proposed it consists of repeatedly observing pairs over a collection of design points principles are derived for choosing the number of these design points on the basis of an observation budget and for allocating the observations between these points in both prescheduled and adaptive settings on the basis of these principles large deviation type bounds of the minimizer in terms of sample size are established

weighted means in stochastic approximation of minima weighted averages of kiefer wolfowitz type procedures which are driven by larger step lengths than usual can achieve the optimal rate of convergence a priori knowledge of a lower bound on the smallest eigenvalue of the hessian matrix is avoided the asymptotic mean squared error of the weighted averaging algorithm is the same as would emerge using a newton type adaptive algorithm several different gradient estimates are considered one of them leads to a vanishing asymptotic bias this gradient estimate applied with the weighted averaging algorithm usually yields a better asymptotic mean squared error than applied with the standard algorithm

coherent evolution in a multiply connected space in the presence of a fractional magnetic flux the back reaction effect electric charges on a circle in the presence of a fractional magnetic flux are considered a winding operator and a flux operator are defined and the corresponding phase space is shown to describe the global properties of the system the heisenberg weyl group of discrete displacements and the sl 2 z q group of discrete bogoliubov transformations in the s w phase space are studied they describe coherent evolution of the system going beyond the external field approximation and taking into account the back reaction of the electric charge on the magnetic flux when q is the power of a prime z q is a galois field and stronger results can be proved

analytic representations based on su 1 1 coherent states and their applications we consider two analytic representations of the su 1 1 lie group the representation in the unit disc based on the su 1 1 perelomov coherent states and the barut girardello representation based on the eigenstates of the su 1 1 lowering generator we show that these representations are related through a laplace transform a weak resolution of the identity in terms of the perelomov su 1 1 coherent states is presented which is valid even when the bargmann index k is smaller than various applications of these results in the context of the two photon realization of su 1 1 in quantum optics are also discussed

asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects global minimization via monte carlo the asymptotic behavior of the systems x n 1 x n a n b x n xi n a n sigma x n psi n and dy bar b y dt sqrt a t sigma y dw is studied where psi n is i i d gaussian xi n is a correlated bounded sequence of random variables and a n approx a 0 log a 1 n without xi n such algorithms are versions of the simulated annealing method for global optimization when the objective function values can only be sampled via monte carlo the discrete algorithm is a combination of stochastic approximation and simulated annealing our forms are appropriate the psi n are the annealing variables and xi n is the sampling noise for large a 0 a full asymptotic analysis is presented via the theory of large deviations mean escape time after arbitrary time n from neighborhoods of stable sets of the algorithm mean transition times after arbitrary time n from a neighborhood of one stable set to another a

resolutions of the identity in terms of line integrals of su 1 1 coherent states resolutions of the identity in terms of contour integrals that involve coherent states and their complementary states are derived the complementary states are auxiliary states that help in the formulation of these resolutions of the identity since the coherent states are normalizable inside the unit disc and the complementary states are normalizable outside the unit disc enlargements of the hilbert space are considered which allow the construction of resolutions of the identity in terms of contour integrals in rings several examples of our formalism are presented

convergence guarantees for generalized adaptive stochastic search methods for continuous global optimization this paper presents some simple technical conditions that guarantee the convergence of a general class of adaptive stochastic global optimization algorithms by imposing some conditions on the probability distributions that generate the iterates these stochastic algorithms can be shown to converge to the global optimum in a probabilistic sense these results also apply to global optimization algorithms that combine local and global stochastic search strategies and also those algorithms that combine deterministic and stochastic search strategies this makes the results applicable to a wide range of global optimization algorithms that are useful in practice moreover this paper provides convergence conditions involving the conditional densities of the random vector iterates that are easy to verify in practice it also provides some convergence conditions in the special case when the iterates are generated by elliptical distributions such as the multivariate normal and cauchy distributions these results are then used to prove the convergence of some practical stochastic global optimization algorithms including an evolutionary programming algorithm in addition this paper introduces the notion of a stochastic algorithm being probabilistically dense in the domain of the function and shows that under simple assumptions this is equivalent to seeing any point in the domain with probability 1 this in turn is equivalent to almost sure convergence to the global minimum finally some simple results on convergence rates are also proved

a simulated annealing algorithm with constant temperature for discrete stochastic optimization we present a modification of the simulated annealing algorithm designed for solving discrete stochastic optimization problems like the original simulated annealing algorithm our method has the hill climbing feature so it can find global optimal solutions to discrete stochastic optimization problems with many local solutions however our method differs from the original simulated annealing algorithm in that it uses a constant rather than decreasing temperature we consider two approaches for estimating the optimal solution the first approach uses the number of visits the algorithm makes to the different states divided by a normalizer to estimate the optimal solution the second approach uses the state that has the best average estimated objective function value as estimate of the optimal solution we show that both variants of our method are guaranteed to converge almost surely to the set of global optimal solutions and discuss how our work applies in the discrete deterministic optimization setting we also show how both variants can be applied for solving discrete optimization problems when the objective function values are estimated using either transient or steady state simulation finally we include some encouraging numerical results documenting the behavior of the two variants of our algorithm when applied for solving two versions of a particular discrete stochastic optimization problem and compare their performance with that of other variants of the simulated annealing algorithm designed for solving discrete stochastic optimization problems

global random optimization by simultaneous perturbation stochastic approximation we examine the theoretical and numerical global convergence properties of a certain ldquogradient freerdquo stochastic approximation algorithm called the ldquosimultaneous perturbation stochastic approximation spsa rdquo that has performed well in complex optimization problems we establish two theorems on the global convergence of spsa the first involving the well known method of injected noise the second theorem establishes conditions under which ldquobasicrdquo spsa without injected noise can achieve convergence in probability to a global optimum a result with important practical benefits

annealing of iterative stochastic schemes annealing of general iterative stochastic schemes is studied by using wentzell s large deviation theory the convergence of this algorithm is shown under some conditions the closed form of the critical constant is specified in terms of its potential function of the mean model

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

diffusion for global optimization in r n we seek a global minimum of u mathbb r n to mathbb r the solution to d dt x t nabla u x t will find local minima using the idea of simulated annealing we consider th

a geometric spectral theory for n tuples of self adjoint operators in finite von neumann algebras abstract suppose b 1 b n are self adjoint elements in a finite von neumann algebra m with trace and define a map from m to complex n 1 space by the formula x x b 1 x b n x next let b denote the image of the positive unit ball of m under the map b is called the spectral scale of b 1 b n it is clearly compact and convex the main theme of this work is that the geometry of the spectral scale b reflects spectral data for the b i s for example in the finite dimensional case the operators commute if and only if the spectral scale is a polytope thus one can see that the operators commute from the shape of spectral scale in the case of a single operator where the scale lies in the plane the slopes of the boundary fill out the spectrum of the operator corners correspond to gaps in the spectrum and flat sports indicate eigenvalues analogous results hold when there is more than one operator in the commutative setting the spectral scale determines the n 1 tuple b 1 b n however an example is given that shows this is not generally true in the noncommutative case finally a matricial version of the spectral scale is shown to be sufficient to completely determine the n 1 tuple b 1 b n

renewal monte carlo renewal theory based reinforcement learning an online reinforcement learning algorithm called renewal monte carlo rmc is presented rmc works for infinite horizon markov decision processes with a designated start state rmc is a monte carlo algorithm that retains the key advantages of monte carlo viz simplicity ease of implementation and low bias while circumventing the main drawbacks of monte carlo viz high variance and delayed updates given a parameterized policy pi theta the algorithm consists of three parts estimating the expected discounted reward mathsf r theta and the expected discounted time mathsf t theta over a regenerative cycle estimating the derivatives nabla theta mathsf r theta and nabla theta mathsf t theta and updating the policy parameters using stochastic approximation to find the roots of mathsf r theta nabla theta mathsf t theta mathsf t theta nabla theta mathsf r theta it is shown that under mild technical conditions rmc converges to a locally optimal policy it is also shown that rmc works for postdecision state models as well an approximate version of rmc is proposed where a regenerative cycle is defined as successive visits to a prespecified renewal set it is shown that if the value function of the system is locally lipschitz on the renewal set then rmc converges to an approximate locally optimal policy three numerical experiments are presented to illustrate rmc and compare it with other state of the art reinforcement learning algorithms

algebraic treatment of some systems with spin like interactions with applications in quantum optics and vibronic spectroscopy a unified scheme based on algebraic techniques is proposed to solve the eigenvalue equation for a class of systems involving spin like interactions from general assumptions hamiltonian models are built from selected elements in the enveloping algebra of the harmonic oscillator su 2 and su 1 1 algebras our method is next illustrated through examples taken in the areas of quantum optics and dynamical jahn teller systems in orbital doublets

recursive stochastic algorithms for global optimization in r d an algorithm of the form x k 1 x k a k nabla u x k xi k b k w k where u cdot is a smooth function on mathbb r d xi k is a sequence of mathbb r d valued random variables w k is a sequence of independent standard d dimensional gaussian random variables a k a k and b k sqrt b sqrt k log log k for k large is considered an algorithm of this type arises by adding slowly decreasing white gaussian noise to a stochastic gradient algorithm it is shown under suitable conditions on u cdot xi k a and b that x k converges in probability to the set of global minima of u cdot no prior information is assumed as to what bounded region contains a global minimum the analysis is based on the asymptotic behavior of the related diffusion process dy t nabla u y t dt c t dw t where w cdot is a standard d dimensional wiener process and c t sqrt c sqrt log t for t large

factorization of analytic representations in the unit disc and number phase statistics of a quantum harmonic oscillator the inner outer part factorization of analytic representations in the unit disc is used for an effective characterization of the number phase statistical properties of a quantum harmonic oscillator it is shown that the factorization is intimately connected with the number phase weyl semigroup and its properties in the barut girardello analytic representation the factorization is implemented as a convolution several examples are given which demonstrate the physical significance of the factorization and its role for quantum statistics in particular we study the effect of phase space interference on the factorization properties of a superposition state

matrix convexity operator analogues of the bipolar and hahn banach theorems abstract several basic results of convexity theory are generalized to the quantized matrix convex sets of wittstock these include the bipolar theorem a gauge version of the hahn banach theorem and the existence theorem for support functionals

facial structure in operator algebra theory we give a complete description of the closed faces in twelve kinds of convex sets that appear in operator algebra theory these consist of positive parts of unit balls for c algebras and their dual spaces and for von neumann algebras and their pre duals of self adjoint parts of unit balls in the same four classes and finally of general unit balls in the four classes all these faces are shown to be semi exposed and naturally paired with a polar face in the dual or pre dual space

the construction of some important classes of generalized coherent states the nonlinear coherent states method considering some important classes of generalized coherent states known in the literature we demonstrated that all of them can be created via conventional methods i e the lowering operator eigenstate and the displacement operator techniques using the nonlinear coherent states approach as a result we obtained a unified method to construct a large class of coherent states which have already been introduced by different prescriptions

generalised coherent states and group representations on hilbert spaces of analytic functions using the example for the group su 1 1 the author obtains the basis functions of a hilbert space of analytic functions from the perelomov 1972 definition of generalised coherent states the lie algebra in this space has the form of a holstein primakoff representation appropriate for su 1 1

adaptive stochastic approximation by the simultaneous perturbation method stochastic approximation sa has long been applied for problems of minimizing loss functions or root finding with noisy input information as with all stochastic search algorithms there are adjustable algorithm coefficients that must be specified and that can have a profound effect on algorithm performance it is known that choosing these coefficients according to an sa analog of the deterministic newton raphson algorithm provides an optimal or near optimal form of the algorithm however directly determining the required hessian matrix or jacobian matrix for root finding to achieve this algorithm form has often been difficult or impossible in practice the paper presents a general adaptive sa algorithm that is based on a simple method for estimating the hessian matrix while concurrently estimating the primary parameters of interest the approach applies in both the gradient free optimization kiefer wolfowitz and root finding stochastic gradient based robbins monro settings and is based on the simultaneous perturbation sp idea introduced previously the algorithm requires only a small number of loss function or gradient measurements per iteration independent of the problem dimension to adaptively estimate the hessian and parameters of primary interest aside from introducing the adaptive sp approach the paper presents practical implementation guidance asymptotic theory and a nontrivial numerical evaluation also included is a discussion and numerical analysis comparing the adaptive sp approach with the iterate averaging approach to accelerated sa

coherent states on the m sheeted covering group of su 1 1 coherent states associated with the m sheeted covering group of su 1 1 are considered and such properties as overcompleteness and resolution of the identity are studied they are used to define analytic representations in which the transformations of the m sheeted covering group of su 1 1 are implemented as extended mobius conformal mappings applications of this formalism in the study of hamiltonians that describe m particle clustering are also considered

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

metropolis type annealing algorithms for global optimization in r d the convergence of a class of metropolis type markov chain annealing algorithms for global optimization of a smooth function u cdot on mathbb r d is established no prior information is assumed as to what bounded region contains a global minimum the analysis contained herein is based on writing the metropolis type algorithm in the form of a recursive stochastic algorithm x k 1 x k a k nabla u x k xi k b k w k where w k is a standard white gaussian sequence xi k are random variables and a k a k b k sqrt b sqrt k log log k for k large convergence results for x k are then applied from our previous work siam journal on control and optimization 29 1991 pp 999 1018 since the analysis of x k is based on the asymptotic behavior of the related langevin type markov diffusion annealing algorithm dy t nabla u y t dt c t dw t where w cdot is a standard wiener process and c t

infinite statistics and the su 1 1 phase operator a few years ago agarwal 1991 phys rev a 44 8398 showed that the susskind glogower phase operators expressible in terms of bose operators provide a realization of the algebra for particles obeying infinite statistics in this paper we show that the su 1 1 phase operators constructed in terms of the elements of the su 1 1 lie algebra also provide a realization of the algebra for infinite statistics there are many realizations of the su 1 1 algebra in terms of single or multimode bose operators three of which are discussed along with their corresponding phase states the susskind glogower phase operator is a special case of the su 1 1 phase operator associated with the holstein primakoff realization of su 1 1

multivariate stochastic approximation using a simultaneous perturbation gradient approximation the problem of finding a root of the multivariate gradient equation that arises in function minimization is considered when only noisy measurements of the function are available a stochastic approximation sa algorithm for the general kiefer wolfowitz type is appropriate for estimating the root the paper presents an sa algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite difference approximation of keifer wolfowitz type procedures theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large dimensional problems

group theoretical quantization and the example of a phase space s1 r the group theoretical quantization scheme is reconsidered by means of elementary systems already the quantization of a particle on a circle shows that the standard procedure has to be supplemented by an additional condition on the admissibility of group actions a systematic strategy for finding admissible group actions for particular subbundles of cotangent spaces is developed two dimensional prototypes of which are t r and s s1 r interpreted as restrictions of t r and t s1 to positive coordinate and momentum respectively in this framework and under an additional natural condition an so 1 2 action on s results as the unique admissible group action furthermore for symplectic manifolds which are specific parts of phase spaces with known quantum theory a simple projection method of quantization is formulated for t r and s equivalent results to those of more established but more involved quantization schemes are obtained the approach may be of interest e g in attempts to quantize g

feedback and weighting mechanisms for improving jacobian estimates in the adaptive simultaneous perturbation algorithm it is known that a stochastic approximation sa analogue of the deterministic newton raphson algorithm provides an asymptotically optimal or near optimal form of stochastic search however directly determining the required jacobian matrix or hessian matrix for optimization has often been difficult or impossible in practice this paper presents a general adaptive sa algorithm that is based on a simple method for estimating the jacobian matrix while concurrently estimating the primary parameters of interest relative to prior methods for adaptively estimating the jacobian matrix the paper introduces two enhancements that generally improve the quality of the estimates for underlying jacobian hessian matrices thereby improving the quality of the estimates for the primary parameters of interest the first enhancement rests on a feedback process that uses previous jacobian estimates to reduce the error in the current estimate the second enhancement is based on an optimal weighting of per iteration jacobian estimates from the use of simultaneous perturbations the algorithm requires only a small number of loss function or gradient measurements per iteration independent of the problem dimension to adaptively estimate the jacobian matrix and parameters of primary interest

a globally convergent stochastic approximation by combining a constrained kiefer wolfowitz search with an automatic learning algorithm it is shown that asymptotically normal convergence of an estimator to a global optimum under reasonably lenient assumptions can be attained it is enough that the objective function be smooth and locally strictly convex at its minima the central conclusion is that if theta n is the estimate produced by the method shown at the nth decision epoch then for some global minimizer theta n 1 3 theta n theta is asymptotically normally distributed this coincides with the conventional kiefer wolfowitz convergence rate to a local optimum whereas this study was motivated by needs of machine learning the basic plan would seem applicable to root finding tasks and to other types of stochastic approximation algorithms

generally deformed oscillator isospectral oscillator system and hermitian phase operator the generally deformed oscillator gdo and its multiphoton realization as well as the coherent and squeezed vacuum states are studied we discuss in particular the gdo depending on a complex parameter q therefore we call it q gdo together with the finite dimensional cyclic representations as a realistic physical system of a gdo the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator it is pointed out that starting from the q gdo with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically the new creation and annihilation operators of the pegg barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit

coherent states on the m sheeted complex plane coherent states on the m sheeted complex plane are introduced and properties like overcompleteness and resolution of the identity are studied operators am am which create and annihilate clusters of m particles are considered and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the ordinary glauber coherent states with respect to a a an extended bargmann representation in the m sheeted complex plane and p and q representations with respect to our coherent states are studied applications of this formalism in the study of hamiltonians that describe m particle clustering are also considered