mismatch in high rate entropy constrained vector quantization bucklew s 1984 high rate vector quantizer mismatch result is extended from fixed rate coding to variable rate coding using a lagrangian formulation it is shown that if an asymptotically high rate optimal sequence of variable rate codes is designed for a k dimensional probability density function pdf g and then applied to another pdf f for which f g is bounded then the resulting mismatch or loss of performance from the optimal possible is given by the relative entropy or kullback leibler 1968 divergence i f spl par g it is also shown that under the same assumptions an asymptotically optimal code sequence for g can be converted to an asymptotically optimal code sequence for a mismatched source f by modifying only the lossless component of the code applications to quantizer design using uniform and gaussian densities are described including a high rate analog to the shannon rate distortion result of sakrison 1975 and lapidoth 1997 showing that the gaussian is the worst case for lossy compression of a source with known covariance by coupling the mismatch result with composite quantizers the worst case properties of uniform and gaussian densities are extended to conditionally uniform and gaussian densities which provides a lloyd clustering algorithm for fitting mixtures to general densities

isospectrality of spherical mhd dynamo operators pseudo hermiticity and a no go theorem the isospectrality problem is studied for the operator of the spherical hydromagnetic 2 dynamo it is shown that this operator is formally pseudo hermitian j symmetric and lives in a krein space based on the j symmetry an operator intertwining ansatz with first order differential intertwining operators is tested for its compatibility with the structure of the 2 dynamo operator matrix an intrinsic structural inconsistency is obtained in the set of associated matrix riccati equations this inconsistency is interpreted as a no go theorem which forbids the construction of isospectral 2 dynamo operator classes with the help of first order differential intertwining operators

data compression with low distortion and finite blocklength this paper considers lossy source coding of n dimensional memoryless sources and shows an explicit approximation to the minimum source coding rate required to sustain the probability of exceeding distortion d no greater than epsilon which is simpler than known dispersion based approximations our approach takes inspiration in the celebrated classical result stating that the shannon lower bound to rate distortion function becomes tight in the limit d to 0 we formulate an abstract version of the shannon lower bound that recovers both the classical shannon lower bound and the rate distortion function itself as special cases likewise we show that a nonasymptotic version of the abstract shannon lower bound recovers all previously known nonasymptotic converses a necessary and sufficient condition for the shannon lower bound to be attained exactly is presented it is demonstrated that whenever that condition is met the rate dispersion function is given simply by the varentropy of the source remarkably all finite alphabet sources with balanced distortion measures satisfy that condition in the range of low distortions most continuous sources violate that condition still we show that lattice quantizers closely approach the nonasymptotic shannon lower bound provided that the source density is smooth enough and the distortion is low this implies that fine multidimensional lattice coverings are nearly optimal in the rate distortion sense even at finite n the achievability proof technique is based on a new bound on the output entropy of lattice quantizers in terms of the differential entropy of the source the lattice cell size and a smoothness parameter of the source density the technique avoids both the usual random coding argument and the simplifying assumption of the presence of a dither signal

multiterminal source coding with high resolution we consider separate encoding and joint decoding of correlated continuous information sources subject to a difference distortion measure we first derive a multiterminal extension of the shannon lower bound for the rate region then we show that this shannon outer bound is asymptotically tight for small distortions these results imply that the loss in the sum of the coding rates due to the separation of the encoders vanishes in the limit of high resolution furthermore lattice quantizers followed by slepian wolf lossless encoding are asymptotically optimal we also investigate the high resolution rate region in the remote coding case where the encoders observe only noisy versions of the sources for the quadratic gaussian case we establish a separation result to the effect that multiterminal coding aimed at reconstructing the noisy sources subject to the rate constraints followed by estimation of the remote sources from these reconstructions is optimal under certain regularity conditions on the structure of the coding scheme

quantization on the grassmann manifold the problem of quantization in an euclidean space with unitary constraints can be formulated as an unconstrained problem on a grassmann manifold such constraints arise in areas such as wireless communication with multiple antennas at the transmitter and at the receiver due to the constraints the distortion rate analysis developed for euclidean spaces cannot be applied directly this paper extends gersho s asymptotic large rate small distortion distortion bounds to the case when the source is distributed on the complex grassmann manifold the special structure of the grassmann manifold and the distortion measures defined on it differentiate this problem from the traditional vector quantization in euclidean spaces

asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization two results are given first using a result of csiszar 1973 the asymptotic i e high resolution low distortion performance for entropy constrained tessellating vector quantization heuristically derived by gersho 1979 is proven for all sources with finite differential entropy this implies using gersho s conjecture and zador s formula that tessellating vector quantizers are asymptotically optimal for this broad class of sources and generalizes a rigorous result of gish and pierce 1968 from the scalar to the vector case second the asymptotic performance is established for zamir and feder s 1992 randomized lattice quantization with the only assumption that the source has finite differential entropy it is proven that the low distortion performance of the zamir feder universal vector quantizer is asympotically the same as that of the deterministic lattice quantizer

on invertibility of nonsquare generalized bezoutians abstract conditions for inverting h bezoutians of nonsquare matrix polynomials are studied necessary and sufficient conditions for invertibility of the anderson jury bezoutian are obtained an application to the inversion of block hankel matrices is given

resolvent convergence and spectral approximations of sequences of self adjoint subspaces abstract this paper studies resolvent convergence and spectral approximations of sequences of self adjoint subspaces relations in complex hilbert spaces concepts of strong resolvent convergence norm resolvent convergence spectral inclusion and spectral exactness are introduced fundamental properties of resolvents of subspaces are studied by applying these properties several equivalent and sufficient conditions for convergence of sequences of self adjoint subspaces in the strong and norm resolvent senses are given it is shown that a sequence of self adjoint subspaces is spectrally inclusive under the strong resolvent convergence and spectrally exact under the norm resolvent convergence a sufficient condition is given for spectral exactness of a sequence of self adjoint subspaces in an open interval lacking essential spectral points in addition criteria are established for spectral inclusion and spectral exactness of a sequence of self adjoint subspaces that are defined on proper closed subspaces

lagrangian vector quantization with combined entropy and codebook size constraints in this paper the lagrangian formulation of variable rate vector quantization is extended to quantization with simultaneous constraints on entropy and codebook size including variable and fixed rate quantization as special cases the formulation leads to a lloyd quantizer design algorithm and generalizations of gersho s approximations characterizing optimal performance for asymptotically large rate a variation of gersho s approach is shown to yield rigorous results partially characterizing the asymptotically optimal performance

lattice based robust distributed source coding for three correlated sources the problem of robust distributed source coding for three correlated sources is studied in this work a lattice based scheme is proposed and the analysis of its performance is provided in the high resolution regime special attention is paid to the degenerate case where the three sources are gaussian and identical in this case our scheme is shown to achieve within an asymptotic gap of 0 069 bits in terms of rate per description from the information theoretic limit of quadratic symmetric gaussian multiple description coding with central and individual decoders when the side distortion and the ratio between the central and side distortions both approach 0

a note on eigenvalues of a class of singular continuous and discrete linear hamiltonian systems abstract in this paper we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear hamiltonian systems are equal where both endpoints are in the limit circle cases the proof is fundamental and is given for both continuous and discrete hamiltonian systems the method used in this paper also works for both endpoints are regular or one endpoint is regular and the other is in the limit circle case

perturbations of spectra of closed subspaces in banach spaces abstract this paper studies the variation of spectra of closed subspaces i e linear relations in banach spaces under some perturbations a concept of commutativity for subspaces in banach spaces is introduced and its properties are studied it is shown that boundedness of the inverse of a closed subspace is preserved under relatively bounded and small gap perturbations and perturbation about spectral condition separately by using these results upper semi continuity of spectra of closed subspaces is obtained and an error estimate of spectra of closed subspaces is given under bounded perturbation in the special case that the space is a hilbert space invariance of self adjointness of subspaces under relatively bounded and small gap perturbations and the variation of spectrum of a self adjoint subspace under bounded and hermitian perturbation are discussed the results obtained in the present paper generalize the corresponding results for closed operators to closed subspaces and some of which weaken certain assumptions of the related existing results

the linearization of boundary eigenvalue problems and reproducing kernel hilbert spaces the boundary eigenvalue problems for the adjoint of a symmetric relation s in a hilbert space with finite not necessarily equal defect numbers which are related to the selfadjoint hilbert space extensions of s are characterized in terms of boundary coefficients and the reproducing kernel hilbert spaces they induce 2001 elsevier science inc all rights reserved

rate distortion performance in coding bandlimited sources by sampling and dithered quantization the rate distortion characteristics of a scheme for encoding continuous time band limited stationary sources with a prescribed band is considered in this coding procedure the input is sampled at nyquist s rate or faster the samples undergo dithered uniform or lattice quantization using subtractive dither and the quantizer output is entropy coded the rate distortion performance and the tradeoff between the sampling rate and the quantization accuracy is investigated utilizing the observation that the coding scheme is equivalent to an additive noise channel it is shown that the mean square error of the scheme is fixed as long as the product of the sampling period and the quantizer second moment is kept constant while for a fixed distortion the coding rate generally increases when the sampling rate exceeds the nyquist rate finally as the lattice quantizer dimension becomes large the equivalent additive noise channel of the scheme tends to be white gaussian and both the rate and the distortion performance become invariant to the sampling rate

a rigorous approach to high resolution entropy constrained vector quantization the nonnegativity of relative entropy implies that the differential entropy of a random vector x with probability density function pdf f is upper bounded by mathsf e left log g text x right for any arbitrary pdf g using this inequality with a cleverly chosen g we derive a lower bound on the asymptotic excess rate of entropy constrained vector quantization for d dimensional sources and r th power distortion where the asymptotic excess rate is defined as the difference between the smallest output entropy of a vector quantizer satisfying the distortion constraint and the rate distortion function in the limit as the distortion tends to zero specialized to the one dimensional case this lower bound coincides with the asymptotic excess rate achieved by a uniform quantizer thereby recovering the result by gish and pierce that uniform quantizers are asymptotically optimal as the allowed distortion tends to zero furthermore in the one dimensional case the derivation of the lower bound reveals a necessary condition for a sequence of quantizers to be asymptotically optimal this condition implies that any sequence of asymptotically optimal almost regular quantizers must converge to a uniform quantizer as the distortion tends to zero while the obtained lower bound itself is not novel to the best of our knowledge we present the first rigorous derivation that follows the direct approach by gish and pierce without resorting to heuristic high resolution approximations commonly found in the quantization literature furthermore our derivation holds for all d dimensional sources having finite differential entropy and whose integer part has finite entropy in contrast to gish and pierce we do not require additional constraints on the continuity or decay of the source pdf

stability of essential spectra of self adjoint subspaces under compact perturbations abstract this paper studies stability of essential spectra of self adjoint subspaces i e self adjoint linear relations under finite rank and compact perturbations in hilbert spaces relationships between compact perturbation of closed subspaces and relatively compact perturbation of their operator parts are first established this gives a characterization of compact perturbation in terms of difference between the operator parts of perturbed and unperturbed subspaces it is shown that a self adjoint subspace is still self adjoint under either relatively bounded perturbation with relative bound less than one or relatively compact perturbation or compact perturbation with a certain additional condition by using these results invariance of essential spectra of self adjoint subspaces is proved under relatively compact and compact perturbations separately as a special case finite rank perturbation is discussed the results obtained in this paper generalize the corresponding results for self adjoint operators to self adjoint subspaces

the validity of the additive noise model for uniform scalar quantizers a uniform scalar quantizer with small step size large support and midpoint reconstruction levels is frequently modeled as adding orthogonal noise to the quantizer input this paper rigorously demonstrates the asymptotic validity of this model when the input probability density function pdf is continuous and satisfies several other mild conditions specifically as step size decreases the correlation between input and quantization error becomes negligible relative to the mean squared error mse the model is even valid when the input density is discontinuous at the origin but discontinuities elsewhere can prevent the correlation from being negligible though this invalidates the additive model an asymptotic formula for the correlation is found in terms of the step size and the heights and positions of the discontinuities for a finite support input density such as uniform it is shown that the support of the uniform quantizer can be matched to that of the density in ways that make the correlation approach a variety of limits the derivations in this paper are based on an analysis of the asymptotic convergence of cell centroids to cell midpoints this convergence is fast enough that the centroids and midpoints induce the same asymptotic mse but not fast enough to induce the same correlations

asymptotic analysis of optimal fixed rate uniform scalar quantization studies the asymptotic characteristics of uniform scalar quantizers that are optimal with respect to mean squared error mse when a symmetric source density with infinite support is sufficiently well behaved the optimal step size spl delta sub n for symmetric uniform scalar quantization decreases as 2 spl sigma n sup 1 v sup 1 1 6n sup 2 where n is the number of quantization levels spl sigma sup 2 is the source variance and v sup 1 spl middot is the inverse of v y y sup 1 spl int sub y sup spl infin p spl sigma sup 1 x x dx equivalently the optimal support length n spl delta sub n increases as 2 spl sigma v sup 1 1 6n sup 2 granular distortion is asymptotically well approximated by spl delta sub n sup 2 12 and the ratio of overload to granular distortion converges to a function of the limit spl tau spl equiv lim sub y spl rarr spl infin y sup 1 e x x y provided as usually happens that spl tau exists when it does its value is related to the number of finite moments of the source density an asymptotic formula for the overall distortion d sub n is obtained and spl tau 1 is both necessary and sufficient for the overall distortion to be asymptotically well approximated by spl delta sub n sup 2 12 applying these results to the class of two sided densities of the form b x sup spl beta e spl alpha x sup spl alpha which includes gaussian laplacian gamma and generalized gaussian it is found that spl tau 1 that spl delta sub n decreases as ln n sup 1 spl alpha n that d sub n is asymptotically well approximated by spl delta sub n sup 2 12 and decreases as ln n sup 2 spl alpha n sup 2 and that more accurate approximations to spl delta sub n are possible the results also apply to densities with one sided infinite support such as rayleigh and weibull and to densities whose tails are asymptotically similar to those previously mentioned

bandwidth consecutive multicolorings of graphs abstract let g be a simple graph in which each vertex v has a positive integer weight b v and each edge v w has a nonnegative integer weight b v w a bandwidth consecutive multicoloring of g assigns each vertex v a specified number b v of consecutive positive integers so that for each edge v w all integers assigned to vertex v differ from all integers assigned to vertex w by more than b v w the maximum integer assigned to a vertex is called the span of the coloring in the paper we first investigate fundamental properties of such a coloring we then obtain a pseudo polynomial time exact algorithm and a fully polynomial time approximation scheme for the problem of finding such a coloring of a given series parallel graph with the minimum span we finally extend the results to the case where a given graph g is a partial k tree that is g has a bounded tree width

high resolution source coding for non difference distortion measures multidimensional companding entropy coded vector quantization is studied using high resolution multidimensional companding over a class of nondifference distortion measures for distortion measures which are locally quadratic a rigorous derivation of the asymptotic distortion and entropy coded rate of multidimensional companders is given along with conditions for the optimal choice of the compressor function this optimum compressor when it exists depends on the distortion measure but not on the source distribution the rate distortion performance of the companding scheme is studied using an asymptotic expression for the rate distortion function which parallels the shannon lower bound for difference distortion measures it is proved that the high resolution performance of the scheme is arbitrarily close to the rate distortion limit for large quantizer dimensions if the compressor function and the lattice quantizer used in the companding scheme are optimal extending an analogous statement for entropy coded lattice quantization and mse distortion the companding approach is applied to obtain a high resolution quantizing scheme for noisy sources

spectral properties of self adjoint subspaces abstract this paper focuses on spectral properties of self adjoint subspaces in the product space of a hilbert space several classifications of the spectrum of a self adjoint subspace are introduced including discrete spectrum essential spectrum continuous spectrum singular continuous spectrum absolutely continuous spectrum and singular spectrum their relationships between the subspace and its operator part are established respectively it is shown that all self adjoint subspace extensions of a closed hermitian subspace have the same essential spectrum as a simple application it is shown that every self adjoint subspace extension of the corresponding minimal subspace to a singular second order symmetric linear difference equation has a bounded spectrum under some conditions this shows that there is an essential difference between spectral properties of singular symmetric difference operators and classical regular or singular symmetric differential operators

generalized bezoutian and matrix equations abstract a natural generalization of the classical bezout matrix of two polynomials is introduced for a family of several matrix polynomials the main aim of the paper is to show that this generalized bezoutian serves as an adequate connecting link between the class of equations in matrix polynomials m y z l r and the class of linear matrix equations ax xb c each equation in one of these classes is coupled with a certain equation in the other class so that for each couple the generalized bezoutian corresponding to a solution y z of the equation in matrix polynomials is a solution of the matrix equation and conversely any solution x of the matrix equation is a generalized bezoutian corresponding to a certain solution of the equation in matrix polynomials in particular either both equations are solvable or both have no solutions explicit formulas connecting the solutions of the two equations are given also various representation formulas for the generalized bezoutian are derived and its relation to the resultant matrix and the greatest common divisor of several matrix polynomials is discussed

bezoutian and schur cohn problem for operator polynomials abstract a generalized bezout operator bezoutian for a pair of operator polynomials is introduced and its kernel is described in terms of common spectral data of the underlying polynomials the location of the spectrum of an operator polynomial with compact spectrum with respect to the unit circle infinite dimensional version of the schur cohn problem is expressed via the inertia of a suitable bezoutian an application to the geometric dichotomy problem for difference equations with operator coefficients is given as well

on universal quantization the quantization of n dimensional vectors in r n with an arbitrary probability measure under a mean square error constraint is discussed it is demonstrated that a uniform one dimensional quantizer followed by a noiseless digital variable rate encoder entropy encoding can yield a rate that is for any n no more than 0 754 bit per sample higher than the rate associated with the optimal n dimensionai quantizer regardless of the probabilistic characterization of the input n vector for the allowable mean square error

on entropy constrained vector quantization using gaussian mixture models a flexible and low complexity entropy constrained vector quantizer ecvq scheme based on gaussian mixture models gmms lattice quantization and arithmetic coding is presented the source is assumed to have a probability density function of a gmm an input vector is first classified to one of the mixture components and the karhunen loeve transform of the selected mixture component is applied to the vector followed by quantization using a lattice structured codebook finally the scalar elements of the quantized vector are entropy coded sequentially using a specially designed arithmetic coder the computational complexity of the proposed scheme is low and independent of the coding rate in both the encoder and the decoder therefore the proposed scheme serves as a lower complexity alternative to the gmm based ecvq proposed by gardner subramaniam and rao the performance of the proposed scheme is analyzed under a high rate assumption and quantified for a given gmm the practical performance of the scheme was evaluated through simulations on both synthetic and speech line spectral frequency lsf vectors for lsf quantization the proposed scheme has a comparable performance to at rates relevant for speech coding 20 28 bits per vector with lower computational complexity

efficient approximation algorithms for bandwidth consecutive multicolorings of graphs let g be a graph in which each vertex v has a positive integer weight b v and each edge v w has a nonnegative integer weight b v w a bandwidth consecutive multicoloring simply called a b coloring of g assigns each vertex v a specified number b v of consecutive positive integers as colors of v so that for each edge v w all integers assigned to vertex v differ from all integers assigned to vertex w by more than b v w the maximum integer assigned to vertices is called the span of the coloring the b coloring problem asks to find a b coloring of a given graph g with the minimum span in the paper we present four efficient approximation algorithms for the problem which have theoretical performance guarantees for the computation time the span of a found b coloring and the approximation ratio we also obtain several upper bounds on the minimum span expressed in terms of the maximum b degrees one of which is an extension of brooks theorem on an ordinary coloring

optimization of lattices for quantization a training algorithm for the design of lattices for vector quantization is presented the algorithm uses a steepest descent method to adjust a generator matrix in the search for a lattice whose voronoi regions have minimal normalized second moment the numerical elements of the found generator matrices are interpreted and translated into exact values experiments show that the algorithm is stable in the sense that several independent runs reach equivalent lattices the obtained lattices reach as low second moments as the best previously reported lattices or even lower specifically we report lattices in nine and ten dimensions with normalized second moments of 0 0716 and 0 0708 respectively and nonlattice tessellations in seven and nine dimensions with 0 0727 and 0 0711 which improves on previously known values the new nine and ten dimensional lattices suggest that conway and sloane s 1993 conjecture on the duality between the optimal lattices for packing and quantization might be false a discussion of the application of lattices in vector quantizer design for various sources uniform and nonuniform is included

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

distributed scalar quantization for computing high resolution analysis and extensions communication of quantized information is frequently followed by a computation we consider situations of distributed functional scalar quantization distributed scalar quantization of possibly correlated sources followed by centralized computation of a function under smoothness conditions on the sources and function companding scalar quantizer designs are developed to minimize mean squared error mse of the computed function as the quantizer resolution is allowed to grow striking improvements over quantizers designed without consideration of the function are possible and are larger in the entropy constrained setting than in the fixed rate setting as extensions to the basic analysis we characterize a large class of functions for which regular quantization suffices consider certain functions for which asymptotic optimality is achieved without arbitrarily fine quantization and allow limited collaboration between source encoders in the entropy constrained setting a single bit per sample communicated between encoders can have an arbitrarily large effect on functional distortion in contrast such communication has very little effect in the fixed rate setting

on bezoutians of nonsquare matrix polynomials and inversion of matrices with nonsquare blocks abstract the notion of bezoutian of nonsquare matrix polynomials is defined it is used to establish formulae for the inverse of square matrices partitioned into nonsquare blocks in particular a generalization of the gohberg semencul formula for the inverse of a generalized toeplitz matrix is proved

the theory of v bezoutians and its applications abstract some propositions of the theory of v bezoutians are presented as an application the following problem is considered to construct an appropriate pontryagin space in which a given matrix boundary value problem containing the eigenvalue parameter in the boundary conditions is self adjoint the solution of this problem is given in three special cases with the help of v bezoutians of matrix polynomials entering the boundary conditions

an evolutionary approach for bandwidth multicoloring problems in this paper we consider some generalizations of the vertex coloring problem where distance constraints are imposed between adjacent vertices bandwidth coloring problem and each vertex has to be colored with more than one color bandwidth multicoloring problem we propose an evolutionary metaheuristic approach for the first problem combining an effective tabu search algorithm with population management procedures the approach can be applied to the second problem as well after a simple transformation computational results on instances from the literature show that the overall algorithm is able to produce high quality solutions in a reasonable amount of time outperforming the most effective algorithms proposed for the bandwidth coloring problem and improving the best known solution of many instances of the bandwidth multicoloring problem