some proximity and sensitivity results in quadratic integer programming we show that for any optimal solution bar z for a given separable quadratic integer programming problem there exist an optimal solution bar x for its continuous relaxation such that parallel bar z bar x parallel infty leqslant n delta a wheren is the number of variables and a is the largest absolute subdeterminant of the integer constraint matrixa also for any feasible solutionz which is not optimal for the separable quadratic integer programming problem there exists a feasible solution bar z having greater objective function value and with parallel bar z z parallel infty leqslant n delta a we further prove under some additional assumptions that the distance between a pair of optimal solutions to an integer quadratic programming problem with right hand side vectorsb andb respectively depends linearly on b b 1 finally the validity of all the results for nonseparable mixed integer quadratic programs is established the proximity results obtained in this paper are extensions of some of the results described in cook et al 1986 for linear integer programming

lower and upper bounds for the allocation problem and other nonlinear optimization problems we demonstrate the impossibility of strongly polynomial algorithms for the allocation problem in the comparison model and in the algebraic tree computation model that follow from lower bound results consequently there are no strongly polynomial algorithms for nonlinear concave separable optimization over a totally unimodular constraint matrix this is in contrast to the case when the objective is linear we present scaling based algorithms that use a greedy algorithm as a subroutine the algorithms are polynomial for the allocation problem and its extensions and are also optimal for the sample allocation problem and the generalized upper bounds allocation problem in that the complexity meets the lower bound derived from the comparison model for other extensions of the allocation problem the scaling based algorithms presented here are the fastest known these algorithms are also polynomial time algorithms for solving with e accuracy the allocation problem and its extension in continuous variables

a pyramid vector quantizer the geometric properties of a memoryless laplacian source are presented and used to establish a source coding theorem motivated by this geometric structure a pyramid vector quantizer pvq is developed for arbitrary vector dimension the pvq is based on the cubic lattice points that lie on the surface of an l dimensional pyramid and has simple encoding and decoding algorithms a product code version of the pvq is developed and generalized to apply to a variety of sources analytical expressions are derived for the pvq mean square error mse and simulation results are presented for pvq encoding of several memoryless sources for large rate and dimension pvq encoding of memoryless laplacian gamma and gaussian sources provides rose improvements of 5 64 8 40 and 2 39 db respectively over the corresponding optimum scalar quantizer although suboptimum in a rate distortion sense because the pvq can encode large dimensional vectors it offers significant reduction in rose distortion compared with the optimum lloyd max scalar quantizer and provides an attractive alternative to currently available vector quantizers

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

a structured fixed rate vector quantizer derived from a variable length scalar quantizer i memoryless sources for pt i see ibid vol 39 no 3 p 851 67 1993 the fixed rate scalar vector quantizer svq for quantizing stationary memoryless sources is extended to a specific type of vector source in which each component is a stationary memoryless scalar subsource independent of the other components algorithms for the design and implementation of the original svq are modified to apply to this case the resulting svq referred to as the extended svq esvq is then used to quantize stationary sources with memory with known autocorrelation function numerical results are presented for the quantization of first order gauss markov sources using this scheme it is shown that the esvq based scheme performs very close to entropy coded transform quantization while maintaining a fixed rate output and outperforms the fixed rate scheme that uses scalar lloyd max quantization of the transform coefficients it is also shown that this scheme performs better than implementable vector quantizers especially at high rates

a separable frechet space of almost universal disposition abstract the gurari space is the unique separable banach space g which is of almost universal disposition for finite dimensional banach spaces which means that for every e 0 for all finite dimensional normed spaces e f for every isometric embedding e e g there exists an e isometric embedding f f g such that f e e we show that g n with a special sequence of semi norms is of almost universal disposition for finite dimensional graded frechet spaces the construction relies heavily on the universal operator on the gurari space recently constructed by garbulinska wegrzyn and the third author in addition we consider a non graded sequence of semi norms on g n with which the space g n is of almost universal disposition for finite dimensional frechet spaces with a fixed sequence of semi norms in both cases this yields in particular that g n is universal in the class of all separable frechet spaces

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

a new trellis source code for memoryless sources the trellis based scalar vector quantizer tb svq for memoryless sources combines the scalar vector quantizer svq with trellis coded quantization tcq to achieve both boundary gain and granular gain the resulting quantizer outperforms all other reasonable complexity fixed rate quantizers unfortunately the resulting code is catastrophic a single bit error within a block can propagate indefinitely into other blocks this correspondence presents a new trellis source code that can achieve essentially the same or in some cases better performance as the tb svq for error free channels but limits the propagation of channel errors

trellis coded vector quantization trellis coded quantization is generalized to allow a vector reproduction alphabet three encoding structures are described several encoder design rules are presented and two design algorithms are developed it is shown that for a stationary ergodic vector source if the optimized trellis coded vector quantization reproduction process is jointly stationary and ergodic with the source then the quantization noise is zero mean and of a variance equal to the difference between the source variance and the variance of the reproduction sequence several examples illustrate the encoder design procedure and performance

splitting of exact sequences of frechet spaces in the absence of continuous norms abstract in the present paper we study the splitting of a short exact sequence 0 g e s f 0 of nuclear frechet spaces where f need not admit a continuous norm i e e or f may be spaces of continuous functions on an open subset of r n or on a compact c manifold if e is such a space then we give a necessary and sufficient condition for the splitting of the sequence in terms of a condition on s and of linear topological invariants loc and dnloc this is used to give a structure theory of the space s n a space which is isomorphic to many spaces of smooth functions and to study the splitting of differential complexes the paper offers a different approach to problems which have been studied in domanski vogt j funct anal 153 1998 203

lattice and trellis quantization with lattice and trellis bounded codebooks high rate theory for memoryless sources high rate lattice and trellis quantizers for nonuniform sources are introduced and analyzed the performance of these quantizers is determined by two separable quantities the granular gain and the boundary gain which are determined by the shapes of the granular cells and of the support region respectively the granular gain and boundary gain are the duals of shaping and coding gain in data transmission applications using this duality it is shown for gaussian sources that the ultimate achievable boundary gain with high rate lattice bounded lattice codebooks is the same as the ultimate gain that can be obtained from variable rate entropy coding it is observed that if lattice codebooks can achieve the ultimate granular gain of 0 255 b per dimension then lattice bounded lattice codebooks can approach the rate distortion limit the performance of lattice quantizers is compared to that of optimum vector quantizers

m convex function minimization by continuous relaxation approach proximity theorem and algorithm the concept of m convexity for functions in integer variables introduced by murota adv math 124 1996 pp 272 311 plays a primary role in the theory of discrete convex analysis in this paper we consider the problem of minimizing an m convex function which is a natural generalization of the separable convex resource allocation problem under a submodular constraint and contains some classes of nonseparable convex function minimization on integer lattice points we propose a new approach for m convex function minimization based on continuous relaxation by establishing proximity theorems we develop a new algorithm based on continuous relaxation we apply the approach to some special cases of the separable convex quadratic resource allocation problem and the convex quadratic tree resource allocation problem to obtain faster algorithms

trellis based scalar vector quantizer for memoryless sources the paper describes a structured vector quantization approach for stationary memoryless sources that combines the scalar vector quantizer svq ideas laroia and farvardin 1993 with trellis coded quantization marcellin and fischer 1990 the resulting quantizer is called the trellis based scalar vector quantizer tb svq the svq structure allows the tb svq to realize a large boundary gain while the underlying trellis code enables it to achieve a significant portion of the total granular gain for large block lengths and powerful possibly complex trellis codes the tb svq can in principle achieve the rate distortion bound as indicated by the results obtained even for reasonable block lengths and relatively simple trellis codes the tb svq outperforms all other fixed rate quantizers at reasonable complexity

trellis coded quantization of memoryless and gauss markov sources trellis coded quantization tcq is developed and applied to the encoding of memoryless and gauss markov sources the theoretical justification for the approach is alphabet constrained rate distortion theory which is a dual to the channel capacity argument that motivates trellis coded modulation tcm the authors adopt the notions of signal set expansion set partitioning and branch labeling of tcm but modify the techniques to account for the source distribution to design tcq coders of low complexity with excellent mean squared error mse performance for a memoryless uniform source tcq provides an mse within 0 21 db of the distortion rate bound at all positive integral rates the performance is superior to that promised by the coefficient of quantization for all of the best lattices known in dimensions 24 or less for a memoryless gaussian source the tcq performance at rates of 0 5 1 and 2 b sample is superior to all previous results the authors found in the literature the encoding complexity of tcq is very modest tcq is incorporated into a predictive coding structure for the encoding of gauss markov sources simulation results for first second and third order gauss markov sources are presented

a hierarchical dynamic programming approach to fixed rate entropy coded quantization in quantization of any source with a nonuniform probability density function the entropy coding of the quantizer output can result in a substantial decrease in bit rate a straightforward entropy coding scheme faces us with the problem of variable data rate a solution in a space of dimensionality n is to select an appropriate subset of elements in the n fold cartesian product of a scalar quantizer and represent its elements with codewords of the same length the drawback is that the search addressing of this scheme can no longer be achieved independently along the one dimensional subspaces a reasonable rule is to select the n fold symbols of the highest probability for a memoryless source this is equivalent to selecting the n fold symbols with the lowest additive self information in this case due to the additivity property of the self information the selected subset has a high degree of structure which can be used to substantially decrease the search addressing complexity a dynamic programming approach is used to exploit this structure we build our recursive structure required for the dynamic programming in a hierarchy of levels this results in several benefits over the conventional trellis based approaches using this structure we develop efficient rules based on aggregating the states to substantially reduce the search addressing complexities while keeping the degradation in performance negligible

a splitting theorem for the space of smooth functions we will show using purely linear functional analytic methods that each exact complex0 f c s0 p0 c s1 p1 c s2 p2 wherepiare matrices of convolution operators in particular linear differential operators with constant coefficients splits fromp1on in the category of topological vector spaces moreover we characterize when these complexes split completely the obtained result covers all known particular cases as well as some more general cases independently of the analytic nature ofpi the result is a consequence of studying splitting of short exact sequences0 f jx qg 0 wheref x gare graded frechet spaces i e frechet spaces equipped with fixed reduced spectra representing them andj qare consistent with the graded structures we characterize splitting of in case i forgare spacesc where is an open subset in rn or ii gis the space of all sequences

a low complexity method for fixed rate entropy constrained vector quantization this paper describes a new approach to fixed rate entropy constrained vector quantization fevq for stationary memoryless sources where the structure of codewords are derived from a variable length scalar quantizer we formulate the quantization search operation as a zero one integer optimization problem and show that the resulting integer program can be closely approximated by solving a simple linear program the result is a lagrange formulation which adjoins the constraint on the entropy codeword length to the distortion unlike the previously known methods with a fixed lagrange multiplier we use an iterative algorithm to optimize the underlying objective function while updating the lagrange multiplier until the constraint on the overall rate is satisfied the key feature of the new method is the substantial reduction in the number of iterations in comparison with previous related methods in order to achieve some packing gain we combine the process of trellis coded quantization with that of fevq this results in an iterative application of the viterbi algorithm on the underlying trellis for selecting the lagrange multiplier numerical results are presented which demonstrate substantial improvement in comparison with the alternative methods reported in the literature

entropy constrained trellis coded quantization trellis coded quantization is generalized to allow noiseless coding of the trellis branch reproduction symbols an entropy constrained trellis coded quantization ectcq design algorithm is presented based on the generalized lloyd algorithm for trellis code design and the entropy constrained vector quantization design algorithm ectcq systems are designed for memoryless gaussian and laplacian sources and for first and second order gauss markov sources with an 8 state trellis the mean squared error ectcq encoding performance for memoryless sources with smooth densities is within about 0 5 db of the rate distortion function small additional improvements are achieved with larger trellises similar though not quite as good performance is achieved for the gauss markov sources by using the ectcq in a predictive coding structure

trellis coded vector quantization a vector generalization of trellis coded quantization tcq called trellis coded vector quantization tcvq and experimental results showing its performance for an i i d gaussian source are presented for a given rate tcvq yields a lower distortion that tcq at the cost of an increase in implementation complexity in addition tcvq allows fractional rates which tcq does not