the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

the fast hartley transform algorithm the fast hartley transform fht is similar to the cooley tukey fast fourier transform fft but performs much faster because it requires only real arithmetic computations compared to the complex arithmetic computations required by the fft through use of the fht discrete cosine transforms dct and discrete fourier transforms dft can be obtained the recursive nature of the fht algorithm derived in this paper enables us to generate the next higher order fht from two identical lower order fht s in practice this recursive relationship offers flexibility in programming different sizes of transforms while the orderly structure of its signal flow graphs indicates an ease of implementation in vlsi

on computing the split radix fft this paper presents an efficient fortran program that computes the duhamel hollmann split radix fft an indexing scheme is used that gives a three loop structure for the split radix fft that is very similar to the conventional cooley tukey fft both a decimation in frequency and a decimation in time program are presented an arithmetic analysis is made to compare the operation count of the cooley tukey fft fo several different radixes to that of the split radix fft the split radix fft seems to require the least total arithmetic of any power of two dft algorithm

traveling salesman path problems in the traveling salesman path problem we are given a set of cities traveling costs between city pairs and fixed source and destination cities the objective is to find a minimum cost path from the source to destination visiting all cities exactly once in this paper we study polyhedral and combinatorial properties of a variant we call the traveling salesman walk problem in which the objective is to find a minimum cost walk from the source to destination visiting all cities at least once we first characterize traveling salesman walk perfect graphs graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities we show these graphs have a description by way of forbidden minors and also characterize them constructively we also address the asymmetric traveling salesman path problem atspp and give a factor o sqrt n approximation algorithm for this problem

svd compression unitary transforms and computational complexity the search for fast unitary transforms and the need for data compression in linear systems are complementary issues compression requires the definition of a threshold dependent on the condition number which is invariant over the unitary group with respect to this threshold it is shown that the svd is the optimal tool considerations in connection with the kronecker product and direct sum of unitary matrices show that the computational complexity of unitary transforms is entropy like in nature thereby indicating that the o n log n complexity unitary transforms are dense over the unitary group

classical sampling theorems in the context of multirate and polyphase digital filter bank structures the recovery of a signal from so called generalized samples is a problem of designing appropriate linear filters called reconstruction or synthesis filters this relationship is reviewed and explored novel theorems for the subsampling of sequences are derived by direct use of the digital filter bank framework these results are related to the theory of perfect reconstruction in maximally decimated digital filter bank systems one of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error the reconstruction filters turn out to be multiplierless and of the fir finite impulse response type these ideas are extended to the case of two dimensional signals by use of a kronecker formalism the subsampling of bandlimited sequences is also considered a sequence x n with a fourier transform vanishes for mod omega mod or l pi m where l and m are integers with l

recursive fast algorithm and the role of the tensor product the use of the tensor product for modeling and designing fft algorithms is addressed the benefit of the tensor product approach lies in the strong connection between certain tensor product constructs and important computer architectures the scope of the tensor product approach is generalized to include a much larger class of fast recursive algorithms this greatly enhances the versatility of the tensor product technique and brings many different algorithms to the level of understanding and flexibility enjoyed by the fft

survey design under the regression superpopulation model abstract the construction of sample designs and estimators under a linear regression superpopulation model is considered the anticipated variance the variance of the predictor computed with respect to the sampling design and the superpopulation model is used as a criterion for evaluating probability designs and model unbiased predictors regression predictors that are model unbiased and design consistent are constructed

a fast recursive algorithm for computing the discrete cosine transform the discrete cosine transform dct is widely applied in various fields including image data compression because it operates like the karhunen loeve transform for stationary random data this paper presents a recursive algorithm for dct with a structure that allows the generation of the next higher order dct from two identical lower order dct s as a result the method for implementing this recursive dct requires fewer multipliers and adders than other dct algorithms

bcr method for solving generalized coupled sylvester equations over centrosymmetric or anti centrosymmetric matrix abstract this paper introduces another version of biconjugate residual method bcr for solving the generalized coupled sylvester matrix equations over centrosymmetric or anti centrosymmetric matrix we prove this version of bcr algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round off errors furthermore a method is provided for choosing the initial matrices to obtain the least norm solution of the problem at last some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed

split manageable efficient algorithm for fourier and hadamard transforms in this paper a general efficient manageable split algorithm to compute one dimensional 1 d unitary transforms by using the special partitioning in the frequency domain is introduced the partitions determine fast transformations that split the n point unitary transform into a set of n sub i point transforms i 1 n n sub 1 n sub n n here we introduce a class of splitting transformations the so called paired transforms based on these transforms the decompositions of the fourier transforms of arbitrary orders are given and the corresponding algorithms are considered comparative estimates revealing the efficiency of the proposed algorithms with respect to the known ones are given in particular a proposed method of calculating the 2 sup r point fourier transform requires 2 sup r 1 r 3 2 multiplications and 2 sup r 1 r 9 r sup 2 3r 6 additions in terms of the paired transforms the splitting of the 2 sup r point hadamard transform is described as a result the proposed algorithm for computing this transform uses on the average no more than six operations of additions per sample

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

an improved integrality gap for asymmetric tsp paths the asymmetric traveling salesperson path problem atspp is one where given an asymmetric metric space v d with specified vertices s and t the goal is to find an s t path of minimum length that passes through all the vertices in v this problem is closely related to the asymmetric tsp atsp which seeks to find a tour instead of an s t path visiting all the nodes for atsp a approximation guarantee implies an o approximation for atspp however no such connection is known for the integrality gaps of the linear programming lp relaxations for these problems the current best approximation algorithm for atspp is o ln n ln ln n whereas the best bound on the integrality gap of the natural lp relaxation the subtour elimination lp for atspp is o ln n in this paper we close this gap and improve the current best bound on the integrality gap from o ln n to o ln n ln ln n the resulting algorithm uses the structure of narrow s t cuts in the lp solution to construct a random tree spanning

generalized sliding fft and its application to implementation of block lms adaptive filters this paper has two contributions first the concept of the generalized sliding fast fourier transform gsfft as an efficient implementation of the hopping fft is introduced application of the gsfft is broad and not limited to what has been considered in this paper the frequency domain block lms fblms adaptive filters are then revised and their implementations for block lengths less than the length of the adaptive filter are studied the gsfft and the available pruned ffts are used to give an efficient implementation of these filters in the particular case of the block length equal to one where the fblms algorithm reduces to the frequency domain lms flms algorithm it is shown that the latter can be implemented with the order of m complexity where m is the length of the adaptive filter

preconditioners based on the alternating direction implicit algorithm for the 2d steady state diffusion equation with orthotropic heterogeneous coefficients in this paper we combine the alternating direction implicit adi algorithm with the concept of preconditioning and apply it to linear systems discretized from the 2d steady state diffusion equations with orthotropic heterogeneous coefficients by the finite element method assuming tensor product basis functions specifically we adopt the compound iteration idea and use adi iterations as the preconditioner for the outside krylov subspace method that is used to solve the preconditioned linear system an efficient algorithm to perform each adi iteration is crucial to the efficiency of the overall iterative scheme we exploit the kronecker product structure in the matrices inherited from the tensor product basis functions to achieve high efficiency in each adi iteration meanwhile in order to reduce the number of krylov subspace iterations we incorporate partially the coefficient information into the preconditioner by exploiting the local support property of the finite element basis functions numerical results demonstrated the efficiency and quality of the proposed preconditioner

a new approach to the minimum cut problem this paper present a new approach to finding minimum cuts in undirected graphs the fundamental principle is simple the edges in a graph s minimum cut form an extremely small fraction of the graph s edges using this idea we give a randomized strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability the algorithm runs in o n 2 log 3 n time a significant improvement over the previous o mn time bounds based on maximum flows it is simple and intuitive and uses no complex data structures our algorithm can be parallelized to run in rnc with n 2 processors this gives the first proof that the minimum cut problem can be solved in rnc the algorithm does more than find a single minimum cut it finds all of them with minor modifications our algorithm solves two other problems of interest our algorithm finds all cuts with value within a multiplicative factor of of the minimum cut s in expected o n 2 time or in rnc with n 2 processors the problem of finding a minimum multiway cut of graph into r pieces is solved in expected o n 2 r 1 time or in rnc with n 2 r 1 processors the trace of the algorithm s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph this data structure can be efficiently transformed into the more standard cactus representing for minimum cuts

transform domain lms algorithm the concept of transform domain adaptive filtering is introduced in certain applications filtering in the transform domain results in great improvements in convergence rate over the conventional time domain adaptive filtering the relationship between several existing frequency domain adaptive filtering algorithms is established applications of the discrete fourier transform dft and the discrete cosine transform dct domain adaptive filtering algorithms in the areas of speech processing and adaptive line enhancers are discussed

m adic polyadic invariance and its applications the concept of m adic invariance allows approximation of a linear time invariant lti system by a linear m adic invariant lmi system or equivalently approximation of a circulant matrix by a super circulant matrix the approximation reduces the number of multiplies required for computing n point cyclic convolution to 2 log m n 1 n where n mn the error introduced by the approximation can be removed if desired by subsequent processing in one concrete case determination of a small number of noncontiguous frequencies this approach approximation and subsequent correction can effect substantial savings in a number of multiplies compared to both fast fourier transform fft algorithm and direct discrete fourier transform ddft these applications are preceded by a tutorial presentation of concepts which are basic to m adic invariant systems

optimal auxiliary covariate based two phase sampling design for semiparametric efficient estimation of a mean or mean difference with application to clinical trials to address the objective in a clinical trial to estimate the mean or mean difference of an expensive endpoint y one approach employs a two phase sampling design wherein inexpensive auxiliary variables w predictive of y are measured in everyone y is measured in a random sample and the semiparametric efficient estimator is applied this approach is made efficient by specifying the phase two selection probabilities as optimal functions of the auxiliary variables and measurement costs while this approach is familiar to survey samplers it apparently has seldom been used in clinical trials and several novel results practicable for clinical trials are developed we perform simulations to identify settings where the optimal approach significantly improves efficiency compared to approaches in current practice we provide proofs and r code the optimality results are developed to design an hiv vaccine trial with objective to compare the mean importance weighted breadth y of the t cell response between randomized vaccine groups the trial collects an auxiliary response w highly predictive of y and measures y in the optimal subset we show that the optimal design estimation approach can confer anywhere between absent and large efficiency gain up to 24 in the examples compared to the approach with the same efficient estimator but simple random sampling where greater variability in the cost standardized conditional variance of y given w yields greater efficiency gains accurate estimation of e y w is important for realizing the efficiency gain which is aided by an ample phase two sample and by using a robust fitting method copyright 2013 john wiley sons ltd

optimal sampling and estimation strategies under the linear model in some cases model based and model assisted inferences can lead to very different estimators these two paradigms are not so different if we search for an optimal strategy rather than just an optimal estimator a strategy being a pair composed of a sampling design and an estimator we show that under a linear model the optimal model assisted strategy consists of a balanced sampling design with inclusion probabilities that are proportional to the standard deviations of the errors of the model and the horvitz thompson estimator if the heteroscedasticity of the model is fully explainable by the auxiliary variables then this strategy is also optimal in a model based sense moreover under balanced sampling and with inclusion probabilities that are proportional to the standard deviation of the model the best linear unbiased estimator and the horvitz thompson estimator are equal finally it is possible to construct a single estimator for both the design and model variance the inference can thus be valid under the sampling design and under the model

architecture oriented regular algorithms for discrete sine and cosine transforms we propose fast algorithms for a class of trigonometric transforms these algorithms have a unified structure and simple data exchange similar to the constant geometry pease 1968 algorithm derived from the cooley tukey 1965 fft we can easily extend many of the parallel fft approaches for these algorithms the idea of the method is to localize the nonregularities peculiar to fast algorithms for other trigonometric transforms into the nodes of the cooley tukey fft type computational graph only the basic operation in the nodes of the computational graph will be different for different transforms thus a simple programmable processor element for executing the node functions can be the basis for parallel constructs

computational methods for modifying seemingly unrelated regressions models computational efficient methods for updating seemingly unrelated regressions models with new observations are proposed a recursive algorithm to solve a series of updating problems is developed the algorithm is based on orthogonal transformations and has as main computational tool the updated generalized qr decomposition ugqrd strategies to compute the orthogonal factorizations by exploiting the block sparse structure of the matrices are designed the problems of adding and deleting exogenous variables from the seemingly unrelated regressions model have also been investigated the solution of these problems utilize the strategies for computing the ugqrd

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

semiparametric regression estimation in the presence of dependent censoring summary we propose a semiparametric estimation procedure for estimating the regression of an outcome y measured at the end of a fixed follow up period on baseline explanatory variables x measured prior to start of follow up in the presence of dependent censoring given x the proposed estimators are consistent when the data are missing at random but not missing completely at random rubin 1976 and do not require full specification of the complete data likelihood specifically we assume that the probability of censoring at time t is independent of the outcome y conditional on the recorded history up to t of a vector of time dependent covariates that are correlated with y our estimators can be used to adjust for dependent censoring and nonrandom noncompliance in randomised trials studying the effect of a treatment on the mean of a response variable of interest even with independent censoring our methods allow the investigator to increase efficiency by exploiting the correlation of the outcome with a vector of time dependent covariates

on hermitian and skew hermitian splitting iteration methods for the linear matrix equation axb c in this paper a hermitian and skew hermitian splitting hss iteration method for solving the linear matrix equation axb c is proposed this method is formed by extending the corresponding hss iteration method for solving ax b the analysis shows that the hss iteration method will converge under certain assumptions moreover the optimal parameter of this iteration method is presented numerical results show that this new method is more efficient and robust than existing methods

finite population sampling with multivariate auxiliary information abstract this article examines strategies that are approximately design unbiased and nearly optimal assuming a large sample survey and a regression superpopulation model a new class of predictors is proposed to link certain features of optimal design unbiased and model unbiased predictors generalized regression predictors are shown to pervade the subclass of asymptotically design unbiased adu predictors generalized regression predictors are combined with model based stratification to construct highly efficient adu strategies

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

randomized distributed edge coloring via an extension of the chernoff hoeffding bounds certain types of routing scheduling and resource allocation problems in a distributed setting can be modeled as edge coloring problems we present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed point to point model of computation our algorithms compute an edge coloring of a graph g with n nodes and maximum degree delta with at most 1 6 delta o log 1 delta n colors with high probability arbitrarily close to 1 for any fixed delta 0 they run in polylogarithmic time the upper bound on the number of colors improves upon the 2 delta 1 coloring achievable by a simple reduction to vertex coloring to analyze the performance of our algorithms we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables the chernoff hoeffding bounds are fundamental tools that are used very frequently in estimating tail probabilities however they assume stochastic independence among certain random variables which may not always hold our results extend the chernoff hoeffding bounds to certain types of random variables which are not stochastically independent we believe that these results are of independent interest and merit further study

parameterized preconditioned hermitian and skew hermitian splitting iteration method for a class of linear matrix equations recently wang et al 2013 proposed an efficient hermitian and skew hermitian splitting hss iteration method to approximate the solution of linear matrix equation to further improve the efficiency of the hss method we in this paper establish a parameterized preconditioned hss pphss iteration method by introducing two hermitian positive definite preconditioners and two more different iteration parameters for the hss method the convergence properties of the pphss method used for solving linear matrix equation a x b c are strictly analyzed and the quasi optimal values of the iteration parameters for the pphss method are also derived numerical results demonstrate the feasibility and the efficiency of the pphss iteration method

stationary splitting iterative methods for the matrix equation axb c abstract stationary splitting iterative methods for solving a x b c are considered in this paper the main tool to derive our new method is the induced splitting of a given nonsingular matrix a m n by a matrix h such that i h 1 exists convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants in particular for certain surface fitting applications our method is much more efficient than the progressive iterative approximation pia a conventional iterative method often used in computer aided geometric design cagd

optimal systolic array algorithms for tensor product in this paper we examine the computational complexity of optimal systolic array algorithms for tensor product we provide a time minimal schedule that meets the computed processor lower and upper bounds including one for tensor product our principle contribution is to find lower and upper bounds in the algorithm and its implementation for generating function and find processor time minimal schedule

on the integrality ratio for the asymmetric traveling salesman problem we improve the lower bound on the integrality ratio of the held karp bound for asymmetric tsp with triangle inequality from 4 3 to 2

the jacobi and gauss seidel type iteration methods for the matrix equation a x b c in this paper the jacobi and gauss seidel type iteration methods are proposed for solving the matrix equation a x b c which are based on the splitting schemes of the matrices a and b the convergence and computational cost of these iteration methods are discussed furthermore we give the preconditioned jacobi and gauss seidel type iteration methods numerical examples are given to demonstrate the efficiency of these methods proposed in this paper

radix r k ffts matricial representation and sdc sdf pipeline implementation this paper proposes to use the discrete fourier transform dft matrix factorization based on the kronecker product to express the family of radix rk single path delay commutator single path delay feedback sdc sdf pipeline fast fourier transform fft architectures the matricial expressions of the radix r r 2 r 3 and r 4 decimation in frequency dif sdc sdf pipeline architectures are derived these expressions can be written using a small set of operators resulting in a compact representation of the algorithms the derived expressions are general in terms of r and the number of points of the fft n expressions are given where it is not necessary that n is a power of rk the proposed set of operators can be mapped to equivalent hardware circuits thus the designer can easily go from the matricial representations to their implementations and vice versa as an example the mapping of the operators is shown for radix 2 22 23 and 24 and the details of the corresponding sdc sdf pipeline fft architectures are presented furthermore a general expression is given for the sdc sdf radix rk pipeline architectures when k 4 this general expression helps the designer to efficiently handle a wider design exploration space and select the optimum single path architecture for a given value of n

approximation algorithms for asymmetric tsp by decomposing directed regular multigraphs a directed multigraph is said to be d regular if the indegree and outdegree of every vertex is exactly d by hall s theorem one can represent such a multigraph as a combination of at most n2 cycle covers each taken with an appropriate multiplicity we prove that if the d regular multigraph does not contain more than d 2 copies of any 2 cycle then we can find a similar decomposition into n2 pairs of cycle covers where each 2 cycle occurs in at most one component of each pair our proof is constructive and gives a polynomial algorithm to find such a decomposition since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs we also give an alternative simpler algorithm to extract a single such pair this combinatorial theorem then comes handy in rounding a fractional solution of an lp relaxation of the maximum traveling salesman problem tsp problem the first stage of the rounding procedure obtains two cycle covers that do not share a 2 cycle with weight at least twice the weight of the optimal solution then we show how to extract a tour from the 2 cycle covers whose weight is at least 2 3 of the weight of the longest tour this improves upon the previous 5 8 approximation with a simpler algorithm utilizing a reduction from maximum tsp to the shortest superstring problem we obtain a 2 5 approximation algorithm for the latter problem which is again much simpler than the previous one for minimum asymmetric tsp the same technique gives two cycle covers not sharing a 2 cycle with weight at most twice the weight of the optimum assuming triangle inequality we then show how to obtain from this pair of cycle covers a tour whose weight is at most 0 842 log 2 n larger than optimal this improves upon a previous approximation algorithm with approximation guarantee of 0 999 log 2 n other applications of the rounding procedure are approximation algorithms for maximum 3 cycle cover factor 2 3 previously 3 5 and maximum asymmetric tsp with triangle inequality factor 10 13 previously 3 4

noniterative map reconstruction using sparse matrix representations we present a method for noniterative maximum a posteriori map tomographic reconstruction which is based on the use of sparse matrix representations our approach is to precompute and store the inverse matrix required for map reconstruction this approach has generally not been used in the past because the inverse matrix is typically large and fully populated i e not sparse in order to overcome this problem we introduce two new ideas the first idea is a novel theory for the lossy source coding of matrix transformations which we refer to as matrix source coding this theory is based on a distortion metric that reflects the distortions produced in the final matrix vector product rather than the distortions in the coded matrix itself the resulting algorithms are shown to require orthonormal transformations of both the measurement data and the matrix rows and columns before quantization and coding the second idea is a method for efficiently storing and computing the required orthonormal transformations which we call a sparse matrix transform smt the smt is a generalization of the classical fft in that it uses butterflies to compute an orthonormal transform but unlike an fft the smt uses the butterflies in an irregular pattern and is numerically designed to best approximate the desired transforms we demonstrate the potential of the noniterative map reconstruction with examples from optical tomography the method requires offline computation to encode the inverse transform however once these offline computations are completed the noniterative map algorithm is shown to reduce both storage and computation by well over two orders of magnitude as compared to a linear iterative reconstruction methods

a new construction framework for polyphase complete complementary codes with various lengths complete complementary codes ccc with equal sequence lengths have found numerous applications in wireless communications owing to their ideal correlation properties in modern multi user multi rate communication systems cccs with various sequence lengths for different complementary codes are desirable in this paper we propose a new construction of polyphase ccc with various lengths by using unimodular paraunitary pu matrices our key idea is to decompose a large square pu matrix into direct sum of matrices with smaller sizes which is equivalent to block diagonalization of matrices in order to construct ccc with various lengths we show that our proposed constructions not only generalize several previously known methods but also lead us to new cccs with both equal and various lengths