distribution of generalized fermat prime numbers numbers of the form fb n b2n 1 are called generalized fermat numbers gfn a computational method for testing the probable primality of a gfn is described which is as fast as testing a number of the form 2m 1 the theoretical distributions of gfn primes for fixed n are derived and compared to the actual distributions the predictions are surprisingly accurate and can be used to support bateman and horn s quantitative form of hypothesis h of schinzel and sierpinski a list of the current largest known gfn primes is included

l kernel classification nonparametric kernel methods are widely used and proven to be successful in many statistical learning problems well known examples include the kernel density estimate kde for density estimation and the support vector machine svm for classification we propose a kernel classifier that optimizes the l2 or integrated squared error ise of a difference of densities we focus on the gaussian kernel although the method applies to other kernels suitable for density estimation like a support vector machine svm the classifier is sparse and results from solving a quadratic program we provide statistical performance guarantees for the proposed l2 kernel classifier in the form of a finite sample oracle inequality and strong consistency in the sense of both ise and probability of error a special case of our analysis applies to a previously introduced ise based method for kernel density estimation for dimensionality greater than 15 the basic l2 kernel classifier performs poorly in practice thus we extend the method through the introduction of a natural regularization parameter which allows it to remain competitive with the svm in high dimensions simulation results for both synthetic and real world data are presented

f divergence estimation and two sample homogeneity test under semiparametric density ratio models a density ratio is defined by the ratio of two probability densities we study the inference problem of density ratios and apply a semiparametric density ratio estimator to the two sample homogeneity test in the proposed test procedure the f divergence between two probability densities is estimated using a density ratio estimator the f divergence estimator is then exploited for the two sample homogeneity test we derive an optimal estimator of f divergence in the sense of the asymptotic variance in a semiparametric setting and provide a statistic for two sample homogeneity test based on the optimal estimator we prove that the proposed test dominates the existing empirical likelihood score test through numerical studies we illustrate the adequacy of the asymptotic theory for finite sample inference

a new representation of elements of finite fields gf 2 sup m yielding small complexity arithmetic circuits let f sub 2 denote the binary field and f sub 2m an algebraic extension of degree m 1 over f sub 2 traditionally elements of f sub 2m are either represented as powers of a primitive element of f sub 2m together with 0 or by an expansion in a basis of the vector space f sub 2m over f sub 2 we propose a new representation based on an isomorphism from f sub 2m into the residue polynomial ring module x sup n 1 the new representation simultaneously satisfies the properties of various traditional representations which leads in some cases to architectures of parallel in parallel out arithmetic circuits adder multiplier exponentiator inverter squarer divider with average to small complexity we show that the implementation of all the arithmetic circuits designed for the new representation on an integrated circuit sometimes has smaller complexity than the implementation of all the arithmetic circuits designed for other representations in addition we derive a serial multiplier for the field f sub 2m which comprises the least number of gates of all the serial multipliers known to the author when m 1 is a prime such that 2 is primitive in the field z sub m 1

smoothed cross validation for bandwidth selection of a kernel density estimator a generalization of the widely studied least squares cross validation method is considered the essential idea is to do a particular type of presmoothing of the data this is seen to be essentially the same as using the smoothed bootstrap estimate of the mean integrated squared error analysis reveals that a rather large amount of presmoothing yields excellent asymptotic performance the rate of convergence to the optimum is known to be best possible under a wide range of smoothness conditions the method is more appealing than other selectors with this property because its motivation is not heavily dependent on precise asymptotic analysis and because its form is simple and intuitive theory is also given for choice of the amount of presmoothing and this is used to derive a data based method for this choice

on some perturbation inequalities for operators abstract an estimate for the norm of the solution to the equation ax xb s obtained by r bhatia c davis and a mcintosh for normal operators a and b is shown to be valid for a larger class some other inequalities in the same spirit are obtained including a sin theorem for singular vectors some inequalities concerning the continuity of the map a ita obtained recently by kittaneh and kosaki are extended using these ideas

mean shift mode seeking and clustering mean shift a simple interactive procedure that shifts each data point to the average of data points in its neighborhood is generalized and analyzed in the paper this generalization makes some k means like clustering algorithms its special cases it is shown that mean shift is a mode seeking process on the surface constructed with a shadow kernal for gaussian kernels mean shift is a gradient mapping convergence is studied for mean shift iterations cluster analysis if treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data applications in clustering and hough transform are demonstrated mean shift is also considered as an evolutionary strategy that performs multistart global optimization

estimating divergence functionals and the likelihood ratio by convex risk minimization we develop and analyze m estimation methods for divergence functionals and the likelihood ratios of two probability distributions our method is based on a nonasymptotic variational characterization of f divergences which allows the problem of estimating divergences to be tackled via convex empirical risk optimization the resulting estimators are simple to implement requiring only the solution of standard convex programs we present an analysis of consistency and convergence for these estimators given conditions only on the ratios of densities we show that our estimators can achieve optimal minimax rates for the likelihood ratio and the divergence functionals in certain regimes we derive an efficient optimization algorithm for computing our estimates and illustrate their convergence behavior and practical viability by simulations

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

generalised mersenne numbers revisited generalised mersenne numbers gmns were defined by solinas in 1999 and feature in the nist fips 186 2 and secg standards for use in elliptic curve cryptography their form is such that modular reduction is extremely efficient thus making them an attractive choice for modular multiplication implementation however the issue of residue multiplication efficiency seems to have been overlooked asymptotically using a cyclic rather than a linear convolution residue multiplication modulo a mersenne number is twice as fast as integer multiplication this property does not hold for prime gmns unless they are of mersenne s form in this work we exploit an alternative generalisation of mersenne numbers for which an analogue of the above property and hence the same efficiency ratio holds even at bitlengths for which schoolbook multiplication is optimal while also maintaining very efficient reduction moreover our proposed primes are abundant at any bitlength whereas gmns are extremely rare our multiplication and reduction algorithms can also be easily parallelised making our arithmetic particularly suitable for hardware implementation furthermore the field representation we propose also naturally protects against side channel attacks including timing attacks simple power analysis and differential power analysis which is essential in many cryptographic scenarios in constrast to gmns

bandwidth choice for density derivatives an adaptation of least squares cross validation is proposed for bandwidth choice in the kernel estimation of the derivatives of a probability density the practicality of the method is demonstrated by an example and a simulation study theoretical justification is provided by an asymptotic optimality result

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

structure of parallel multipliers for a class of fields gf 2 m abstract this paper presents a configuration of parallel multipliers for gf 2 m based on canonical bases the possible parallel multipliers by the proposed configuration are limited to a class of fields gf 2 m however they can be constructed by o m 2 and gates and o m 2 eor gates with the structural modularity this is a desirable feature for the hardware implementation and their operation time is about log m t where m is the dimension of gf 2 m and t is the delay time of an eor gate in order to construct such parallel multipliers we define two types of polynomials of special form over gf 2 one is called all one polynomial denoted by aop and the other is called equally spaced polynomial denoted by esp furthermore we show a necessary and sufficient condition for esps to be irreducible over gf 2 and the uniqueness of the irreducible esps over gf 2 finally we propose the configuration of parallel multipliers for a class of fields gf 2 m based on irreducible aops and esps over gf 2

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

robust and efficient estimation by minimising a density power divergence a minimum divergence estimation method is developed for robust parameter estimation the proposed approach uses new density based divergences which unlike existing methods of this type such as minimum hellinger distance estimation avoid the use of nonparametric density estimation and associated complications such as bandwidth selection the proposed class of density power divergences is indexed by a single parameter which controls the trade off between robustness and efficiency the methodology affords a robust extension of maximum likelihood estimation for which 0 choices of near zero afford considerable robustness while retaining efficiency close to that of maximum likelihood

a new finite field multiplier using redundant representation this article presents simple and highly regular architectures for finite field multipliers using a redundant representation the basic idea is to embed a finite field into a cyclotomic ring which is based on the elegant multiplicative structure of a cyclic group one important feature of our architectures is that they provide area time trade offs which enable us to implement the multipliers in a partial parallel hybrid fashion this hybrid architecture has great significance in its vlsi implementation in very large fields the squaring operation using the redundant representation is simply a permutation of the coordinates it is shown that when there is an optimal normal basis the proposed bit serial and hybrid multiplier architectures have very low space complexity constant multiplication is also considered and is shown to have an advantage in using the redundant representation

direct density derivative estimation estimating the derivatives of probability density functions is an essential step in statistical data analysis a naive approach to estimate the derivatives is to first perform density estimation and then compute its derivatives however this approach can be unreliable because a good density estimator does not necessarily mean a good density derivative estimator to cope with this problem in this letter we propose a novel method that directly estimates density derivatives without going through density estimation the proposed method provides computationally efficient estimation for the derivatives of any order on multidimensional data with a hyperparameter tuning method and achieves the optimal parametric convergence rate we further discuss an extension of the proposed method by applying regularized multitask learning and a general framework for density derivative estimation based on bregman divergences applications of the proposed method to nonparametric kullback leibler divergence approximation and bandwidth matrix selection in kernel density estimation are also explored

matrix factorizations and their perturbations there are several ways in which a matrix can be factorized as a product of two special matrices these factorizations are often used in numerical analysis and many perturbation bounds useful in such analysis have been proved by various authors in this paper a simple method which leads to these results and several new ones is discussed in the last section related results on the lipschitz continuity of the matrix absolute value are surveyed

a least squares approach to direct importance estimation we address the problem of estimating the ratio of two probability density functions which is often referred to as the importance the importance values can be used for various succeeding tasks such as covariate shift adaptation or outlier detection in this paper we propose a new importance estimation method that has a closed form solution the leave one out cross validation score can also be computed analytically therefore the proposed method is computationally highly efficient and simple to implement we also elucidate theoretical properties of the proposed method such as the convergence rate and approximation error bounds numerical experiments show that the proposed method is comparable to the best existing method in accuracy while it is computationally more efficient than competing approaches

low weight polynomial form integers for efficient modular multiplication in 1999 solinas introduced families of moduli called the generalized mersenne numbers gmns which are expressed in low weight polynomial form p f t where t is limited to a power of 2 gmns are very useful in elliptic curve cryptosystems over prime fields since modular reduction by a gmn requires only integer additions and subtractions however since there are not many gmns and each gmn requires a dedicated implementation gmns are hardly useful for other cryptosystems here we modify gmn by removing restriction on the choice of t and restricting the coefficients of f t to 0 and plusmn1 we call such families of moduli low weight polynomial form integers lwpfis we show an efficient modular multiplication method using lwpfi moduli lwpfis allow general implementation and there exist many lwpfi moduli one may consider lwpfis as a trade off between general integers and gmns

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

bias reduction and metric learning for nearest neighbor estimation of kullback leibler divergence asymptotically unbiased nearest neighbor estimators for kl divergence have recently been proposed and demonstrated in a number of applications with small sample sizes however these nonparametric methods typically suer from high estimation bias due to the non local statistics of empirical nearest neighbor information in this paper we show that this non local bias can be mitigated by changing the distance metric and we propose a method for learning an optimal mahalanobis type metric based on global information provided by approximate parametric models of the underlying densities in both simulations and experiments we demonstrate that this interplay between parametric models and nonparametric estimation methods signicantly improves the accuracy of the nearest neighbor kl divergence estimator

manopt a matlab toolbox for optimization on manifolds optimization on manifolds is a rapidly developing branch of nonlinear optimization its focus is on problems where the smooth geometry of the search space can be leveraged to design efficient numerical algorithms in particular optimization on manifolds is well suited to deal with rank and orthogonality constraints such structured constraints appear pervasively in machine learning applications including low rank matrix completion sensor network localization camera network registration independent component analysis metric learning dimensionality reduction and so on the manopt toolbox available at www manopt org is a user friendly documented piece of software dedicated to simplify experimenting with state of the art riemannian optimization algorithms by dealing internally with most of the differential geometry the package aims particularly at lowering the entrance barrier

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal

inequalities for the schatten p norm iv we present some inequalities for the schattenp norm of operators on a hilbert space it is shown among other things that ifa is an operator such that rea a 0 then for any operatorx ax xa p 2a x p also for any two operatorsa andb a b 2 2 a b 2 2 2 a b 2 2

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho