dc programming and dca thirty years of developments the year 2015 marks the 30th birthday of dc difference of convex functions programming and dca dc algorithms which constitute the backbone of nonconvex programming and global optimization in this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools the survey is comprised of three parts in the first part we present a brief history of the field while in the second we summarize the state of the art results and recent advances we focus on main theoretical results and dca solvers for important classes of difficult nonconvex optimization problems and then give an overview of real world applications whose solution methods are based on dca the third part is devoted to new trends and important open issues as well as suggestions for future developments

large scale molecular optimization from distance matrices by a d c optimization approach a so called dca method based on a d c difference of convex functions optimization approach algorithm for solving large scale distance geometry problems is developed different formulations of equivalent d c programs in the l 1 approach are stated via the lagrangian duality without gap relative to d c programming and new nonstandard nonsmooth reformulations in the l infty approach resp the l 1 l infty approach are introduced substantial subdifferential calculations permit us to compute sequences of iterations in the dca quite simply the computations actually require matrix vector products and only one cholesky factorization resp with an additional solution of a convex program in the l 1 approach resp the l 1 l infty approach and allow the exploitation of sparsity in the large scale setting two techniques respectively using shortest paths between all pairs of atoms to generate the complete dissimilarity matrix and the spanning trees procedure are investigated in order to compute a good starting point for the dca finally many numerical simulations of the molecular optimization problems with up to 12567 variables are reported which prove the practical usefulness of the nonstandard nonsmooth reformulations the globality of found solutions and the robustness and efficiency of our algorithms

the structured sensitivity of vandermonde like systems we consider a general class of structured matrices that includes possibly confluent vandermonde and vandermonde like matrices here the entries in the matrix depend nonlinearly upon a vector of parameters we define condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right hand side convenient expressions are derived for the infinity norm based condition numbers and order of magnitude estimates are given for condition numbers defined in terms of a general vector norm we then discuss the computation of the corresponding backward errors after linearising the constraints we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity norm solution of an underdetermined linear system exact componentwise condition numbers are also derived for matrix inversion and the least squares problem and the linearised least squares backward error is characterised

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

a new matrix free algorithm for the large scale trust region subproblem we present a new method for the large scale trust region subproblem the method is matrix free in the sense that only matrix vector products are required we recast the trust region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter we then find the solution of the trust region subproblem from the eigenvectors associated with two of the smallest eigenvalues of the parameterized eigenvalue problem corresponding to the optimal parameter the new algorithm uses a different interpolating scheme than existing methods and introduces a unified iteration that naturally includes the so called hard case we show that the new iteration is well defined and convergent at a superlinear rate we present computational results to illustrate convergence properties and robustness of the method

estimating the matrixp norm the holderp norm of anm n matrix has no explicit representation unlessp 1 2 or it is shown here that thep norm can be estimated reliably ino mn operations a generalization of the power method is used with a starting vector determined by a technique with a condition estimation flavour the algorithm nearly always computes ap norm estimate correct to the specified accuracy and the estimate is always within a factorn 1 1 p of a p as a by product a new way is obtained to estimate the 2 norm of a rectangular matrix this method is more general and produces better estimates in practice than a similar technique of cline conn and van loan

solving the trust region subproblem by a generalized eigenvalue problem the state of the art algorithms for solving the trust region subproblem trs are based on an iterative process involving solutions of many linear systems eigenvalue problems subspace optimization or line search steps a relatively underappreciated fact due to gander golub and von matt linear algebra appl 114 1989 pp 815 839 is that trss can be solved by one generalized eigenvalue problem with no outer iterations in this paper we rediscover this fact and discover its great practicality which exhibits good performance both in accuracy and efficiency moreover we generalize the approach in various directions namely by allowing for an ellipsoidal constraint dealing with the so called hard case and obtaining approximate solutions efficiently when high accuracy is unnecessary we demonstrate that the resulting algorithm is a general purpose trs solver effective both for dense and large sparse problems including the so called hard case our algorithm is easy to implement its essence i

on the extremal structure of least upper bound norms and their dual abstract given finite dimensional real or complex banach spaces e and f with norms e r and f r we denote by n the least upper bound norm induced on l e f some results are given on the extremal structures of b the unit ball of n of its polar b and of b which is the polar of the unit ball of the least upper bound norm n the exposed faces the extreme points and a large family of other faces of b and b are presented it turns out that b is a subset of b the set of tangency points of the surfaces of b and b is completely determined and represented as the union of the exposed faces of b which are normal to rank one mappings we determine sharp bounds on the ranks of mappings in these exposed faces

new and efficient dca based algorithms for minimum sum of squares clustering the purpose of this paper is to develop new efficient approaches based on dc difference of convex functions programming and dca dc algorithm to perform clustering via minimum sum of squares euclidean distance we consider the two most widely used models for the so called minimum sum of squares clustering mssc in short that are a bilevel programming problem and a mixed integer program firstly the mixed integer formulation of mssc is carefully studied and is reformulated as a continuous optimization problem via a new result on exact penalty technique in dc programming dca is then investigated to the resulting problem secondly we introduce a gaussian kernel version of the bilevel programming formulation of mssc named gkmssc the gkmssc problem is formulated as a dc program for which a simple and efficient dca scheme is developed a regularization technique is investigated for exploiting the nice effect of dc decomposition and a simple procedure for finding good starting points of dca is developed the proposed dca schemes are original and very inexpensive because they amount to computing at each iteration the projection of points onto a simplex and or onto a ball and or onto a box which are all determined in the explicit form numerical results on real word datasets show the efficiency the scalability of dca and its great superiority with respect to k means and kernel k means standard methods for clustering highlightsthe purpose of this paper is to develop new efficient approaches based on dc programming and dca for clustering we consider the two models for the minimum sum of squares clustering mssc a mixed integer program and a bilevel programming formulation the mixed integer program is reformulated as a dc program via an exact penalty technique for which dca is investigated a gaussian kernel version of the bilevel programming formulation is introduced and a simple and efficient dca scheme is developed the two main features of dca say the effect of dc decomposition and of starting points are carefully studied

double regularization methods for robust feature selection and svm classification via dc programming abstract in this work two novel formulations for embedded feature selection are presented a second order cone programming approach for support vector machines is extended by adding a second regularizer to encourage feature elimination the one and the zero norm penalties are used in combination with the tikhonov regularization under a robust setting designed to correctly classify instances up to a predefined error rate even for the worst data distribution the use of the zero norm leads to a nonconvex formulation which is solved by using difference of convex dc functions extending dc programming to second order cones experiments on high dimensional microarray datasets were performed and the best performance was obtained with our approaches compared with well known feature selection methods for support vector machines

a d c optimization algorithm for solving the trust region subproblem this paper is devoted to difference of convex functions d c optimization d c duality local and global optimality conditions in d c programming the d c algorithm dca and its application to solving the trust region problem the dca is an iterative method that is quite different from well known related algorithms thanks to the particular structure of the trust region problem the dca is very simple requiring only matrix vector products and in practice converges to the global solution the inexpensive implicitly restarted lanczos method of sorensen is used to check the optimality of solutions provided by the dca when a nonglobal solution is found a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the dca at this point it is shown that in the nonconvex case the dca converges to the global solution of the trust region problem using only matrix vector products and requiring at most 2m 2 restarts where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem numerical simulations establish the robustness and efficiency of the dca compared to standard related methods especially for large scale problems

convergence of a subgradient method for computing the bound norm of matrices abstract let and be any norms on r m and r n respectively we study a subgradient method for computing the associated bound norm s a sup ax x 1 a nonconvex optimization problem it is proved that homodual method converges when one of the norms and is polyhedral

two by two random matrix theory with matrix representations of octonions the recently established pseudo real matrix representations of octonions y tian adv appl clifford algebras 10 61 2000 are discussed in connection with 2 2 random matrix theory a random matrix model that can be tuned to produce level repulsion from linear to octic is developed and nearest neighbor spacing distributions for this model are presented

a proof of convergence of the concave convex procedure using zangwill s theory the concave convex procedure cccp is an iterative algorithm that solves d c difference of convex functions programs as a sequence of convex programs in machine learning cccp is extensively used in many learning algorithms including sparse support vector machines svms transductive svms and sparse principal component analysis though cccp is widely used in many applications its convergence behavior has not gotten a lot of specific attention yuille and rangarajan analyzed its convergence in their original paper however we believe the analysis is not complete the convergence of cccp can be derived from the convergence of the d c algorithm dca proposed in the global optimization literature to solve general d c programs whose proof relies on d c duality in this note we follow a different reasoning and show how zangwill s global convergence theory of iterative algorithms provides a natural framework to prove the convergence of cccp this underlines zangwill s theory as a powerful and general framework to deal with the convergence issues of iterative algorithms after also being used to prove the convergence of algorithms like expectation maximization and generalized alternating minimization in this note we provide a rigorous analysis of the convergence of cccp by addressing two questions when does cccp find a local minimum or a stationary point of the d c program under consideration and when does the sequence generated by cccp converge we also present an open problem on the issue of local convergence of cccp

globally solving the trust region subproblem using simple first order methods we consider the trust region subproblem which is given by a minimization of a quadratic not necessarily convex function over the euclidean ball based on the well known second order necessary and

estimating hadamard operator norms with application to triangular truncation abstract it is shown how the power method can be used to estimate hadamard operator norms its application to the problem of finding the norm of the triangular truncation operator is considered

spacing distributions for some gaussian ensembles of hermitian matrices the probability e n s that an interval of length s contains exactly n eigenvalues of a random matrix is expressed in terms of their correlation functions for the gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and antisymmetric parts the authors can thus write e n s as a convergent infinite product multiplied by an infinite sum

eigenvalue spacing statistics of a four matrix model of some four by four random matrices a simple four matrix model consisting of some 4 4 real and imaginary random matrices and three parameters is introduced it is shown that the eigenvalue spacing statistics of an ensemble of such matrices can be used to describe transitions between all the wigner surmises of random matrix theory formulae for the nearest neighbour spacing distributions of the ensemble are given for various parameter values

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

estimating the largest elements of a matrix we derive an algorithm for estimating the largest p a a a 1 values a ij or a ij for an m a a n matrix a along with their locations in the matrix the matrix is accessed using only matrixa a a vector or matrixa a a matrix products for p 1 the algorithm estimates the norm a m max i j a ij or max i j a ij the algorithm is based on a power method for mixed subordinate matrix norms and iterates on n a a t matrices where t a a a p is a parameter for p t 1 we show that the algorithm is essentially equivalent to rook pivoting in gaussian elimination we also obtain a bound for the expected number of matrixa a a vector products for random matrices and give a class of counter examples our numerical experiments show that for p 1 the algorithm usually converges in just two iterations requiring the equivalent of 4t matrixa a a vector products and for t 2 the algorithm already provides excellent estimates that are usually within a factor 2 of the largest element and frequently exact for p 1 we incorporate deflation to improve the performance of the algorithm experiments on real life datasets show that the algorithm is highly effective in practice

algorithmes de calcul du maximum des formes quadratiques sur la boule unite de la norme du maximum certaines methodes directes et indirectes pour le calcul de max x t ax x 1 sont etudiees les methodes directes sont basees sur les proprietes particulieres des normes 1 2 et ces methodes sont tres simples mais ne s appliquent qu a certaines familles de matrices la methode indirecte est la methode autoduale introduite dans 25 26 avec 1 dans ce cas le choix du vecteur initial pour qu il y ait convergence vers une solution optimale est largement discute some direct and indirect methods are studied for computing max x t ax x 1 wherea is symmetric definite positive direct methods are constructed using particular properties of 1 2 norms these methods are very simple but uniquely suitable to certains families of matrices the indirect method is the autodual method introduced in 25 26 29 with 1 in this case the problem of choosing an initial vector so that convergence of the iterative sequence occurs to an optimal solution is largely discussed

minimization of a large scale quadratic function subject to a spherical constraint an important problem in linear algebra and optimization is the trust region subproblem minimize a quadratic function subject to an ellipsoidal or spherical constraint this basic problem has several important large scale applications including seismic inversion and forcing convergence in optimization methods existing methods to solve the trust region subproblem require matrix factorizations which are not feasible in the large scale setting this paper presents an algorithm for solving the large scale trust region subproblem that requires a fixed size limited storage proportional to the order of the quadratic and that relies only on matrix vector products the algorithm recasts the trust region subproblem in terms of a parameterized eigenvalue problem and adjusts the parameter with a superlinearly convergent iteration to find the optimal solution from the eigenvector of the parameterized problem only the smallest eigenvalue and corresponding eigenvector of the parameterized problem needs to be computed the implicitly restarted lanczos method is well suited to this subproblem

stability of linear quadratic minimization over euclidean balls stability properties of the problem of minimizing a nonconvex linear quadratic function over a euclidean ball known as the trust region subproblem are studied in this paper we investigate in detail the case where the linear part of the objective function is perturbed and obtain necessary and sufficient conditions for the upper lower semicontinuity of the karush kuhn tucker kkt point set map and the global solution map explicit formulas for computing the directional derivative and the frechet derivative of the optimal value function stability of the karush kuhn tucker point set under the perturbation of the quadratic form is also studied

maximum margin clustering made practical motivated by the success of large margin methods in supervised learning maximum margin clustering mmc is a recent approach that aims at extending large margin methods to unsupervised learning however its optimization problem is nonconvex and existing mmc methods all rely on reformulating and relaxing the nonconvex optimization problem as semidefinite programs sdp though sdp is convex and standard solvers are available they are computationally very expensive and only small data sets can be handled to make mmc more practical we avoid sdp relaxations and propose in this paper an efficient approach that performs alternating optimization directly on the original nonconvex problem a key step to avoid premature convergence in the resultant iterative procedure is to change the loss function from the hinge loss to the laplacian square loss so that overconfident predictions are penalized experiments on a number of synthetic and real world data sets demonstrate that the proposed approach is more accurate much faster hundreds to tens of thousands of times faster and can handle data sets that are hundreds of times larger than the largest data set reported in the mmc literature

the mclennan zubarev steady state distribution and fluctuation theorems abstract the mclennan zubarev steady state distribution is studied in the connection with fluctuation theorems we derive the mclennan zubarev steady state distribution from the nonequilibrium detailed balance relation then considering the cumulant function or cumulant functional two fluctuation theorems for entropy and for currents are proved using the fluctuation theorem for currents the current is expanded in terms of thermodynamic forces in the lowest order of the thermodynamic force we find that the transport coefficient satisfies the onsager s reciprocal relation in the next order we derived the correction term to the green kubo formula

a continuous dc programming approach to the strategic supply chain design problem from qualified partner set we present a new continuous approach based on the dc difference of convex functions programming and dc algorithms dca to the problem of supply chain design at the strategic level when production of a new market opportunity has to be launched among a set of qualified partners a well known formulation of this problem is the mixed integer linear program in this paper we reformulate this problem as a dc program by using an exact penalty technique the proposed algorithm is a combination of dca and branch and bound scheme it works in a continuous domain but provides mixed integer solutions numerical simulations on many empirical data sets show the efficiency of our approach with respect to the standard branch and bound algorithm

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

scaling limit of vicious walks and two matrix model we consider the diffusion scaling limit of the one dimensional vicious walker model of fisher and derive a system of nonintersecting brownian motions the spatial distribution of n particles is studied and it is described by use of the probability density function of eigenvalues of nxn gaussian random matrices the particle distribution depends on the ratio of the observation time t and the time interval t in which the nonintersecting condition is imposed as t t is going on from 0 to 1 there occurs a transition of distribution which is identified with the transition observed in the two matrix model of pandey and mehta despite of the absence of matrix structure in the original vicious walker model in the diffusion scaling limit accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution

on some properties of quadratic programs with a convex quadratic constraint in this paper we consider the problem of minimizing a possibly nonconvex quadratic function with a quadratic constraint we point out some new properties of the problem in particular in the first part of the paper we show that i given a kkt point that is not a global minimizer it is easy to find a better feasible point ii strict complementarity holds at the local nonglobal minimizer in the second part of this paper we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function using the unconstrained formulation we give in the nonconvex case a new second order necessary condition for global minimizers in the third part of this paper algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported

local minimizers of quadratic functions on euclidean balls and spheres in this paper a characterization of the local nonglobal minimizes of a quadratic function defined on a euclidean ball or a sphere is given it is proven that there exists at most one local nonglobal minimizes and that the lagrange multiplier that corresponds to this minimizes is the largest solution of a nonlinear scalar equation an algorithm is proposed for computing the local nonglobal minimizes

a new nonconvex approach for image restoration with gamma noise abstract in this paper a new total generalized variational tgv model for restoring images with multiplicative noise is proposed which contains a nonconvex fidelity term and a tgv term we use a difference of convex functions algorithm dca to deal with the proposed model for multiplicative noise removal there exist many models and algorithms most of which focus on convex approximation so that numerical algorithms with guaranteed convergence can be designed unlike these algorithms we use the dca algorithm to remove multiplicative noise by numerical experiments it is shown that the proposed approach leads to a better solution compared with the gradient projection algorithm for solving the classic multiplicative noise removal models we prove that the sequence generated by the dca algorithm converges to a stationary point which satisfies the first order optimality condition finally we demonstrate the performance of our whole scheme by numerical examples a comparison with other methods is provided as well numerical results demonstrate that the proposed algorithm significantly outperforms some previous methods for multiplicative gamma noise removal

efficient nonnegative matrix factorization by dc programming and dca in this letter we consider the nonnegative matrix factorization nmf problem and several nmf variants two approaches based on dc difference of convex functions programming and dca dc algorithm are developed the first approach follows the alternating framework that requires solving at each iteration two nonnegativity constrained least squares subproblems for which dca based schemes are investigated the convergence property of the proposed algorithm is carefully studied we show that with suitable dc decompositions our algorithm generates most of the standard methods for the nmf problem the second approach directly applies dca on the whole nmf problem two algorithms one computing all variables and one deploying a variable selection strategy are proposed the proposed methods are then adapted to solve various nmf variants including the nonnegative factorization the smooth regularization nmf the sparse regularization nmf the multilayer nmf the convex convex hull nmf and the symmetric nmf we also show that our algorithms include several existing methods for these nmf variants as special versions the efficiency of the proposed approaches is empirically demonstrated on both real world and synthetic data sets it turns out that our algorithms compete favorably with five state of the art alternating nonnegative least squares algorithms

collusive game solutions via optimization a nash based collusive game among a finite set of players is one in which the players coordinate in order for each to gain higher payoffs than those prescribed by the nash equilibrium solution in this paper we study the optimization problem of such a collusive game in which the players collectively maximize the nash bargaining objective subject to a set of incentive compatibility constraints we present a smooth reformulation of this optimization problem in terms of a nonlinear complementarity problem we establish the convexity of the optimization problem in the case where each player s strategy set is unidimensional in the multivariate case we propose upper and lower bounding procedures for the collusive optimization problem and establish convergence properties of these procedures computational results with these procedures for solving some test problems are reported

coderivatives of a karush kuhn tucker point set map and applications abstract the problem of minimizing a linear quadratic function over the euclidean ball is encountered frequently in the theory of trust region methods in nonlinear programming by some tools from variational analysis we investigate the stability of the karush kuhn tucker point set map of that problem with respect to total perturbations of its data verifiable sufficient conditions for the local lipschitz like property of the map are obtained and the connection of our results with the existing criteria for the lower semicontinuity of this karush kuhn tucker point set map is shown

combination between global and local methods for solving an optimization problem over the efficient set abstract in this paper we propose an algorithm for globally solving optimization problems over efficient sets the algorithm is established based on a branch and bound scheme in which the bounding procedure is performed by using the well known weak duality theorem in lagrange duality a suitable combination of this algorithm with a local search procedure in d c optimization named dca leads to a promising global algorithm whose efficiency is more or less confirmed by computational experiments on a large set of test problems

on some gaussian ensembles of hermitian matrices a gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and anti symmetric parts is studied similarly a gaussian ensemble of hermitian quaternion matrices with an arbitrary ratio of their self dual and anti self dual parts is studied analytical expressions for the correlation and cluster functions as well as their limits when the order of the matrices is large are derived for both ensembles

a constrained eigenvalue problem in this paper we consider the following mathematical and computational problem given the quantities a n m by n m matrix symmetric n 0 n n m by m matrix with full rank t vector of dimension m with nt t 1 determine an x such that operatorname x t ax min subject to the constraints n t x t i x t x 1 ii variants of this problem occur in many applications 1 5 7 8 11 the problem has been studied previously when t 0 the null vector cf 4 6 when t 0 then the problem becomes more complicated

optimization based dc programming and dca for hierarchical clustering one of the most promising approaches for clustering is based on methods of mathematical programming in this paper we propose new optimization methods based on dc difference of convex functions programming for hierarchical clustering a bilevel hierarchical clustering model is considered with different optimization formulations they are all nonconvex nonsmooth optimization problems for which we investigate attractive dc optimization algorithms called dca numerical results on some artificial and real world databases are reported the results demonstrate that the proposed algorithms are more efficient than related existing methods

accelerating the dc algorithm for smooth functions we introduce two new algorithms to minimise smooth difference of convex dc functions that accelerate the convergence of the classical dc algorithm dca we prove that the point computed by dca can be used to define a descent direction for the objective function evaluated at this point our algorithms are based on a combination of dca together with a line search step that uses this descent direction convergence of the algorithms is proved and the rate of convergence is analysed under the ojasiewicz property of the objective function we apply our algorithms to a class of smooth dc programs arising in the study of biochemical reaction networks where the objective function is real analytic and thus satisfies the ojasiewicz property numerical tests on various biochemical models clearly show that our algorithms outperform dca being on average more than four times faster in both computational time and the number of iterations numerical experiments show that the algorithms are globally convergent to a non equilibrium steady state of various biochemical networks with only chemically consistent restrictions on the network topology

sparse covariance matrix estimation by dca based algorithms this letter proposes a novel approach using the formula see text norm regularization for the sparse covariance matrix estimation scme problem the objective function of scme problem is composed of a nonconvex part and the formula see text term which is discontinuous and difficult to tackle appropriate dc difference of convex functions approximations of formula see text norm are used that result in approximation scme problems that are still nonconvex dc programming and dca dc algorithm powerful tools in nonconvex programming framework are investigated two dc formulations are proposed and corresponding dca schemes developed two applications of the scme problem that are considered are classification via sparse quadratic discriminant analysis and portfolio optimization a careful empirical experiment is performed through simulated and real data sets to study the performance of the proposed algorithms numerical results showed their efficiency and their superiority compared with seven state of the art methods

spacing distributions for point processes on a regular fractal the homogeneous poisson point process in rd denoted by pd is a basic model of stochastic geometry and modern statistical physics using ideas from fractal geometry geometrical statistics and random matrix theory we introduce the model of random points on a self similar fractal as a model of intermediate statistics in the sense that the interpoint spacing statistics of the model are intermediate between those of p1 and p2 when the fractal dimension is in between 1 and 2 and intermediate between those of p2 and p3 when the fractal dimension is in between 2 and 3 and so on we also introduce the idea of using a continuous family of such models to interpolate between p1 and p2 and thereby effectuate crossover transitions between p1 statistics and p2 statistics we first derive the kth nearest neighbor spacing distribution for the general model and then study the interpoint spacing statistics of several realizations of the model involving sierpinski fractals in r2 and r3 we also study a realization of a continuous interpolation between p1 and p2 in particular a continuous interpolation between a point process on a line and a point process on a plane filling curve using the continuous family of self similar koch curves in r2 in the latter study we specifically analyze the second nearest neighbor interpoint spacing statistics which undergo a crossover transition between semi poisson and ginibre statistics

wigner surmises and the two dimensional homogeneous poisson point process we derive a set of identities that relate the higher order interpoint spacing statistics of the two dimensional homogeneous poisson point process to the wigner surmises for the higher order spacing distributions of eigenvalues from the three classical random matrix ensembles we also report a remarkable identity that equates the second nearest neighbor spacing statistics of the points of the poisson process and the nearest neighbor spacing statistics of complex eigenvalues from ginibre s ensemble of 2 x 2 complex non hermitian random matrices

gaussian point processes and two by two random matrix theory the statistics of the multidimensional gaussian point process are discussed in connection with the spacing statistics of eigenvalues of 2x2 random matrices we consider the three dimensional gaussian point process when two of the coordinates of a point are randomly chosen from a gaussian distribution having a mean of zero and a variance of sigma 2 1 but the third coordinate is chosen from a gaussian distribution having a variance in the range of 0 or sigma 3 2 or 1 the probability density function associated with a random point being at a distance r from the origin is shown to be closely related to the nearest neighbor spacing distribution of eigenvalues coming from an ensemble of 2x2 matrices defined by the french kota pandey mehta two matrix model of random matrix theory an elementary explanation of this result is given

hamiltonian derivation of a detailed fluctuation theorem we analyze the microscopic evolution of a system undergoing a far from equilibrium thermodynamic process explicitly accounting for the degrees of freedom of participating heat reservoirs we derive a hybrid result similar in form to both the fluctuation theorem and a statement of detailed balance we relate this result to the steady state fluctuation theorem and to a free energy relation valid far from equilibrium

gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones a gaussian ensemble of hermitian matrices depending on a parameter is considered when 0 the ensemble is gaussian orthogonal and when 1 it is gaussian unitary an analytic expression for then level correlation and cluster functions is given for anyn and 0 1 this ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions

entropy and nonlinear nonequilibrium thermodynamic relation for heat conducting steady states among various possible routes to extend entropy and thermodynamics to nonequilibrium steady states ness we take the one which is guided by operational thermodynamics and the clausius relation in our previous study we derived the extended clausius relation for ness where the heat in the original relation is replaced by its renormalized counterpart called the excess heat and the gibbs shannon expression for the entropy by a new symmetrized gibbs shannon like expression here we concentrate on markov processes describing heat conducting systems and develop a new method for deriving thermodynamic relations we first present a new simpler derivation of the extended clausius relation and clarify its close relation with the linear response theory we then derive a new improved extended clausius relation with a nonlinear nonequilibrium contribution which is written as a correlation between work and heat we argue that the nonlinear nonequilibrium contribution is unavoidable and is determined uniquely once we accept the very natural definition of the excess heat moreover it turns out that to operationally determine the difference in the nonequilibrium entropy to the second order in the temperature difference one may only use the previous clausius relation without a nonlinear term or must use the new relation depending on the operation i e the path in the parameter space this peculiar twist may be a clue to a better understanding of thermodynamics and statistical mechanics of ness

nonconvex regularizations for feature selection in ranking with sparse svm feature selection in learning to rank has recently emerged as a crucial issue whereas several preprocessing approaches have been proposed only a few have focused on integrating feature selection into the learning process in this paper we propose a general framework for feature selection in learning to rank using support vector machines with a sparse regularization term we investigate both classical convex regularizations such as ell 1 or weighted ell 1 and nonconvex regularization terms such as log penalty minimax concave penalty or ell p pseudo norm with p two algorithms are proposed the first an accelerated proximal approach for solving the convex problems and the second a reweighted ell 1 scheme to address nonconvex regularizations we conduct intensive experiments on nine datasets from letor 3 0 and letor 4 0 corpora numerical results show that the use of nonconvex regularizations we propose leads to more sparsity in the resulting models while preserving the prediction performance the number of features is decreased by up to a factor of 6 compared to the ell 1 regularization in addition the software is publicly available on the web 1 available at http remi flamary com soft soft ranksvm nc html

combining dc algorithms dcas and decomposition techniques for the training of nonpositive semidefinite kernels today decomposition methods are one of the most popular methods for training support vector machines svms with the use of kernels that do not satisfy mercer s condition new techniques must be designed to handle nonpositive semidefinite kernels resulting to this choice in this work we incorporate difference of convex dc functions optimization techniques into decomposition methods to tackle this difficulty the new approach needs no problem modification and we show that the only use of a truncated dc algorithms dcas in the decomposition scheme produces a sufficient decrease of the objective function at each iteration thanks to this property an asymptotic convergence proof of the new algorithm is produced without any blockwise convexity assumption on the objective function we also investigate a working set selection rule using second order information for sequential minimal optimization smo type decomposition in the spirit of dc optimization numerical results show the robustness and the efficiency of the new methods compared with state of the art software

fluctuation theorem for currents and schnakenberg network theory a fluctuation theorem is proved for the macroscopic currents of a system in a nonequilibrium steady state by using schnakenberg network theory the theorem can be applied in particular in reaction systems where the affinities or thermodynamic forces are defined globally in terms of the cycles of the graph associated with the stochastic process describing the time evolution