a random matrix approach to the crossover of energy level statistics from wigner to poisson we analyze a class of parametrized random matrix models introduced by rosenzweig and porter which is expected to describe the energy level statistics of quantum systems whose classical dynamics varies from regular to chaotic as a function of a parameter we compute the generating function for the correlations of energy levels in the limit of infinite matrix size the crossover between poisson and wigner statistics is measured by a renormalized coupling constant the model is exactly solved in the sense that in the limit of infinite matrix size the energy level correlation functions and their generating function are given in terms of a finite set of integrals

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

proximal thresholding algorithm for minimization over orthonormal bases the notion of soft thresholding plays a central role in problems from various areas of applied mathematics in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis using convex analytical tools we extend this notion to that of proximal thresholding and investigate its properties providing in particular several characterizations of such thresholders we then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems and we establish the strong convergence of a proximal thresholding algorithm to solve it numerical applications to signal recovery are demonstrated

contour integral method for transitions to the circular unitary ensemble the representation of correlation functions as a contour integral has been useful in the study of transitions to the gaussian unitary ensemble gue we develop the formalism for transitions to the circular unitary ensemble cue and consider the general lcue to cue transition where lcue denotes a superposition of l independent cue spectra in an arbitrary ratio for large matrices we derive the two level correlation function for all l including l the poisson case the results are useful in the study of weakly broken partitioning symmetries and weakly coupled mesoscopic cavities

higher symplectic reflection algebras and non homogeneous n koszul property abstract we study a class of associative algebras associated to finite groups acting on a vector space these algebras are non homogeneous n koszul algebra generalizations of symplectic reflection algebras p etingof v ginzburg symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism invent math 147 2002 243 348 the koszul property was generalized to homogeneous algebras of degree n 2 in r berger koszulity for nonquadratic algebras j algebra 239 2001 705 734 in the present paper the extension of the koszul property to non homogeneous algebras is realized through a pbw theorem this pbw theorem is the generalization to the n case of a quadratic result obtained by braverman and gaitsgory a braverman d gaitsgory poincare birkhoff witt theorem for quadratic algebras of koszul type j algebra 181 1996 315 328 see also a polishchuk l positselski quadratic algebras univ lecture ser vol 37 amer math soc providence ri 2005 we work in the general setting where the ground ring is an arbitrary von neumann regular ring

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

a primal dual proximal algorithm for sparse template based adaptive filtering application to seismic multiple removal unveiling meaningful geophysical information from seismic data requires to deal with both random and structured noises as their amplitude may be greater than signals of interest primaries additional prior information is especially important in performing efficient signal separation we address here the problem of multiple reflections caused by wave field bouncing between layers since only approximate models of these phenomena are available we propose a flexible framework for time varying adaptive filtering of seismic signals using sparse representations based on inaccurate templates we recast the joint estimation of adaptive filters and primaries in a new convex variational formulation this approach allows us to incorporate plausible knowledge about noise statistics data sparsity and slow filter variation in parsimony promoting wavelet frames the designed primal dual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters the approach demonstrates significantly good performance in low signal to noise ratio conditions both for simulated and real field data

dualization of generalized equations of maximal monotone type this paper develops a simple duality framework for generalized equations defined by set valued mappings from a linear space to another the original problem is related to two auxiliary problems of the similar form corresponding to lagrangian and dual problems in the theory of convex programming as in convex programming the alternative formulations can be used to obtain information about a given problem and then used to solve it numerically in particular dualization can be used in deriving existence criteria for a given problem indirectly by considering one of the alternative formulations also a given problem can often be solved more easily by way of a dual method the strongest results of this paper concern monotone mappings in this context the duality framework yields several new criteria for maximal monotonicity of composite mappings these results are useful theoretically as well as in numerical solution of generalized equations the duality framework can also be used in problem decomposition since dualization can lead to reformulations to which operator splitting methods and other special methods can be applied

on the convergence rate improvement of a primal dual splitting algorithm for solving monotone inclusion problems we present two modified versions of the primal dual splitting algorithm relying on forward backward splitting proposed in v tilde mathrm u u adv comput math 38 3 667 681 2013 for solving monotone inclusion problems under strong monotonicity assumptions for some of the operators involved we obtain for the sequences of iterates that approach the solution orders of convergence of mathcal o frac 1 n o 1 n and mathcal o omega n o n for omega in 0 1 0 1 respectively the investigated primal dual algorithms are fully decomposable in the sense that the operators are processed individually at each iteration we also discuss the modified algorithms in the context of convex optimization problems and present numerical experiments in image processing and pattern recognition in cluster analysis

recursive construction for a class of radial functions ii superspace we extend the recursion formula for matrix bessel functions which we obtained previously to superspace it is sufficient to do this for the unitary orthosymplectic supergroup by direct computations we show that fairly explicit results can be obtained at least up to dimension 8 8 for the supermatrices since we introduce a new technique we discuss various of its aspects in some detail

linearized bregman iterations for compressed sensing finding a solution of a linear equation au f with various minimization properties arises from many applications one such application is compressed sensing where an efficient and robust to noise algorithm to find a minimal l 1 norm solution is needed this means that the algorithm should be tailored for large scale and completely dense matrices a while au and a t u can be computed by fast transforms and the solution we seek is sparse recently a simple and fast algorithm based on linearized bregman iteration was proposed in 28 32 for this purpose this paper is to analyze the convergence of linearized bregman iterations and the minimization properties of their limit based on our analysis here we derive also a new algorithm that is proven to be convergent with a rate furthermore the new algorithm is simple and fast in approximating a minimal l 1 norm solution of au f as shown by numerical simulations hence it can be used as another choice of an efficient tool in compressed sensing

dual methods in entropy maximization application to some problems in crystallography this paper is devoted to some infinite dimensional optimization problems with finitely many constraints these deal with entropy maximization and this paper is particularly concerned with those or

asynchronous block iterative primal dual decomposition methods for monotone inclusions we propose new primal dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators the principal innovation in these algorithms is that they are block iterative in the sense that at each iteration only a subset of the monotone operators needs to be processed as opposed to all operators as in established methods flexible strategies are used to select the blocks of operators activated at each iteration in addition we allow lags in operator processing permitting asynchronous implementation the decomposition phase of each iteration of our methods is to generate points in the graphs of the selected monotone operators in order to construct a half space containing the kuhn tucker set associated with the system the coordination phase of each iteration involves a projection onto this half space we present two related methods the first method provides weakly convergent primal and dual sequences under general conditions while the second is a variant in which strong convergence is guaranteed without additional assumptions neither algorithm requires prior knowledge of bounds on the linear operators involved or the inversion of linear operators our algorithmic framework unifies and significantly extends the approaches taken in earlier work on primal dual projective splitting methods

perspective functions proximal calculus and applications in high dimensional statistics abstract perspective functions arise explicitly or implicitly in various forms in applied mathematics and in statistical data analysis to date no systematic strategy is available to solve the associated typically nonsmooth optimization problems in this paper we fill this gap by showing that proximal methods provide an efficient framework to model and solve problems involving perspective functions we study the construction of the proximity operator of a perspective function under general assumptions and present important instances in which the proximity operator can be computed explicitly or via straightforward numerical operations these results constitute central building blocks in the design of proximal optimization algorithms we showcase the versatility of the framework by designing novel proximal algorithms for state of the art regression and variable selection schemes in high dimensional statistics

convergence rate analysis of primal dual splitting schemes primal dual splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces they decompose problems that are built from sums linear compositions and infimal convolutions of simple functions so that each simple term is processed individually via proximal mappings gradient mappings and multiplications by the linear maps this leads to easily implementable and highly parallelizable or distributed algorithms which often obtain nearly state of the art performance in this paper we analyze a monotone inclusion problem that captures a large class of primal dual splittings as a special case we introduce a unifying scheme and use some abstract analysis of the algorithm to prove convergence rates of the proximal point algorithm forward backward splitting peaceman rachford splitting and forward backward forward splitting applied to the model problem our ergodic convergence rates are deduced under var

a weak to strong convergence principle for feje monotone methods in hilbert spaces we consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly exploiting an idea originally proposed by haugazeau we present a simple modification of these methods that makes them strongly convergent without additional assumptions several applications are discussed

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

the use of noise properties in set theoretic estimation in most digital signal processing problems the goal is to estimate an object from noise corrupted observations of a physical system the authors describe how a wide range of probabilistic information pertaining to the noise process can be used in a general set theoretic estimation framework the basic principle is to constrain the sample statistics of the estimation residual to be consistent with those probabilistic properties of the noise which are available and to construct sets accordingly in the solution space adding these sets to the collection of sets describing the solution will yield a smaller feasibility set and hence more reliable estimates pieces of information relative to quantities such as range moments absolute moments and second and higher order probabilistic attributes are considered and properties of the corresponding sets are established simulations are provided to illustrate the theoretical developments

projective splitting with forward steps this work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators for the projective splitting framework recently proposed by combettes and eckstein we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps the resulting algorithms have the same kind of coordination procedure and can be implemented in the same block iterative and highly flexible manner but may perform backward steps on some operators and forward steps on others prior algorithms in the projective splitting family have used only backward steps forward steps can be used for any lipschitz continuous operators provided the stepsize is bounded by the inverse of the lipschitz constant if the lipschitz constant is unknown a simple backtracking linesearch procedure may be used for affine operators the stepsize can be chosen adaptively without knowledge of the lipschitz constant and without any additional forward steps we close the paper by empirically studying the performance of several kinds of splitting algorithms on a large scale rare feature selection problem

a unified approach to duality in convex programming abstract bifunctional duality lagrange duality and fenchel duality in convex programming are presented under a common point of view stability criteria weaker and more natural than the usual slater type constraint qualification hypotheses are obtained for any of these types of duality

a pseudo heuristic parameter selection rule for l1 regularized minimization problems abstract this paper considers the regularization parameter determination of l 1 regularized minimization problem we solve the l 1 regularized problem using iterative reweighted least squares irls which involves solving a linear system whose coefficient matrix has the form m 1 n 0 1 the aim of this paper is to find an efficient and computationally inexpensive algorithm to both choose the regularization parameter and solve the l 1 regularized problem in order to achieve this we propose an irls algorithm with adaptive regularization parameter selection based on a heuristic parameter determination rule de boor s parameter selection criterion compared with some of the state of the art algorithms and parameter selection rules the numerical experiments show the efficiency and robustness of the proposed method

nonconvex nonsmooth optimization via convex nonconvex majorization minimization the class of majorization minimization algorithms is based on the principle of successively minimizing upper bounds of the objective function each upper bound or surrogate function is locally tight at the current estimate and each minimization step decreases the value of the objective function we present a majorization minimization approach based on a novel convex nonconvex upper bounding strategy for the solution of a certain class of nonconvex nonsmooth optimization problems we propose an efficient algorithm for minimizing the convex surrogate function based on the alternating direction method of multipliers a preliminary convergence analysis for the proposed approach is provided numerical experiments show the effectiveness of the proposed method for the solution of nonconvex nonsmooth minimization problems

nonlinear wavelet image processing variational problems compression and noise removal through wavelet shrinkage this paper examines the relationship between wavelet based image processing algorithms and variational problems algorithms are derived as exact or approximate minimizers of variational problems in particular we show that wavelet shrinkage can be considered the exact minimizer of the following problem given an image f defined on a square i minimize over all g in the besov space b11 l1 i the functional f g l2 i 2 g b11 l1 i we use the theory of nonlinear wavelet image compression in l2 i to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i i d mean zero gaussian noise a new signal to noise ratio snr which we claim more accurately reflects the visual perception of noise in images arises in this derivation we present extensive computations that support the hypothesis that near optimal shrinkage parameters can be derived if one knows or can estimate only two parameters about an image f the largest for which f bq lq i 1 q 2 1 2 and the norm f bq lq i both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than donoho and johnstone s visushrink procedure an example suggests however that donoho and johnstone s 1994 1995 1996 sureshrink method which uses a different shrinkage parameter for each dyadic level achieves a lower error than our procedure

systems of structured monotone inclusions duality algorithms and applications a general primal dual splitting algorithm for solving systems of structured coupled monotone inclusions in hilbert spaces is introduced and its asymptotic behavior is analyzed each inclusion in the primal system features compositions with linear operators parallel sums and lipschitzian operators all the operators involved in this structured model are used separately in the proposed algorithm most steps of which can be executed in parallel this provides a flexible solution method applicable to a variety of problems beyond the reach of the state of the art several applications are discussed to illustrate this point

proximity for sums of composite functions we propose an algorithm for computing the proximity operator of a sum of composite convex functions in hilbert spaces and investigate its asymptotic behavior applications to best approximation and image recovery are described

on the equivalence of soft wavelet shrinkage total variation diffusion total variation regularization and sides soft wavelet shrinkage total variation tv diffusion tv regularization and a dynamical system called sides are four useful techniques for discontinuity preserving denoising of signals and images in this paper we investigate under which circumstances these methods are equivalent in the one dimensional case first we prove that haar wavelet shrinkage on a single scale is equivalent to a single step of space discrete tv diffusion or regularization of two pixel pairs in the translationally invariant case we show that applying cycle spinning to haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of tv diffusion we prove that space discrete tv diffusion and tv regularization are identical and that they are also equivalent to the sides system when a specific force function is chosen afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the laplacian pyramid of the signal we analyze possibilities to avoid gibbs like artifacts for multiscale haar wavelet shrinkage by scaling the thresholds finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and pde variational approaches these methods are based on iterated shift invariant wavelet shrinkage at multiple scales with scaled thresholds

a predual proximal point algorithm solving a non negative basis pursuit denoising model this paper develops an implementation of a predual proximal point algorithm pppa solving a non negative basis pursuit denoising model the model imposes a constraint on the l 2 norm of the residual instead of penalizing it the pppa solves the predual of the problem with a proximal point algorithm ppa moreover the minimization that needs to be performed at each iteration of ppa is solved with a dual method we can prove that these dual variables converge to a solution of the initial problem our analysis proves that we turn a constrained non differentiable convex problem into a short sequence of nice concave maximization problems by nice we mean that the functions which are maximized are differentiable and their gradient is lipschitz the algorithm is easy to implement easier to tune and more general than the algorithms found in the literature in particular it can be applied to the basis pursuit denoising bpdn and the non negative basis pursuit denoising nnbpdn and it does not make any assumption on the dictionary we prove its convergence to the set of solutions of the model and provide some convergence rates experiments on image approximation show that the performances of the pppa are at the current state of the art for the bpdn

should penalized least squares regression be interpreted as maximum a posteriori estimation penalized least squares regression is often used for signal denoising and inverse problems and is commonly interpreted in a bayesian framework as a maximum a posteriori map estimator the penalty function being the negative logarithm of the prior for example the widely used quadratic program with an l1 penalty associated to the lasso basis pursuit denoising is very often considered as map estimation under a laplacian prior in the context of additive white gaussian noise awgn reduction this paper highlights the fact that while this is one possible bayesian interpretation there can be other equally acceptable bayesian interpretations therefore solving a penalized least squares regression problem with penalty x need not be interpreted as assuming a prior c exp x and using the map estimator in particular it is shown that for any prior px the minimum mean square error mmse estimator is the solution of a penalized least square problem with some penalty x which can be interpreted as the map estimator with the prior c exp x vice versa for certain penalties x the solution of the penalized least squares problem is indeed the mmse estimator with a certain prior px in general dpx x c exp x dx

the feasible solution in signal restoration the feasible solution to the signal restoration problem is defined as the one which satisfies all constraints which can be imposed on the true solution a very important set of constraints can be obtained by examining the statistics of the noise these and other constraints can be described as closed convex sets thus projection onto closed convex sets is the numerical method used to obtain a feasible solution examples of this method demonstrate its usefulness in one and two dimensional signal restoration the limitations of the method are discussed

a projection based algorithm for consistent and inconsistent constraints signal synthesis and reconstruction is considered when the signal is to be determined by n constraint sets ci the solution sought is required to minimize a weighted quadratic cost functional hat j emphasis is on cases in which the intersection of the sets ci is empty our proposed procedure employs a suitably weighted simultaneous projection iteration method it is shown that the iterates generated by the algorithm converge weakly to a global minimizer of hat j provided the set of fixed points of the algorithm is nonempty if the problem is consistent c o bigcap c i neq emptyset weak convergence is to an element in co however it is indicated that large classes of inconsistent problems which could not be treated by existing methods admit a solution as well

a semismooth newton method with multidimensional filter globalization for l_1 optimization due to their property of enhancing the sparsity of solutions l 1 regularized optimization problems have developed into a highly dynamic research area with a wide range of applications we present a class of methods for l 1 regularized optimization problems that are based on a combination of semismooth newton steps a filter globalization and shrinkage thresholding steps a multidimensional filter framework is used to control the acceptance and to evaluate the quality of the semismooth newton steps if the current newton iterate is rejected a shrinkage thresholding based step with quasi armijo stepsize rule is used instead global convergence and transition to local q superlinear convergence for both convex and nonconvex objective functions are established we present numerical results and comparisons with several state of the art methods that show the efficiency and competitiveness of the proposed method

parallel proximal algorithm for image restoration using hybrid regularization regularization approaches have demonstrated their effectiveness for solving ill posed problems however in the context of variational restoration methods a challenging question remains namely how to find a good regularizer while total variation introduces staircase effects wavelet domain regularization brings other artefacts e g ringing however a tradeoff can be made by introducing a hybrid regularization including several terms not necessarily acting in the same domain e g spatial and wavelet transform domains while this approach was shown to provide good results for solving deconvolution problems in the presence of additive gaussian noise an important issue is to efficiently deal with this hybrid regularization for more general noise models to solve this problem we adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms for spatial domain regularization isotropic or anisotropic total variation definitions using various gradient filters are considered an accelerated version of the parallel proximal algorithm is proposed to perform the minimization some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper numerical experiments performed in the context of poisson data recovery show the good behavior of the algorithm as well as promising results concerning the use of hybrid regularization techniques

sparse signal estimation by maximally sparse convex optimization this paper addresses the problem of sparsity penalized least squares for applications in sparse signal processing e g sparse deconvolution this paper aims to induce sparsity more strongly than l1 norm regularization while avoiding non convex optimization for this purpose this paper describes the design and use of non convex penalty functions regularizers constrained so as to ensure the convexity of the total cost function f to be minimized the method is based on parametric penalty functions the parameters of which are constrained to ensure convexity of f it is shown that optimal parameters can be obtained by semidefinite programming sdp this maximally sparse convex msc approach yields maximally non convex sparsity inducing penalty functions constrained such that the total cost function f is convex it is demonstrated that iterative msc imsc can yield solutions substantially more sparse than the standard convex sparsity inducing approach i e l1 norm minimization

general projective splitting methods for sums of maximal monotone operators we describe a general projective framework for finding a zero of the sum of n maximal monotone operators over a real hilbert space unlike prior methods for this problem we neither assume n 2 nor first reduce the problem to the case n 2 our analysis defines a closed convex extended solution set for which we can construct a separating hyperplane by individually evaluating the resolvent of each operator at the cost of a single computationally simple projection step this framework gives rise to a family of splitting methods of unprecedented flexibility numerous parameters including the proximal stepsize may vary by iteration and by operator the order of operator evaluation may vary by iteration and may be either serial or parallel the analysis essentially generalizes our prior results for the case n 2 we also include a relative error criterion for approximately evaluating resolvents which was not present in our earlier work

filtered wavelet thresholding methods when working with nonlinear filtering algorithms for image denoising problems there are two crucial aspects namely the choice of the thresholding parameter and the use of a proper filter function both greatly influence the quality of the resulting denoised image in this paper we propose two new filters which are a piecewise quadratic and an exponential function of respectively arid we show how they can be successfully used instead of the classical donoho and johnstone s soft thresholding filter we exploit the increased regularity and flexibility of the new filters to improve the quality of the final results moreover we prove that our filtered approximation is a near minimizer of the functional which has to be minimized to solve the denoising problem we also show that the quadratic filter due to its shape yields good results if we choose as the donoho and johnstone universal threshold while the exponential one is more suitable if we use the recently proposed h curve criterion encouraging results in extensive numerical experiments on several test images confirm the effectiveness of our proposal

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

recursive construction for a class of radial functions i ordinary space a class of spherical functions is studied which can be viewed as the matrix generalization of bessel functions we derive a recursive structure for these functions we show that they are only special cases of more general radial functions which also have a properly generalized recursive structure some explicit results are worked out for the first time we identify a subclass of such radial functions which consist of a finite number of terms only

simultaneous low pass filtering and total variation denoising this paper seeks to combine linear time invariant lti filtering and sparsity based denoising in a principled way in order to effectively filter denoise a wider class of signals lti filtering is most suitable for signals restricted to a known frequency band while sparsity based denoising is suitable for signals admitting a sparse representation with respect to a known transform however some signals cannot be accurately categorized as either band limited or sparse this paper addresses the problem of filtering noisy data for the particular case where the underlying signal comprises a low frequency component and a sparse or sparse derivative component a convex optimization approach is presented and two algorithms derived one based on majorization minimization mm and the other based on the alternating direction method of multipliers admm it is shown that a particular choice of discrete time filter namely zero phase noncausal recursive filters for finite length data formulated in terms of banded matrices makes the algorithms effective and computationally efficient the efficiency stems from the use of fast algorithms for solving banded systems of linear equations the method is illustrated using data from a physiological measurement technique i e near infrared spectroscopic time series imaging that in many cases yields data that is well approximated as the sum of low frequency sparse or sparse derivative and noise components

nonlinear wavelet shrinkage with bayes rules and bayes factors abstract wavelet shrinkage the method proposed by the seminal work of donoho and johnstone is a disarmingly simple and efficient way of denoising data shrinking wavelet coefficients was proposed from several optimality criteria in this article a wavelet shrinkage by coherent bayesian inference in the wavelet domain is proposed the methods are tested on standard donoho johnstone test functions

warped proximal iterations for monotone inclusions abstract resolvents of set valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions we propose a generalization of this notion called warped resolvent which is constructed with the help of an auxiliary operator the properties of warped resolvents are investigated and connections are made with existing notions abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems

fixed point continuation for ell_1 minimization methodology and convergence we present a framework for solving the large scale ell 1 regularized convex minimization problem min x 1 mu f x our approach is based on two powerful algorithmic ideas operator splitting and continuation operator splitting results in a fixed point algorithm for any given scalar mu continuation refers to approximately following the path traced by the optimal value of x as mu increases in this paper we study the structure of optimal solution sets prove finite convergence for important quantities and establish q linear convergence rates for the fixed point algorithm applied to problems with f x convex but not necessarily strictly convex the continuation framework motivated by our convergence results is demonstrated to facilitate the construction of practical algorithms

solving coupled composite monotone inclusions by successive fejer approximations of their kuhn tucker set we propose a new class of primal dual fejer monotone algorithms for solving systems of composite monotone inclusions our construction is inspired by a framework used by eckstein and svaiter for the basic problem of finding a zero of the sum of two monotone operators at each iteration points in the graph of the monotone operators present in the model are used to construct a half space containing the kuhn tucker set associated with the system the primal dual update is then obtained via a relaxed projection of the current iterate onto this half space an important feature that distinguishes the resulting splitting algorithms from existing ones is that they do not require prior knowledge of bounds on the linear operators involved or the inversion of linear operators

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles

characteristic polynomials of real symmetric random matrices the correlation functions of the random variables det x in which x is an hermitian n n random matrix are known to exhibit universal local statistics in the large n limit we study here the correlation of those same random variables for real symmetric matrices goe the derivation relies on an exact dual representation of the problem the k point functions are expressed in terms of finite integrals over quaternionic k k matrices however the control of the dyson limit in which the distance of the various parameters s is of the order of the mean spacing requires an integration over the symplectic group it is shown that a generalization of the itzykson zuber method holds for this problem but contrary to the unitary case the semi classical result requires a finite number of corrections to be exact we have also considered the problem of an external matrix source coupled to the random matrix and obtain explicit integral formulae which are useful for the analysis of the large n limit

a fast algorithm for sparse reconstruction based on shrinkage subspace optimization and continuation we propose a fast algorithm for solving the ell 1 regularized minimization problem min x in mathbb r n mu x 1 ax b 2 2 for recovering sparse solutions to an undetermined system of linear equations ax b the algorithm is divided into two stages that are performed repeatedly in the first stage a first order iterative shrinkage method yields an estimate of the subset of components of x likely to be nonzero in an optimal solution restricting the decision variables x to this subset and fixing their signs at their current values reduces the ell 1 norm x 1 to a linear function of x the resulting subspace problem which involves the minimization of a smaller and smooth quadratic function is solved in the second phase our code fpc as embeds this basic two stage algorithm in a continuation homotopy approach by assigning a decreasing sequence of values to mu this code exhibits state of the art performance in terms of both its speed and its ability to recover sparse signals

moreau s decomposition in banach spaces moreau s decomposition is a powerful nonlinear hilbertian analysis tool that has been used in various areas of optimization and applied mathematics in this paper it is extended to reflexive banach spaces and in the context of generalized proximity measures this extension unifies and significantly improves upon existing results

koszulity for nonquadratic algebras abstract it is known that a koszul algebra is defined as being a quadratic algebra with a pure resolution of the ground field in this paper we extend koszulity to algebras whose relations are homogeneous of degree s 2 a cubic artin schelter regular algebra has motivated our work generalized koszulity is connected to lattice distributivity and to confluence a generalized symmetric algebra is proved to be generalized koszul and the bimodule version of the generalized koszul resolution is used for investigating its hochschild homology

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

group sparse signal denoising non convex regularization convex optimization convex optimization with sparsity promoting convex regularization is a standard approach for estimating sparse signals in noise in order to promote sparsity more strongly than convex regularization it is also standard practice to employ non convex optimization in this paper we take a third approach we utilize a non convex regularization term chosen such that the total cost function consisting of data consistency and regularization terms is convex therefore sparsity is more strongly promoted than in the standard convex formulation but without sacrificing the attractive aspects of convex optimization unique minimum robust algorithms etc we use this idea to improve the recently developed overlapping group shrinkage ogs algorithm for the denoising of group sparse signals the algorithm is applied to the problem of speech enhancement with favorable results in terms of both snr and perceptual quality

a monotone skew splitting model for composite monotone inclusions in duality the principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew adjoint operator an algorithmic framework is developed for solving this generic problem in a hilbert space setting new primal dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators and convergence results are established these algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration comparisons with existing methods are made and applications to composite variational problems are demonstrated

convergence of the linearized bregman iteration for l1 norm minimization one of the key steps in compressed sensing is to solve the basis pursuit problem min u n u au f bregman iteration was very successfully used to solve this problem in 40 also a simple and fast iterative algorithm based on linearized bregman iteration was proposed in 40 which is described in detail with numerical simulations in 35 a convergence analysis of the smoothed version of this algorithm was given in 11 the purpose of this paper is to prove that the linearized bregman iteration proposed in 40 for the basis pursuit problem indeed converges

regression analysis of locality preserving projections via sparse penalty recent studies have shown that linear subspace algorithms such as principal component analysis linear discriminant analysis and locality preserving projections have attracted tremendous attention in many fields of information processing however the projection results obtained by these algorithms are linear combination of the original features which is difficult to be interpreted psychologically and physiologically motivated by compressive sensing theory we formulate the generalized eigenvalue problem under cs framework which then allows us to apply a sparsity penalty and minimization procedure to locality preserving projections the proposed algorithm is called sparse locality preserving projections which performs locality preserving projections in the lasso regression framework that dimensionality reduction feature selection and classification are merged into one analysis the method is also extended to its regularized form to improve its generalization the proposed algorithm is a combination of locality preserving with sparse penalty additionally the algorithm can be performed in either supervised or unsupervised tasks experimental results on toy and real data sets show that our methods are effective and demonstrate much higher performance

variable metric quasi fejer monotonicity the notion of quasi fejer monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis in this paper we extend this notion in the context of variable metric algorithms whereby the underlying norm is allowed to vary at each iteration applications to convex optimization and inverse problems are demonstrated

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

non convex sparse regularisation abstract we study the regularising properties of tikhonov regularisation on the sequence space l 2 with weighted non quadratic penalty term acting separately on the coefficients of a given sequence we derive sufficient conditions for the penalty term that guarantee the well posedness of the method and investigate to which extent the same conditions are also necessary a particular interest of this paper is the application to the solution of operator equations with sparsity constraints assuming a linear growth of the penalty term at zero we prove the sparsity of all regularised solutions moreover we derive a linear convergence rate under the assumptions of even faster growth at zero and a certain injectivity of the operator to be inverted these results in particular cover non convex l p regularisation with 0 p 1

enhanced sparsity by non separable regularization this paper develops a convex approach for sparse one dimensional deconvolution that improves upon l1 norm regularization the standard convex approach we propose a sparsity inducing non separable non convex bivariate penalty function for this purpose it is designed to enable the convex formulation of ill conditioned linear inverse problems with quadratic data fidelity terms the new penalty overcomes limitations of separable regularization we show how the penalty parameters should be set to ensure that the objective function is convex and provide an explicit condition to verify the optimality of a prospective solution we present an algorithm an instance of forward backward splitting for sparse deconvolution using the new penalty

a parallel splitting method for coupled monotone inclusions a parallel splitting method is proposed for solving systems of coupled monotone inclusions in hilbert spaces and its convergence is established under the assumption that solutions exist unlike existing alternating algorithms which are limited to two variables and linear coupling our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes the breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions variational problems best approximation and network flows

sparse identification of posynomial models posynomials are nonnegative combinations of monomials with possibly fractional and both positive and negative exponents posynomial models are widely used in various engineering design endeavors such as circuits aerospace and structural design mainly due to the fact that design problems cast in terms of posynomial objectives and constraints can be solved efficiently by means of a convex optimization technique known as geometric programming gp however while quite a vast literature exists on gp based design very few contributions can yet be found on the problem of identifying posynomial models from experimental data posynomial identification amounts to determining not only the coefficients of the combination but also the exponents in the monomials which renders the identification problem hard in this paper we propose an approach to the identification of multivariate posynomial models based on the expansion on a given large scale basis of monomials the model is then identified by seeking coefficients of the combination that minimize a mixed objective composed by a term representing the fitting error and a term inducing sparsity in the representation which results in a problem formulation of the square root lasso type with nonnegativity constraints on the variables we propose to solve the problem via a sequential coordinate minimization scheme which is suitable for large scale implementations a numerical example is finally presented dealing with the identification of a posynomial model for a naca 4412 airfoil

maximum entropy reconstruction using derivative information part 1 fisher information and convex duality maximum entropy spectral density estimation is a technique for reconstructing an unknown density function from some known measurements by maximizing a given measure of entropy of the estimate here we present a variety of new entropy measures which attempt to control derivative values of the densities our models apply among others to the inference problem based on the averaged fisher information measure the duality theory we develop resembles models used in convex optimal control problems we present a variety of examples including relaxed moment matching with fisher information and best interpolation on a strip

restoration of poissonian images using alternating direction optimization much research has been devoted to the problem of restoring poissonian images namely for medical and astronomical applications however the restoration of these images using state of the art regularizers such as those based upon multiscale representations or total variation is still an active research area since the associated optimization problems are quite challenging in this paper we propose an approach to deconvolving poissonian images which is based upon an alternating direction optimization method the standard regularization or maximum a posteriori map restoration criterion which combines the poisson log likelihood with a nonsmooth convex regularizer log prior leads to hard optimization problems the log likelihood is nonquadratic and nonseparable the regularizer is nonsmooth and there is a nonnegativity constraint using standard convex analysis tools we present sufficient conditions for existence and uniqueness of solutions of these optimization problems for several types of regularizers total variation frame based analysis and frame based synthesis we attack these problems with an instance of the alternating direction method of multipliers admm which belongs to the family of augmented lagrangian algorithms we study sufficient conditions for convergence and show that these are satisfied either under total variation or frame based analysis and synthesis regularization the resulting algorithms are shown to outperform alternative state of the art methods both in terms of speed and restoration accuracy

generalizing the notion of koszul algebra we introduce a generalization of the notion of a koszul algebra which includes graded algebras with relations in different degrees and we establish some of the basic properties of these algebras this class is closed under twists twisted tensor products regular central extensions and ore extensions we explore the monomial algebras in this class and we include some well known examples of algebras that fall into this class

a proximal iteration for deconvolving poisson noisy images using sparse representations we propose an image deconvolution algorithm when the data is contaminated by poisson noise the image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms our key contributions are as follows first we handle the poisson noise properly by using the anscombe variance stabilizing transform leading to a nonlinear degradation equation with additive gaussian noise second the deconvolution problem is formulated as the minimization of a convex functional with a data fidelity term reflecting the noise properties and a nonsmooth sparsity promoting penalty over the image representation coefficients e g lscr1 norm an additional term is also included in the functional to ensure positivity of the restored image third a fast iterative forward backward splitting algorithm is proposed to solve the minimization problem we derive existence and uniqueness conditions of the solution and establish convergence of the iterative algorithm finally a gcv based model selection procedure is proposed to objectively select the regularization parameter experimental results are carried out to show the striking benefits gained from taking into account the poisson statistics of the noise these results also suggest that using sparse domain regularization may be tractable in many deconvolution applications with poisson noise such as astronomy and microscopy

projection and proximal point methods convergence results and counterexamples recently hundal has constructed a hyperplane h a cone k and a starting point y0 in 2 such that the sequence of alternating projections pk ph n y0 n n converges weakly to some point in h k but not in norm we show how this construction results in a counterexample to norm convergence for iterates of averaged projections hence we give an a rmative answer to a question raised by reich two decades ago furthermore new counterexamples to norm convergence for iterates of rmly nonexpansive maps a la genel and lindenstrauss and for the proximal point algorithm al a g uler are provided we also present a counterexample along with some weak and norm convergence results for the new framework of string averaging projection methods introduced by censor et al extensions to banach spaces and the situation for the hilbert ball are discussed as well 2003 elsevier ltd all rights reserved msc 47h09 47j25 90c25

monotone operator theory in convex optimization several aspects of the interplay between monotone operator theory and convex optimization are presented the crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized we review the properties of subdifferentials as maximally monotone operators and in tandem investigate those of proximity operators as resolvents in particular we study new transformations which map proximity operators to proximity operators and establish connections with self dual classes of firmly nonexpansive operators in addition new insights and developments are proposed on the algorithmic front

uniqueness of drs as the 2 operator resolvent splitting and impossibility of 3 operator resolvent splitting given the success of douglas rachford splitting drs it is natural to ask whether drs can be generalized are there other 2 operator resolvent splittings sharing the favorable properties of drs can drs be generalized to 3 operators this work presents the answers no and no in a certain sense drs is the unique 2 operator resolvent splitting and generalizing drs to 3 operators is impossible without lifting where lifting roughly corresponds to enlarging the problem size the impossibility result further raises a question how much lifting is necessary to generalize drs to 3 operators this work presents the answer by providing a novel 3 operator resolvent splitting with provably minimal lifting that directly generalizes drs

a note on the paper by eckstein and svaiter on general projective splitting methods for sums of maximal monotone operators in their recent paper from 2009 siam j control optim 48 2009 pp 787 811 j eckstein and b f svaiter proposed a very general and flexible splitting framework for finding a zero of the sum of finitely many maximal monotone operators in this short note we provide a technical result that allows for the removal of eckstein and svaiter s assumption that the sum of the operators be maximal monotone or that the underlying hilbert space be finite dimensional

on the definition of multi koszul modules abstract in 8 we introduced the notion of multi koszul algebra it is an extension of the definition of generalized koszul algebra given by r berger in 1 for homogeneous algebras see also 7 that can be applied to any nonnegatively graded connected algebra over a field k the goal of this article is on the one hand to properly extend the notion of multi koszul algebra to the case where the base ring k is a product of copies of a field k which a priori allows us to treat quiver algebras and on the other hand to introduce the notion of multi koszul module such that it extends the usual definition of generalized koszul module over a generalized koszul algebra we show eventually that multi koszul algebras and multi koszul modules are strongly linked via the notion of one point extensions as in the case of generalized koszul algebras moreover we describe the complete structure of right a module on e x t a m k over e x t a k k where m is a multi koszul module over a multi koszul algebra a extending a result in 9 as a corollary we obtain that the underlying right module structure of e x t a m k over e x t a k k is generated by the component of cohomological degree zero as in the case of generalized koszul modules over generalized koszul algebras

a family of projective splitting methods for the sum of two maximal monotone operators a splitting method for two monotone operators a and b is an algorithm that attempts to converge to a zero of the sum a b by solving a sequence of subproblems each of which involves only the operator a or only the operator b prior algorithms of this type can all in essence be categorized into three main classes the douglas peaceman rachford class the forward backward class and the little used double backward class through a certain extended solution set in a product space we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in hilbert space our algorithms are essentially standard projection methods using splitting decomposition to construct separators we prove convergence through fejer monotonicity techniques but showing fejer convergence of a different sequence to a different set than in earlier splitting methods our projective algorithms converge under more general conditions than prior splitting methods allowing the proximal parameter to vary from iteration to iteration and even from operator to operator while retaining convergence for essentially arbitrary pairs of operators the new projective splitting class also contains noteworthy preexisting methods either as conventional special cases or excluded boundary cases

proximal operator of quotient functions with application to a feasibility problem in query optimization in this paper we determine the proximity functions of the sum and the maximum of componentwise reciprocal quotients of positive vectors for the sum of quotients denoted by q 1 the proximity function is just a componentwise shrinkage function which we call q shrinkage this is similar to the proximity function of the 1 norm which is given by componentwise soft shrinkage for the maximum of quotients q the proximal function can be computed by first order primal dual methods involving epigraphical projections the proximity functions of q 1 are applied to solve convex problems of the form argmin x q a x b subject to x 0 1 x 1 such problems are of interest in selectivity estimation for cost based query optimizers in database management systems

transition from poisson to gaussian unitary statistics the two point correlation function we consider the rosenzweig porter model of random matrix which interpolates between poisson and gaussian unitary statistics and compute exactly the two point correlation function asymptotic formulas for this function are given near the poisson and gaussian limit