on computing the distance to stability for matrices using linear dissipative hamiltonian systems abstract in this paper we consider the problem of computing the nearest stable matrix to an unstable one we propose new algorithms to solve this problem based on a reformulation using linear dissipative hamiltonian systems we show that a matrix a is stable if and only if it can be written as a j r q where j j t r 0 and q 0 that is r is positive semidefinite and q is positive definite this reformulation results in an equivalent optimization problem with a simple convex feasible set we propose three strategies to solve the problem in variables j r q i a block coordinate descent method ii a projected gradient descent method and iii a fast gradient method inspired from smooth convex optimization these methods require o n 3 operations per iteration where n is the size of a we show the effectiveness of the fast gradient method compared to the other approaches and to several state of the art algorithms

stochastic first and zeroth order methods for nonconvex stochastic programming in this paper we introduce a new stochastic approximation type algorithm namely the randomized stochastic gradient rsg method for solving an important class of nonlinear possibly nonconvex stochastic programming problems we establish the complexity of this method for computing an approximate stationary point of a nonlinear programming problem we also show that this method possesses a nearly optimal rate of convergence if the problem is convex we discuss a variant of the algorithm which consists of applying a postoptimization phase to evaluate a short list of solutions generated by several independent runs of the rsg method and we show that such modification allows us to improve significantly the large deviation properties of the algorithm these methods are then specialized for solving a class of simulation based optimization problems in which only stochastic zeroth order information is available

on weyl channels being covariant with respect to the maximum commutative group of unitaries we investigate the weyl channels being covariant with respect to the maximum commutative group of unitary operators this class includes the quantum depolarizing channel and the two pauli channel as well then we show that our estimation of the output entropy for a tensor product of the phase damping channel and the identity channel based upon the decreasing property of the relative entropy allows to prove the additivity conjecture for the minimal output entropy for the quantum depolarizing channel in any prime dimension and for the two pauli channel in the qubit case

harmonic operators the dual perspective the study of harmonic functions on a locally compactgroupghas recently been transferred to a non commutative setting in two different directions chu and lau replaced the algebra l g by the group von neumann algebra vn g and the convolution action of a probability measure on l g by the canonical action of a positive definite function on vn g on the other hand jaworski and the first author replaced l g by b l 2 g to which the convolution action by can be extended in a natural way we establish a link between both approaches the action of on vn g can be extended to b l 2 g we study the corresponding space h

variance based extragradient methods with line search for stochastic variational inequalities in this paper we propose dynamic sampled stochastic approximated ds sa extragradient methods for stochastic variational inequalities svis that are robust with respect to an unknown lipschitz c

the fidelity of density operators in an operator algebraic framework josza s definition of fidelity r jozsa j mod opt 41 12 2315 2323 1994 for a pair of mixed quantum states is studied in the context of two types of operator algebras the first setting is mainly algebraic in that it involves unital c algebras a that possess a faithful trace functional in this context the role of quantum states that is density operators in the classical quantum mechanical framework is assumed by positive elements a for which 1 the second setting is more operator theoretic by fixing a faithful normal semifinite trace on a semifinite von neumann algebra m we define and consider the fidelity of pairs of positive operators in m of unit trace the main results of this paper address monotonicity and preservation of fidelity under the action of certain trace preserving positive linear maps of a or of the predual m our results in the von neumann algebra setting are novel in that we focus on the schrodinger picture rather than the heisenberg picture and they a

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

an optimal method for stochastic composite optimization this paper considers an important class of convex programming cp problems namely the stochastic composite optimization sco whose objective function is given by the summation of general nonsmooth and smooth stochastic components since sco covers non smooth smooth and stochastic cp as certain special cases a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming note however that the optimization algorithms that can achieve this lower bound had never been developed in this paper we show that the simple mirror descent stochastic approximation method exhibits the best known rate of convergence for solving these problems our major contribution is to introduce the accelerated stochastic approximation ac sa algorithm based on nesterov s optimal method for smooth cp nesterov in doklady an sssr 269 543 547 1983 nesterov in math program 103 127 152 2005 and show that the ac sa algorithm can achieve the aforementioned lower bound on the rate of convergence for sco to the best of our knowledge it is also the first universally optimal algorithm in the literature for solving non smooth smooth and stochastic cp problems we illustrate the significant advantages of the ac sa algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems

spatially periodic modulated rayleigh benard convection two dimensional thermal convection in a fluid layer between two rigid walls at different mean temperatures is investigated the top container boundary is undulated and the temperatures at the top and bottom boundaries are spatially periodic modulated with modulation wavelengths large compared to the thickness of the fluid layer the continuous translational invariance in the fluid layer is broken by these spatial modulations consequently phase differences between two periodic modulations give rise to an interesting drifting pattern with the drift direction depending on the sign of the relative phase between the modulations at distinguished ratios between the modulation wave numbers and relative phases the onset of convection changes as function of the modulation amplitudes from a stationary into an oscillatory one we call this phenomenon hopf bifurcation by frustrated drifts possible experiments are described in detail where this phenomenon can be expected textcopyright 1996 the american physical society

noncommutative poisson boundaries of unital quantum operations in this paper poisson boundaries of unital quantum operations also called markov operators are investigated in the case of unital quantum channels compact operators belonging to poisson boundaries are characterized using the characterization of amenable groups by the injectivity of their von neumann algebras we will answer negatively some conjectures appearing in the work of arias et al fixed points of quantum operations j math phys 43 5872 2002 about injectivity of the commuting algebra of the kraus operators of unital quantum operations and their injective envelopes

spatial synchronization of regular optical patterns we investigate an extended nonlinear optical experiment exhibiting the spontaneous formation of hexagonal patterns out of a stationary bifurcation the system is exposed to a two dimensional spatially periodic forcing namely static hexagonal patterns under variation of their spatial periodicity parameters are the strength of the forcing and the distance to pattern forming threshold the system response is quantitatively characterized with different methods we observe several locking regimes where the system is entrained by the forcing most of the locking regimes can be related to resonances between the different critical wave numbers and the forcing wave number or its spatial harmonics one particular locking appears to result from two of these simple resonances in a kind of generalized order m n synchronization the width of the locking regimes increases with forcing strength apparently representing a spatial analog of arnold tongues

coupled nonlocal complex ginzburg landau equations in gasless combustion abstract we consider the evolution of a gasless combustion front we derive coupled complex ginzburg landau type equations for the amplitudes of waves along the front as functions of slow temporal and spatial variables the equations are written in characteristic variables and involve averaged terms which reflect the fact that in the slowest time scale the effect on one wave of a second wave traveling with the group velocity in the opposite direction on the intermediate time scale enters only through its average solutions of the amplitude equations in the form of traveling standing and quasiperiodic waves are found and regions of stability for these solutions are determined in particular we find that the traveling and quasiperiodic including standing waves are not stable simultaneously finally we observe that the stability analysis for coupled complex ginzburg landau equations with averaged terms differs from that for coupled complex ginzburg landau equations with the averages removed

accelerated gradient methods for nonconvex nonlinear and stochastic programming in this paper we generalize the well known nesterov s accelerated gradient ag method originally designed for convex smooth optimization to solve nonconvex and possibly stochastic optimization problems we demonstrate that by properly specifying the stepsize policy the ag method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first order information similarly to the gradient descent method we then consider an important class of composite optimization problems and show that the ag method can solve them uniformly i e by using the same aggressive stepsize policy as in the convex case even if the problem turns out to be nonconvex we demonstrate that the ag method exhibits an optimal rate of convergence if the composite problem is convex and improves the best known rate of convergence if the problem is nonconvex based on the ag method we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization to the best of our knowledge this is the first time that the convergence of the ag method has been established for solving nonconvex nonlinear programming in the literature

iteration complexity of first order augmented lagrangian methods for convex programming this paper considers a special class of convex programming cp problems whose feasible regions consist of a simple compact convex set intersected with an affine manifold we present first order methods for this class of problems based on an inexact version of the classical augmented lagrangian al approach where the subproblems are approximately solved by means of nesterov s optimal method we then establish a bound on the total number of nesterov s optimal iterations i e the inner iterations performed throughout the entire inexact al method to obtain a near primal dual optimal solution we also present variants with possibly better iteration complexity bounds than the original inexact al method which consist of applying the original approach directly to a perturbed problem obtained by adding a strongly convex component to the objective function of the cp problem

synchronized patterns in hierarchical networks of neuronal oscillators with d 3 d 3 symmetry abstract the spatiotemporal patterns generated by systems of nine coupled nonlinear oscillators which are equivariant under the permutation symmetry group d 3 d 3 are determined this system can be interpreted as a hierarchically organized network composed of three interacting systems each of which consists of three coupled oscillators we determine generic synchronized oscillation patterns and transitions between these analytically by numerical simulations and experimentally with an electronic analog network in the theoretical analysis the representative nonlinear ordinary differential equations are reduced to the normal form equations for coupled hopf bifurcations in an eight dimensional center eigenspace whose generic states have been classified previously the results are applied to a specific model system in which the network is formed by a class of oscillators each composed of two asymmetrically coupled hopfield neurons experiments performed on an analog electronic network of such nonlinear oscillators show that most of the states predicted by the theory of the hopf bifurcation with d 3 d 3 symmetry appear in a stable way we find a great variety of periodic and quasiperiodic oscillation patterns of maximal and submaximal symmetry which can be classified in a two level pattern hierarchy in addition to these states we find in simulations homoclinic cycles within the same isotropy class as well as heteroclinic switchings between such cycles

stripe hexagon competition in forced pattern forming systems with broken up down symmetry we investigate the response of two dimensional pattern forming systems with a broken up down symmetry such as chemical reactions to spatially resonant forcing and propose related experiments the nonlinear behavior immediately above threshold is analyzed in terms of amplitude equations suggested for a 1 2 and 1 1 ratio between the wavelength of the spatial periodic forcing and the wavelength of the pattern of the respective system both sets of coupled amplitude equations are derived by a perturbative method from the lengyel epstein model describing a chemical reaction showing turing patterns which gives us the opportunity to relate the generic response scenarios to a specific pattern forming system the nonlinear competition between stripe patterns and distorted hexagons is explored and their range of existence stability and coexistence is determined whereas without modulations hexagonal patterns are always preferred near onset of pattern formation single mode solutions stripes are favored close to threshold for modulation amplitudes beyond some critical value hence distorted hexagons only occur in a finite range of the control parameter and their interval of existence shrinks to zero with increasing values of the modulation amplitude furthermore depending on the modulation amplitude the transition between stripes and distorted hexagons is either subcritical or supercritical

fixed points of quantum operations quantum operations frequently occur in quantum measurement theory quantum probability quantum computation and quantum information theory if an operator a is invariant under a quantum operation we call a a fixed point physically the fixed points are the operators that are not disturbed by the action of our main purpose is to answer the following question if a is a fixed point is a compatible with the operation elements of we shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von neumann algebras

on variance reduction for stochastic smooth convex optimization with multiplicative noise we propose dynamic sampled stochastic approximation sa methods for stochastic optimization with a heavy tailed distribution with finite 2nd moment the objective is the sum of a smooth convex function with a convex regularizer typically it is assumed an oracle with an upper bound sigma 2 on its variance oubv differently we assume an oracle with multiplicative noise this rarely addressed setup is more aggressive but realistic where the variance may not be uniformly bounded our methods achieve optimal iteration complexity and near optimal oracle complexity for the smooth convex class we use an accelerated sa method a la fista which achieves given tolerance varepsilon 0 the optimal iteration complexity of mathscr o varepsilon frac 1 2 with a near optimal oracle complexity of mathscr o varepsilon 2 ln varepsilon frac 1 2 2 this improves upon ghadimi and lan math program 156 59 99 2016 where it is assumed an oubv for the strongly convex class our method achieves optimal iteration complexity of mathscr o ln varepsilon 1 and optimal oracle complexity of mathscr o varepsilon 1 this improves upon byrd et al math program 134 127 155 2012 where it is assumed an oubv in terms of variance our bounds are local they depend on variances sigma x 2 at solutions x and the per unit distance multiplicative variance sigma 2 l for the smooth convex class there exist policies such that our bounds resemble up to absolute constants those obtained in the mentioned papers if it was assumed an oubv with sigma 2 sigma x 2 for the strongly convex class such property is obtained exactly if the condition number is estimated or in the limit for better conditioned problems or for larger initial batch sizes in any case if it is assumed an oubv our bounds are thus sharper since typically max sigma x 2 sigma l 2 ll sigma 2

misspecified nonconvex statistical optimization for sparse phase retrieval existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models to address this issue we take a first step towards taming model misspecification by studying the high dimensional sparse phase retrieval problem with misspecified link functions in particular we propose a simple variant of the thresholded wirtinger flow algorithm that given a proper initialization linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions we further provide extensive numerical experiments to support our theoretical findings

dynamic stochastic approximation for multi stage stochastic optimization in this paper we consider multi stage stochastic optimization problems with convex objectives and conic constraints at each stage we present a new stochastic first order method namely the dynamic stochastic approximation dsa algorithm for solving these types of stochastic optimization problems we show that dsa can achieve an optimal mathcal o 1 epsilon 4 rate of convergence in terms of the total number of required scenarios when applied to a three stage stochastic optimization problem we further show that this rate of convergence can be improved to mathcal o 1 epsilon 2 when the objective function is strongly convex we also discuss variants of dsa for solving more general multi stage stochastic optimization problems with the number of stages t 3 the developed dsa algorithms only need to go through the scenario tree once in order to compute an epsilon solution of the multi stage stochastic optimization problem as a result the memory required by dsa only grows linearly with respect to the number of stages to the best of our knowledge this is the first time that stochastic approximation type methods are generalized for multi stage stochastic optimization with t ge 3

a proximal method for composite minimization we consider minimization of functions that are compositions of convex or prox regular functions possibly extended valued with smooth vector functions a wide variety of important optimization problems fall into this framework we describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem preliminary computational results on both convex and nonconvex examples are promising

approximating the nearest stable discrete time system abstract in this paper we consider the problem of stabilizing discrete time linear systems by computing a nearby stable matrix to an unstable one to do so we provide a new characterization for the set of stable matrices we show that a matrix a is stable if and only if it can be written as a s 1 u b s where s is positive definite u is orthogonal and b is a positive semidefinite contraction that is the singular values of b are less or equal to 1 this characterization results in an equivalent non convex optimization problem with a feasible set on which it is easy to project we propose a very efficient fast projected gradient method to tackle the problem in variables s u b and generate locally optimal solutions we show the effectiveness of the proposed method compared to other approaches

factorization and dilation problems for completely positive maps on von neumann algebras we study factorization and dilation properties of markov maps between von neumann algebras equipped with normal faithful states i e completely positive unital maps which preserve the given states and also intertwine their automorphism groups the starting point for our investigation has been the question of existence of non factorizable markov maps as formulated by c anantharaman delaroche we provide simple examples of non factorizable markov maps on m n mathbb c for all n 3 as well as an example of a one parameter semigroup t t t 0 of markov maps on m 4 mathbb c such that t t fails to be factorizable for all small values of t 0 as applications we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum birkhoff conjecture as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little grothendieck inequality

completely isometric representations of _ and let g be a locally compact group it is shown that there exists a natural completely isometric representation of the completely bounded fourier multiplier algebra m cb a g which is dual to the representation of the measure algebra m g on b l 2 g the image algebras of m g and m cb a g in cb b l 2 g are intrinsically characterized and some commutant theorems are proved it is also shown that for any amenable group g there is a natural completely isometric representation of ucb g on b l 2 g which can be regarded as a duality result of neufang s completely isometric representation theorem for luc g

measurable schur multipliers and completely bounded multipliers of the fourier algebras let g be a locally compact group and a g the fourier algebra of g from 17 in 24 herz de ned the space of herz schur multipliers b2 g he claimed that this space coincided with the space he denoted v g in his article 23 but o2ered no proof of this fact the completely bounded multipliers of the fourier algebra mcba g were de ned by de canniere and haagerup in 8 in was shown by bo zejko and fendler 7 that mcba g 1 4 b2 g isometrically unfortunately this result relies on unpublished work of gilbert entitled convolution operators and banach space tensor products i ii and iii but not widely available however there is a proof of the fact that mcba g 1 4 b2 g due to jolissaint 27 which relies on the representation theorem for completely bounded maps applied to the reduced c algebra c r g in this paper we prove an analogue of the fact that mcba g 1 4 b2 g in fact our result obtains that fact in the case that g is a discrete group and improves upon it in the respect that we obtain a complete isometry where both mcba g and b2 g are given natural operator space structures moreover we show that the natural module action of mcba g on the von neumann algebra generated by the left regular representation vn g extends completely isometrically to a normal action of mcba g on all of b l g the space of bounded operators on l g in fact the image of mcba g in the completely bounded normal maps on b l g consists of l1 g bimodule maps and can be characterised as a certain intrinsic invariant part amongst those maps we feel that our approach is interesting in and of itself since it makes systematic use of modern operator space theoretic techniques pioneered by blecher e2ros haagerup paulsen ruan and smith amongst others it thus gives a starting point for obtaining representations of completely bounded multipliers of kac algebras after 29 or of multipliers of locally compact quantum groups after vaes et al see 47 to support our approach in x 3 we describe the space of measurable schur multipliers v1 x for any suitable measure space x we develop the theory of v1 x fully and systematically our main tools for doing this are the results of smith 41 and blecher and smith 5 which characterise the normal completely bounded maps on the space of all bounded operators on a hilbert space which are module maps over a von neumann algebra we make great use of the weak haagerup tensor product of two von neumann algebras which is

conditional gradient type methods for composite nonlinear and stochastic optimization in this paper we present a conditional gradient type cgt method for solving a class of composite optimization problems where the objective function consists of a weakly smooth term and a strongly convex regularization term while including a strongly convex term in the subproblems of the classical conditional gradient method improves its rate of convergence it does not cost per iteration as much as general proximal type algorithms more specifically we present a unified analysis for the cgt method in the sense that it achieves the best known rate of convergence when the weakly smooth term is nonconvex and possesses nearly optimal complexity if it turns out to be convex while implementation of the cgt method requires explicitly estimating problem parameters like the level of smoothness of the first term in the objective function we also present a few variants of this method which relax such estimation unlike general proximal type parameter free methods these variants of the cgt method do not require any additional effort for computing sub gradients of the objective function and or solving extra subproblems at each iteration we then generalize these methods under stochastic setting and present a few new complexity results to the best of our knowledge this is the first time that such complexity results are presented for solving stochastic weakly smooth nonconvex and strongly convex optimization problems

bounds on the distance between a unital quantum channel and the convex hull of unitary channels motivated by the recent resolution of asymptotic quantum birkhoff conjecture aqbc we attempt to estimate the distance between a given unital quantum channel and the convex hull of unitary channels we provide two lower bounds on this distance by employing techniques from quantum information and operator algebras respectively we then show how to apply these results to construct some explicit counterexamples to aqbc we also point out an interesting connection between the grothendieck s inequality and aqbc

capacity bounds via operator space methods we prove that for generalized dephasing channels the coherent information and reverse coherent information coincides it also implies an alternative approach for the strong super additivity and strong converse of generalized dephasing channels using the operator space technique our argument is based on an improved renyi relative entropy estimate via analyzing the channel s stinespring space we also apply this estimate to new examples of quantum channels arising from quantum group co representation and kitave s quantum computation model in particular we find concrete examples of non degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity we prove that for generalized dephasing channels the coherent information and reverse coherent information coincides it also implies an alternative approach for the strong super additivity and strong converse of generalized dephasing channels using the operator space technique our argument is based on an improved renyi relative entropy estimate via analyzing the channel s stinespring space we also apply this estimate to new examples of quantum channels arising from quantum group co representation and kitave s quantum computation model in particular we find concrete examples of non degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity

spatial forcing of spontaneous optical patterns a nonlinear optical system which spontaneously forms hexagonal patterns is exposed to a weak spatially modulated forcing as forcing stationary hexagonal patterns are used under variation of their transverse wave number in the experiment we observe a locking when the forcing wave number is close to one of the critical wave numbers of the unforced system outside the locking regimes forcing provokes spatiotemporal disorder the system response is characterized quantitatively with respect to its dynamics and to its spatial order

effects of parametric disorder on a stationary bifurcation abstract effects of a frozen random contribution to the control parameter are investigated in terms of the complex ginzburg landau equation with real coefficients the threshold of the bifurcation from the homogeneous basic state is reduced by a random contribution even with a vanishing spatial mean value as shown by three different approaches by a perturbation calculation by a self consistent iteration method and by a fully numerical solution of the linear part of the ginzburg landau equation for arbitrary random contributions the nonlinear stationary solutions are numerically determined and in the limit of small random amplitudes analytical expressions are derived in terms of two different perturbation expansions which cover already several related trends beyond threshold for instance the spatial modulations of the solutions increase with the noise amplitude but decrease with increasing distance from threshold

optimal stochastic approximation algorithms for strongly convex stochastic composite optimization i a generic algorithmic framework in this paper we present a generic algorithmic framework namely the accelerated stochastic approximation ac sa algorithm for solving strongly convex stochastic composite optimization sco problems while the classical stochastic approximation algorithms are asymptotically optimal for solving differentiable and strongly convex problems the ac sa algorithm when employed with proper stepsize policies can achieve optimal or nearly optimal rates of convergence for solving different classes of sco problems during a given number of iterations moreover we investigate these ac sa algorithms in more detail such as by establishing the large deviation results associated with the convergence rates and introducing an efficient validation procedure to check the accuracy of the generated solutions

a smoothing sqp framework for a class of composite l_q lq minimization over polyhedron the composite l q 0 q 1 minimization problem over a general polyhedron has received various applications in machine learning wireless communications image restoration signal reconstruction etc this paper aims to provide a theoretical study on this problem first we derive the karush kuhn tucker kkt optimality conditions for local minimizers of the problem second we propose a smoothing sequential quadratic programming framework for solving this problem the framework requires a approximate solution of a convex quadratic program at each iteration finally we analyze the worst case iteration complexity of the framework for returning an epsilon kkt point i e a feasible point that satisfies a perturbed version of the derived kkt optimality conditions to the best of our knowledge the proposed framework is the first one with a worst case iteration complexity guarantee for solving composite l q minimization over a general polyhedron

on the complexity of steepest descent newton s and regularized newton s methods for nonconvex unconstrained optimization problems it is shown that the steepest descent and newton s methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to o epsilon 2 to drive the norm of the gradient below epsilon this shows that the upper bound of o epsilon 2 evaluations known for the steepest descent is tight and that newton s method may be as slow as the steepest descent method in the worst case the improved evaluation complexity bound of o epsilon 3 2 evaluations known for cubically regularized newton s methods is also shown to be tight

pattern selection in three dimensional magnetoconvection abstract weakly nonlinear boussinesq convection in an imposed vertical magnetic field is studied stress free boundary conditions are used at top and bottom of the layer where the magnetic field is assumed to remain vertical both steady and oscillatory convection are considered among all the spatially periodic patterns steady rolls are the preferred pattern when the onset of convection is steady state in the overstable regime a variety of spatially and temporally periodic patterns may be stable depending on the parameters these include in particular a pattern called alternating rolls the location of stable alternating rolls appears to be consistent with recent numerical simulations of compressible magnetoconvection where such patterns have been found in certain cases the conduction solution loses stability directly to spatially periodic patterns with nonperiodic time dependence the analysis of the possible patterns and their stability takes advantage of existing results from equivariant bifurcation theory and uses the critical wavenumber selected by the linear stability calculation as a function of the imposed magnetic field

run and inspect method for nonconvex optimization and global optimality bounds for r local minimizers many optimization algorithms converge to stationary points when the underlying problem is nonconvex they may get trapped at local minimizers or stagnate near saddle points we propose the run and inspect method which adds an inspect phase to existing algorithms that helps escape from non global stationary points it samples a set of points in a radius r around the current point when a sample point yields a sufficient decrease in the objective we resume an existing algorithm from that point if no sufficient decrease is found the current point is called an approximate r local minimizer we show that an r local minimizer is globally optimal up to a specific error depending on r if the objective function can be implicitly decomposed into a smooth convex function plus a restricted function that is possibly nonconvex nonsmooth therefore for such nonconvex objective functions verifying global optimality is fundamentally easier for high dimensional problems we introduce blockwise inspections to overcome the curse of dimensionality while still maintaining optimality bounds up to a factor equal to the number of blocks we also present the sample complexities of these methods when we apply our method to the existing algorithms on a set of artificial and realistic nonconvex problems we find significantly improved chances of obtaining global minima

accelerated schemes for a class of variational inequalities we propose a novel stochastic method namely the stochastic accelerated mirror prox samp method for solving a class of monotone stochastic variational inequalities svi the main idea of the proposed algorithm is to incorporate a multi step acceleration scheme into the stochastic mirror prox method the developed samp method computes weak solutions with the optimal iteration complexity for svis in particular if the operator in svi consists of the stochastic gradient of a smooth function the iteration complexity of the samp method can be accelerated in terms of their dependence on the lipschitz constant of the smooth function for svis with bounded feasible sets the bound of the iteration complexity of the samp method depends on the diameter of the feasible set for unbounded svis we adopt the modified gap function introduced by monteiro and svaiter for solving monotone inclusion and show that the iteration complexity of the samp method depends on the distance from the initial point to the set of strong solutions it is worth noting that our study also significantly improves a few existing complexity results for solving deterministic variational inequality problems we demonstrate the advantages of the samp method over some existing algorithms through our preliminary numerical experiments

multipliers of the fourier algebras of some simple lie groups and their discrete subgroups for any amenable locally compact group g the space of multipliers ma g of the fourier algebra a g coincides with the space b g of functions on g that are linear combinations of continuous positive definite functions we prove that ma g b g 0 for many non amenable connected groups more specifically we prove that moa g b g 0 for the classical complex lie groups and the general lorentz groups soo n 1 n 2 moa g is a certain subspace of ma g which we call the space of completely bounded multipliers of a g unlike ma g the space moa g has nice stability properties with respect to direct products of groups it is known that the fourier algebra of the free group on n generators n 2 2 admits an unbounded approximate unit pn which is bounded in the multiplier norm we extend this result to any closed subgroup of the general lorentz group soo n 1 moreover we show that for these groups pn can be chosen to be bounded with respect to the moa g norm by a duality argument we obtain that the reduced c algebra of every discrete subgroup of soo n 1 has the completely bounded approximation property in particular this property holds for c f2 the reduced c algebra of the free group on two generators we also prove

stochastic optimization using a trust region method and random models in this paper we propose and analyze a trust region model based algorithm for solving unconstrained stochastic optimization problems our framework utilizes random models of an objective function f x obtained from stochastic observations of the function or its gradient our method also utilizes estimates of function values to gauge progress that is being made the convergence analysis relies on requirements that these models and these estimates are sufficiently accurate with high enough but fixed probability beyond these conditions no assumptions are made on how these models and estimates are generated under these general conditions we show an almost sure global convergence of the method to a first order stationary point in the second part of the paper we present examples of generating sufficiently accurate random models under biased or unbiased noise assumptions lastly we present some computational results showing the benefits of the proposed method compared to existing approaches that are based on sample averaging or stochastic gradients

behavior of accelerated gradient methods near critical points of nonconvex functions we examine the behavior of accelerated gradient methods in smooth nonconvex unconstrained optimization focusing in particular on their behavior near strict saddle points accelerated methods are iterative methods that typically step along a direction that is a linear combination of the previous step and the gradient of the function evaluated at a point at or near the current iterate the previous step encodes gradient information from earlier stages in the iterative process we show by means of the stable manifold theorem that the heavy ball method is unlikely to converge to strict saddle points which are points at which the gradient of the objective is zero but the hessian has at least one negative eigenvalue we then examine the behavior of the heavy ball method and other accelerated gradient methods in the vicinity of a strict saddle point of a nonconvex quadratic function showing that both methods can diverge from this point more rapidly than the steepest descent method

statistical inference for the population landscape via moment adjusted stochastic gradients modern statistical inference tasks often require iterative optimization methods to compute the solution convergence analysis from an optimization viewpoint informs us only how well the solution is approximated numerically but overlooks the sampling nature of the data in contrast recognizing the randomness in the data statisticians are keen to provide uncertainty quantification or confidence for the solution obtained by using iterative optimization methods the paper makes progress along this direction by introducing moment adjusted stochastic gradient descent a new stochastic optimization method for statistical inference we establish non asymptotic theory that characterizes the statistical distribution for certain iterative methods with optimization guarantees on the statistical front the theory allows for model misspecification with very mild conditions on the data for optimization the theory is flexible for both convex and non convex cases remarkably the moment adjusting idea motivated from error standardization in statistics achieves a similar effect to acceleration in first order optimization methods that are used to fit generalized linear models we also demonstrate this acceleration effect in the non convex setting through numerical experiments

the 2 1 steady hopf mode interaction in the two layer be nard problem abstract the interaction of hopf and steady modes with wavenumber ratio 2 1 is investigated for a critical situation two different immiscible liquids lie in layers between horizontal walls and are heated from below a situation with a pair of complex conjugate eigenvalues at wavenumber and a real eigenvalue at wavenumber 2 is at criticality weakly nonlinear amplitude equations are derived for the interaction of these oscillatory and steady modes the two modes generate a two parameter bifurcation the coefficients involved in the equations are determined numerically based on the physical parameters of the system at criticality three obvious equilibrium solutions of the amplitude equations are the steady solution the traveling waves and the mixed standing waves the eigenvalues governing the stability of these solutions are found explicitly numerical results and bifurcation diagrams are given for the critical situation the steady solution and the traveling wave solution are unstable there is a region of stability for the standing wave solution a new equilibrium solution the asymmetric mixed mode is found to be stable in a parameter range bifurcations from the standing wave solution and the asymmetric mixed mode are described

optimal stochastic approximation algorithms for strongly convex stochastic composite optimization ii shrinking procedures and optimal algorithms in this paper we study new stochastic approximation sa type algorithms namely the accelerated sa ac sa for solving strongly convex stochastic composite optimization sco problems specifically by introducing a domain shrinking procedure we significantly improve the large deviation results associated with the convergence rate of a nearly optimal ac sa algorithm presented by ghadimi and lan in siam j optim 22 2012 pp 1469 1492 moreover we introduce a multistage ac sa algorithm which possesses an optimal rate of convergence for solving strongly convex sco problems in terms of the dependence on not only the target accuracy but also a number of problem parameters and the selection of initial points to the best of our knowledge this is the first time that such an optimal method has been presented in the literature from our computational results these ac sa algorithms can substantially outperform the classical sa and some other sa type algorithms for solving certain classes of strongly

translational and nontranslational motion of perturbed turing patterns when the front and rear boundaries of a turing pattern with nonzero flux boundary conditions move synchronously several modes of motion of the pattern are found traveling boundaries can be obtained experimentally by illuminating a turing unstable system through a moving mask consisting of a single dark stripe with a light intensity sufficient to suppress pattern formation all structured moving patterns belong to two general types smooth translation at low mask velocity v x and nontranslational hopping at intermediate v x at high v x turing patterns are unable to form and an unstructured striped image of the mask is seen when the mask width exceeds the turing wavelength bistability between different types of moving patterns can occur as v x is varied

oscillatory convection in a rotating layer abstract the stability of travelling and standing rolls in oscillatory rotating convection is considered from the point of view of equivariant bifurcation theory to study stability with respect to oblique perturbations the problem is formulated on a rotating rhombic lattice all primary solution branches with maximal isotropy are determined together with their stability properties using a truncation of the most general equivariant vector field at third order in addition as many as seven branches of temporally periodic or quasiperiodic solutions with submaximal isotropy may be present instabilities analogous to the kuppers lortz instability of steady rolls in rotating convection are uncovered for both travelling rolls and for standing rolls these instabilities are triggered by the formation of a heteroclinic orbit connecting two travelling roll states or two sets of standing rolls with different wave vectors conditions are given for the formation and asymptotic stability of a structurally stable heteroclinic cycle connecting four travelling roll states the results are compared with recent studies of oscillatory patterns on a rotating square lattice and on a nonrotating rhombic lattice

primal dual optimization algorithms over riemannian manifolds an iteration complexity analysis in this paper we study nonconvex and nonsmooth multi block optimization over euclidean embedded smooth riemannian submanifolds with coupled linear constraints such optimization problems naturally arise from machine learning statistical learning compressive sensing image processing and tensor pca among others by utilizing the embedding structure we develop an admm like primal dual approach based on decoupled solvable subroutines such as linearized proximal mappings where the duality is with respect to the embedded euclidean spaces first we introduce the optimality conditions for the afore mentioned optimization models then the notion of epsilon stationary solutions is introduced as a result the main part of the paper is to show that the proposed algorithms possess an iteration complexity of o 1 epsilon 2 to reach an epsilon stationary solution for prohibitively large size tensor or machine learning models we present a sampling based stochastic algorithm with the same iteration complexity bound in expectation in case the subproblems are not analytically solvable a feasible curvilinear line search variant of the algorithm based on retraction operators is proposed finally we show specifically how the algorithms can be implemented to solve a variety of practical problems such as the np hard maximum bisection problem the ell q regularized sparse tensor principal component analysis and the community detection problem our preliminary numerical results show great potentials of the proposed methods

on birkhoff s theorem for doubly stochastic completely positive maps of matrix algebras abstract a study is made of the extreme points of the convex set of doubly stochastic completely positive maps of the matrix algebra m n if n 2 the extreme points are precisely the unitary maps but if n 3 there are nonunitary extreme points examples of which are exhibited a tilde operation is defined on the linear maps of m n and used to give an elementary derivation of a result of kummerer and maassen

fixed points for luders operations and commutators let b h be the algebra of all bounded linear operators on a complex hilbert space h and let tj j 1 b h be a sequence of operators such that j 1tj tj 1h in the weak operator topology let us define t b h b h by setting t y j 1t jytj for y b h we show that if x b h is an operator such that all the commutators xtj tjx as well as x itself are fixed points for t then xtj tjx 0 for all j 1 in the case when tj j 1 consists of commuting normal operators this shows that every fixed point for t commutes with all tj for j 1

block stochastic gradient iteration for convex and nonconvex optimization the stochastic gradient sg method can quickly solve a problem with a large number of components in the objective or a stochastic optimization problem to a moderate accuracy the block coordinate descent update bcd method on the other hand can quickly solve problems with multiple blocks of variables this paper introduces a method that combines the great features of sg and bcd for problems with many components in the objective and with multiple blocks of variables this paper proposes a block sg bsg method for both convex and nonconvex programs bsg generalizes sg by updating all the blocks of variables in the gauss seidel type updating the current block depends on the previously updated block in either a fixed or randomly shuffled order although bsg has slightly more work at each iteration it typically outperforms sg because of bsg s gauss seidel updates and larger step sizes the latter of which are determined by the smaller per block lipschitz constants the convergence of bsg is es

stochastic quasi newton methods for nonconvex stochastic optimization in this paper we study stochastic quasi newton methods for nonconvex stochastic optimization where we assume that noisy information about the gradients of the objective function is available via a

strong resonances of spatially distributed oscillators a laboratory to study patterns and defects abstract this paper describes phenomena which arise in a continuous system of coupled oscillators subject to a strongly resonant forcing we show that the temporal forcing induces a new type of instability leading to the formation of spatial patterns such as stripes hexagons and spatio temporal intermittent states reminiscent of chemical and magnetical systems we also report the existence of localized states where the instability is confined in the core of a defect leading to an increased mobility and diffusive behaviour of the defect

recursive trust region methods for multiscale nonlinear optimization a class of trust region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems like those arising in systems governed by partial differential equations the algorithms in this class make use of the discretization level as a means of speeding up the computation of the step this use is recursive leading to true multilevel multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial differential equations a simple algorithm of the class is then described and its numerical performance is shown to be numerically promising this observation then motivates a proof of global convergence to first order stationary points on the fine grid that is valid for all algorithms in the class

accelerating variance reduced stochastic gradient methods variance reduction is a crucial tool for improving the slow convergence of stochastic gradient descent only a few variance reduced methods however have yet been shown to directly benefit from nesterov s acceleration techniques to match the convergence rates of accelerated gradient methods such approaches rely on negative momentum a technique for further variance reduction that is generally specific to the svrg gradient estimator in this work we show for the first time that negative momentum is unnecessary for acceleration and develop a universal acceleration framework that allows all popular variance reduced methods to achieve accelerated convergence rates the constants appearing in these rates including their dependence on the number of functions n scale with the mean squared error and bias of the gradient estimator in a series of numerical experiments we demonstrate that versions of saga svrg sarah and sarge using our framework significantly outperform non accelerated versions and compare favourably with algorithms using negative momentum

wavenumber locking and pattern formation in spatially forced systems we study wavenumber locking and pattern formation resulting from weak spatially periodic one dimensional forcing of two dimensional systems we consider systems that produce stationary or traveling stripe patterns when unforced and apply forcing aligned with the stripes forcing at close to twice the pattern wavenumber selects stabilizes or creates resonant stripes locked at half the forcing wavenumber if the mismatch between the forcing and pattern wavenumber is high we find that the pattern still locks but develops a wave vector component perpendicular to the forcing direction and forms rectangular and oblique patterns when the unforced system supports traveling waves resonant rectangular patterns remain stationary but oblique patterns travel in a direction orthogonal to the traveling waves

patterns in hierarchical networks of neuronal oscillators with d3 z3 symmetry abstract in this paper we consider a hierarchically organized network composed of three interacting systems each of which consists of three coupled oscillators this hierarchical network is equivariant under the symmetry group d 3 z 3 using the lattice of isotropy subgroups we study the reduced equations restricted to invariant fixed point subspaces and prove that it is possible for the oscillator network to have 4 distinct equilibria or 45 distinct periodic solutions with maximal isotropy subgroups these are classified and their bifurcation directions are determined in terms of the quadratic coefficients and relevant quantities a center manifold reduction from the hierarchical network to the normal form equations is then performed in order to investigate the codimension two bifurcations using this reduction we find a great variety of equilibria periodic and quasi periodic oscillation patterns of maximal and submaximal symmetry which can be classified in a two level pattern hierarchy

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

symmetry breaking hopf bifurcation in anisotropic systems abstract symmetry breaking hopf bifurcation from a spatially uniform steady state of a spatially extended anisotropic system is considered this work is motivated by the experimental observation of a hopf bifurcation to oblique traveling rolls in electrohydrodynamic convection in planarly aligned nematic liquid crystals symmetry forces four traveling rolls to lose stability simultaneously four coupled complex ordinary differential equations describing the nonlinear interaction of the traveling rolls are analyzed using methods of equivariant bifurcation theory six branches of periodic solutions always bifurcate from the trivial state at the hopf bifurcation these correspond to traveling and standing wave patterns in an open region of coefficient space there is a primary bifurcation to a quasiperiodic standing wave solution the hopf bifurcation can also lead directly to an aperiodic attractor in the form of an asymptotically stable structurally stable heteroclinic cycle the theory is applied to a model for the transition from normal to oblique traveling rolls

additivity of the classical capacity of entanglement breaking quantum channels we show that for the tensor product of an entanglement breaking quantum channel with an arbitrary quantum channel both the minimum entropy of an output of the channel and the holevo schumacher westmoreland capacity are additive in addition for the tensor product of two arbitrary quantum channels we give a bound involving entanglement of formation for the amount of subadditivity for minimum entropy output or superadditivity for classical capacity that can occur

quantum channels arising from abstract harmonic analysis we present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups g beginning with a representation of the measure algebra m g we unify and elaborate on recent counter examples to a conjecture on the structure of fixed point subalgebras in infinite dimensions as well as present an application to the noiseless subsystems method of quantum error correction using a representation of the completely bounded fourier multiplier algebra mcba g we provide a new class of counter examples to the recently solved asymptotic quantum birkhoff conjecture along with a systematic method of producing the examples using a geometric representation of schur maps further properties of our channels including duality quantum capacity and entanglement preservation are discussed along with potential applications to additivity conjectures