ℋ index for continuous time stochastic systems with markov jump and multiplicative noise this paper investigates the problem of index for stochastic linear continuous time systems involving markov jump and multiplicative noise a set of generalized differential riccati equations gdres are presented and it is shown that the solvability of gdres is necessary and sufficient for the feasibility of an index larger than 0 our results extend the corresponding deterministic cases to stochastic systems furthermore the infinite horizon index problem for square systems is considered finally the effectiveness of the obtained results is tested by two examples

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

analytical approach to network inference investigating degree distribution when the network is reconstructed two types of errors can occur false positive and false negative errors about the presence or absence of links in this paper the influence of these two errors on the vertex degree distribution is analytically analyzed moreover an analytic formula of the density of the biased vertex degree distribution is found in the inverse problem we find a reliable procedure to reconstruct analytically the density of the vertex degree distribution of any network based on the inferred network and estimates for the false positive and false negative errors based on e g simulation studies

brief paper stochastic maximum principle in the mean field controls in buckdahn djehiche li and peng 2009 the authors obtained mean field backward stochastic differential equations bsdes in a natural way as a limit of some highly dimensional system of forward and backward sdes corresponding to a great number of particles or agents the objective of the present paper is to deepen the investigation of such mean field bsdes by studying their stochastic maximum principle this paper studies the stochastic maximum principle smp for mean field controls which is different from the classical ones this paper deduces an smp in integral form and it also gets under additional assumptions necessary conditions as well as sufficient conditions for the optimality of a control as an application this paper studies a linear quadratic stochastic control problem of mean field type

probabilistic analysis of mean field games the purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games with mean field interactions we implement the mean field game strategy developed analytically by lasry and lions in a purely probabilistic framework relying on tailor made forms of the stochastic maximum principle while we assume that the state dynamics are affine in the states and the controls and the costs are convex our assumptions on the nature of the dependence of all the coefficients upon the statistical distribution of the states of the individual players remains of a rather general nature our probabilistic approach calls for the solution of systems of forward backward stochastic differential equations of a mckean vlasov type for which no existence result is known and for which we prove existence and regularity of the corresponding value function finally we prove that a solution of the mean field game problem as formulated by lasry and lions does indeed provide appr

indefinite mean field stochastic linear quadratic optimal control this paper is concerned with the discrete time indefinite mean field linear quadratic optimal control problem the so called mean field type stochastic control problems refer to the problem of incorporating the means of the state variables into the state equations and cost functionals such as the mean variance portfolio selection problems a dynamic optimization problem is called to be nonseparable in the sense of dynamic programming if it is not decomposable by a stage wise backward recursion the classical dynamic programming based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies in this paper we show that both the well posedness and the solvability of the indefinite mean field linear quadratic problem are equivalent to the solvability of two coupled constrained generalized difference riccati equations and a constrained linear recursive equation we characterize the optimal control set completely and obtain a set of necessary and sufficient conditions on the mean variance portfolio selection problem the results established in this paper offer a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the mean variance type portfolio selection problems

unbiased risk estimates for singular value thresholding and spectral estimators in an increasing number of applications it is of interest to recover an approximately low rank data matrix from noisy observations this paper develops an unbiased risk estimate holding in a gaussian model for any spectral estimator obeying some mild regularity assumptions in particular we give an unbiased risk estimate formula for singular value thresholding svt a popular estimation strategy that applies a soft thresholding rule to the singular values of the noisy observations among other things our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems in particular we demonstrate the utility of the unbiased risk estimation for svt based denoising of real clinical cardiac mri series data we also give new results concerning the differentiability of certain matrix valued functions

on the set theoretical yang baxter equation we propose a general way of constructing set theoretical solutions of the yangbaxter equation we study the properties of the construction we also show that our construction includes the earlier ones given by weinstein xu and etingof schedlersoloviev

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

discrete time mean field stochastic linear quadratic optimal control problems ii infinite horizon case this paper first presents results on the equivalence of several notions of l2l2 stability for linear mean field stochastic difference equations with random initial value then it is shown that the optimal control of a mean field linear quadratic optimal control with an infinite time horizon uniquely exists and the optimal control can be expressed as a linear state feedback involving the state and its mean via the minimal nonnegative definite solution of two coupled algebraic riccati equations as a byproduct the open loop l2l2 stabilizability is proved to be equivalent to the closed loop l2l2 stabilizability moreover the minimal nonnegative definite solution the maximal solution the stabilizing solution of the algebraic riccati equations and their relations are carefully investigated specifically it is shown that the maximal solution is employed to construct the optimal control and value function to another infinite time horizon mean field linear quadratic optimal control in addition the maximal solution being the stabilizing solution is completely characterized by properties of the coefficients of the controlled system this enriches the existing theory about stochastic algebraic riccati equations finally the notion of exact detectability is introduced with its equivalent characterization of stochastic versions of the popov belevitch hautus criteria it is then shown that the minimal nonnegative definite solution is the stabilizing solution if and only if the uncontrolled system is exactly detectable

mean field games for large population multiagent systems with markov jump parameters in this paper distributed games for large population multiagent systems with random time varying parameters are investigated where the agents are coupled via their individual costs and the structure parameters are a family of independent markov chains with identical generators the cost function of each agent is a long run average tracking type functional with an unknown mean field coupling nonlinear term as reference signal to reduce the computational complexity the mean field approach is applied to construct distributed strategies the population statistics effect pse is used to approximate the average effect of all the agents and the distributed strategies are given through solving a markov jump tracking problem here the pse is a deterministic quantity and can be obtained by solving the stackelberg equilibrium of an auxiliary two player game it is shown that the closed loop system is uniformly stable and the distributed strategies are asymptotically optimal in the sense of nash equilibrium

linear quadratic n person and mean field games with ergodic cost we consider stochastic differential games with n players linear gaussian dynamics in arbitrary state space dimension and long time average cost with quadratic running cost admissible controls

linear quadratic optimal control problems for mean field stochastic differential equations linear quadratic optimal control problems are considered for mean field stochastic differential equations with deterministic coefficients by a variational method the optimality system is derived which is a linear mean field forward backward stochastic differential equation using a decoupling technique two riccati differential equations are obtained which are uniquely solvable under certain conditions then a feedback representation is obtained for the optimal control

stochastic linear quadratic control via primal dual semidefinite programming we study stochastic linear quadratic lq optimal control problems over an infinite time horizon allowing the cost matrices to be indefinite we develop a systematic approach based on semidefinite programming sdp a central issue is the stability of the feedback control we show that this can be effectively examined through the complementary duality of the sdp furthermore we establish several implication relations among the sdp complementary duality the generalized riccati equation and the optimality of the lq control problem based on these relations we propose a numerical procedure that provides a thorough treatment of the lq control problem via primal dual sdp it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution for the latter case we develop an epsilon approximation scheme that is asymptotically optimal

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

indefinite mean field stochastic linear quadratic optimal control from finite horizon to infinite horizon in this paper the finite horizon and the infinite horizon indefinite mean field stochastic linear quadratic optimal control problems are studied firstly the open loop optimal control and the closed loop optimal strategy for the finite horizon problem are introduced and their characterizations difference and relationship are thoroughly investigated the open loop optimal control can be defined for a fixed initial state whose existence is characterized via the solvability of a linear mean field forward backward stochastic difference equation with stationary conditions and a convexity condition on the other hand the existence of a closed loop optimal strategy is shown to be equivalent to any one of the following conditions the solvability of a couple of generalized difference riccati equations the finiteness of the value function for all the initial pairs and the existence of the open loop optimal control for all the initial pairs it is then proved that the solution of the generalized difference riccati equations converges to a solution of a couple of generalized algebraic riccati equations by studying another generalized algebraic riccati equation the existence of the maximal solution of the original ones is obtained together with the fact that the stabilizing solution is the maximal solution finally we show that the maximal solution is employed to express the optimal value of the infinite horizon indefinite mean field linear quadratic optimal control furthermore for the question whether the maximal solution is the stabilizing solution the necessary and the sufficient conditions are presented for several cases

bootstrap based regularization for low rank matrix estimation we develop a flexible framework for low rank matrix estimation that allows us to transform noise models into regularization schemes via a simple bootstrap algorithm effectively our procedure seeks an autoencoding basis for the observed matrix that is stable with respect to the specified noise model we call the resulting procedure a stable autoencoder in the simplest case with an isotropic noise model our method is equivalent to a classical singular value shrinkage estimator for non isotropic noise models e g poisson noise the method does not reduce to singular value shrinkage and instead yields new estimators that perform well in experiments moreover by iterating our stable autoencoding scheme we can automatically generate low rank estimates without specifying the target rank as a tuning parameter

linear matrix inequalities riccati equations and indefinite stochastic linear quadratic controls this paper deals with an optimal stochastic linear quadratic lq control problem in infinite time horizon where the diffusion term in dynamics depends on both the state and the control variables in contrast to the deterministic case we allow the control and state weighting matrices in the cost functional to be indefinite this leads to an indefinite lq problem which may still be well posed due to the deep nature of uncertainty involved the problem gives rise to a stochastic algebraic riccati equation sare which is however fundamentally different from the classical algebraic riccati equation as a result of the indefinite nature of the lq problem to analyze the sare we introduce linear matrix inequalities lmis whose feasibility is shown to be equivalent to the solvability of the sare moreover we develop a computational approach to the sare via a semi definite programming associated with the lmis finally numerical experiments are reported to illustrate the proposed approach

self inverse yang baxter operators from co algebra structures some attempts to find yang baxter operators in a systematic way have led to the theory of quantum groups see 4 many papers in the literature are devoted to the construction of them see 5 p 198 we adopt the terminology of 5 in giving some relevant examples the twist operators on modules over braided bialgebras and operators on comodules over cobraided bialgebras the frt construction showed that any yang baxter operator on a finite dimensional space v can be obtained from a cobraided bialgebra coacting on v other authors might use the term quasi triangular bialgebras instead of braided bialgebras see for example 4 7 all these constructions have proved of great utility in low dimensional topology we present here a new method to construct self inverse yang baxter operators thus to every co algebra structure on a space we associate an operator we describe the algebra respective coalgebra structures producing the same operator yang baxter operators associated to algebras are different from those associated to coalgebras except for a special case we give characterizations for operators which are associated to co algebra structures and construct the structures which produce them we give an

a decomposition theorem for square free unitary solutions of the quantum yang baxter equation abstract it is known that every skew polynomial ring with generating set x and binomial relations in the sense of gateva ivanova trans amer math soc 343 1994 203 is an artin schelter regular domain of global dimension x moreover every such ring gives rise to a non degenerate unitary set theoretical solution r x 2 x 2 of the quantum yang baxter equation which fixes the diagonal of x 2 gateva ivanova s conjecture talk at the international algebra conference miskolc hungary 1996 states that conversely every such solution r comes from a skew polynomial ring with binomial relations an equivalent conjecture duke math j 100 1999 169 says that the underlying set x is r decomposable we prove these conjectures and construct an indecomposable solution r with x which shows that an extension to infinite x is false

a novel coupling of noise reduction algorithms for particle flow simulations proper orthogonal decomposition pod and its extension based on time windows have been shown to greatly improve the effectiveness of recovering smooth ensemble solutions from noisy particle data however to successfully de noise any molecular system a large number of measurements still need to be provided in order to achieve a better efficiency in processing time dependent fields we have combined pod with a well established signal processing technique wavelet based thresholding in this novel hybrid procedure the wavelet filtering is applied within the pod domain and referred to as wavinpod the algorithm exhibits promising results when applied to both synthetically generated signals and particle data in this work the simulations compare the performance of our new approach with standard pod or wavelet analysis in extracting smooth profiles from noisy velocity and density fields numerical examples include molecular dynamics and dissipative particle dynamics simulations of unsteady force and shear driven liquid flows as well as phase separation phenomenon simulation results confirm that wavinpod preserves the dimensionality reduction obtained using pod while improving its filtering properties through the sparse representation of data in wavelet basis this paper shows that wavinpod outperforms the other estimators for both synthetically generated signals and particle based measurements achieving a higher signal to noise ratio from a smaller number of samples the new filtering methodology offers significant computational savings particularly for multi scale applications seeking to couple continuum informations with atomistic models it is the first time that a rigorous analysis has compared de noising techniques for particle based fluid simulations

tensor svd statistical and computational limits in this paper we propose a general framework for tensor singular value decomposition tensor singular value decomposition svd which focuses on the methodology and theory for extracting the hidden low rank structure from high dimensional tensor data comprehensive results are developed on both the statistical and computational limits for tensor svd this problem exhibits three different phases according to the signal to noise ratio snr in particular with strong snr we show that the classical higher order orthogonal iteration achieves the minimax optimal rate of convergence in estimation with weak snr the information theoretical lower bound implies that it is impossible to have consistent estimation in general with moderate snr we show that the non convex maximum likelihood estimation provides optimal solution but with np hard computational cost moreover under the hardness hypothesis of hypergraphic planted clique detection there are no polynomial time algorithms performing consistently in general

low complexity algorithms for low rank clutter parameters estimation in radar systems this paper addresses the problem of the clutter subspace projector estimation in the context of a disturbance composed of a low rank heterogeneous compound gaussian clutter and white gaussian noise in such a context adaptive processing based on an estimated orthogonal projector onto the clutter subspace instead of an estimated covariance matrix requires less samples than classical methods the clutter subspace estimate is usually derived from the eigenvalue decomposition of a covariance matrix estimate however it has been previously shown that a direct maximum likelihood estimator of the clutter subspace projector can be obtained for the considered context in this paper we derive two algorithms based on the block majorization minimization framework to reach this estimator these algorithms are shown to be computationally faster than the state of the art with guaranteed convergence finally the performance of the related estimators is illustrated on realistic space time adaptive processing for airborne radar simulations

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

discrete time mean field stochastic linear quadratic optimal control problems this paper firstly presents necessary and sufficient conditions for the solvability of discrete time mean field stochastic linear quadratic optimal control problems secondly the optimal control within a class of linear feedback controls is investigated using a matrix dynamical optimization method thirdly by introducing several sequences of bounded linear operators the problem is formulated as an operator stochastic linear quadratic optimal control problem by the kernel range decomposition representation of the expectation operator and its pseudo inverse the optimal control is derived using solutions to two algebraic riccati difference equations finally by completing the square the two riccati equations and the optimal control are also obtained

necessary sufficient conditions for pareto optimality in finite horizon mean field type stochastic differential game abstract this paper is concerned with necessary and sufficient conditions for the existence of pareto solutions in finite horizon mean field type stochastic cooperative differential game based on the equivalent characterization of pareto optimality the problem is transformed into a set of constrained mean field type stochastic optimal control problems with a special structure utilizing the mean field type stochastic minimum principle the necessary conditions are put forward under certain convex assumptions it is shown that the necessary conditions are also sufficient ones next the indefinite linear quadratic lq case is studied it is pointed out that the solvability of two related generalized differential riccati equations gdres provides a sufficient condition under which pareto efficient strategies are equivalent to weighted sum optimal controls in addition all pareto solutions are obtained based on the solutions of two generalized differential lyapunov equations gdles at last an example sheds light on the effectiveness of the theoretical results

semi braces and the yang baxter equation abstract in this paper we obtain new solutions of the yang baxter equation that are left non degenerate through left semi braces a generalization of braces introduced by rump in order to provide new solutions we introduce the asymmetric product of left semi braces a generalization of the semidirect product of braces that allows us to produce several examples of left semi braces

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

asymptotically optimal decentralized control for large population stochastic multiagent systems the interaction of interest coupled decision makers and the uncertainty of individual behavior are prominent characteristics of multiagent systems mas how to break through the framework of conventional control theory which aims at single decision maker and single decision objective and to extend the methodology and tools in the stochastic adaptive control theory to analyze mas are of great significance in this paper a preliminary exploration is made in this direction and the decentralized control problem is considered for large population stochastic mas with coupled cost functions different from the deterministic discounted costs in the existing differential game models a time averaged stochastic cost function is adopted for each agent the decentralized control law is constructed based on the state aggregation method and tracking like quadratic optimal control by using probability limit theory the stability and optimality of the closed loop system are analyzed the main contributions of this paper include the following points 1 the concepts of asymptotic nash equilibrium in probability and almost surely respectively are introduced and the relationship between these concepts is illuminated which provide necessary tools for analyzing the optimality of the decentralized control laws 2 the closed loop system is shown to be almost surely uniformly stable and bounded independently of the number of agents n 3 the population state average psa is shown to converge to the infinite population mean ipm trajectory in the sense of both l2 norm and time average almost surely as n increases to infinity 4 the decentralized control law is designed and shown to be almost surely asymptotically optimal the cost of each agent based on local measurements converges to that based on global measurements almost surely as n increases to infinity

comparative numerical analysis using reduced order modeling strategies for nonlinear large scale systems we perform a comparative analysis using three reduced order strategies missing point estimation mpe method gappy pod method and discrete empirical interpolation method deim applied to a biological model describing the spatio temporal dynamics of a predator prey community the comparative study is focused on the efficiency of the reduced order approximations and the complexity reduction of the nonlinear terms different variants are discussed related to the projection based model reduction framework combined with selective spatial sampling to efficiently perform the online computations numerical results are presented

joint covariance estimation with mutual linear structure we consider the problem of joint estimation of structured covariance matrices assuming the structure is unknown estimation is achieved using heterogeneous training sets namely given groups of measurements coming from centered populations with different covariances our aim is to determine the mutual structure of these covariance matrices and estimate them supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices we develop a new efficient algorithm discovering the structure and using it to improve the estimation our technique is based on the application of principal component analysis in the matrix space we also derive an upper performance bound of the proposed algorithm in the gaussian scenario and compare it with the cramer rao lower bound numerical simulations are presented to illustrate the performance benefits of the proposed method

solutions of the yang baxter equation associated with a left brace abstract given a left brace b a method is given to construct explicitly all the non degenerate involutive set theoretic solutions x r of the yang baxter equation such that the associated permutation group g x r is isomorphic as a left brace to b this method depends entirely on the brace structure of b

optimal shrinkage of singular values we consider the recovery of low rank matrices from noisy data by shrinkage of singular values in which a single univariate nonlinearity is applied to each of the empirical singular values we adopt an asymptotic framework in which the matrix size is much larger than the rank of the signal matrix to be recovered and the signal to noise ratio of the low rank piece stays constant for a variety of loss functions including mean square error mse square frobenius norm the nuclear norm loss and the operator norm loss we show that in this framework there is a well defined asymptotic loss that we evaluate precisely in each case in fact each of the loss functions we study admits a unique admissible shrinkage nonlinearity dominating all other nonlinearities we provide a general method for evaluating these optimal nonlinearities and demonstrate our framework by working out simple explicit formulas for the optimal nonlinearities in the frobenius nuclear and operator norm cases for example for a square low rank n by n matrix observed in white noise with level sigma the optimal nonlinearity for mse loss simply shrinks each data singular value y to sqrt y 2 4n sigma 2 or to 0 if y this optimal nonlinearity guarantees an asymptotic mse of 2nr sigma 2 which compares favorably with optimally tuned hard thresholding and optimally tuned soft thresholding providing guarantees of 3nr sigma 2 and 6nr sigma 2 respectively our general method also allows one to evaluate optimal shrinkers numerically to arbitrary precision as an example we compute optimal shrinkers for the schatten p norm loss for any p 0

social optima in mean field lqg control centralized and decentralized strategies we study a class of linear quadratic gaussian lqg control problems with n decision makers where the basic objective is to minimize a social cost as the sum of n individual costs containing mean field coupling the exact socially optimal solution determining a particular pareto optimum requires centralized information for each agent and has high implementational complexity as an alternative we subsequently exploit a mean field structure in the centralized optimal control problem to develop decentralized cooperative optimization so that each agent only uses its own state and a function which may be computed offline the resulting set of strategies asymptotically achieves the social optimum as n a key feature in this scheme is to let each agent optimize a new cost as the sum of its own cost and another component capturing its social impact on all other agents we also discuss the relationship between the decentralized cooperative solution and the so called nash certainty equivalence based solution presented in previous work on mean field lqg games

braces and the yang baxter equation several aspects of relations between braces and non degenerate involutive set theoretic solutions of the yang baxter equation are discussed and many consequences are derived in particular for each positive integer n a finite square free multipermutation solution of the yang baxter equation with multipermutation level n and an abelian involutive yang baxter group is constructed this answers a problem of gateva ivanova and cameron it is proved that finite non degenerate involutive set theoretic solutions of the yang baxter equation whose associated involutive yang baxter group is abelian are multipermutation solutions earlier the authors proved this with the additional square free hypothesis on the solutions it is also proved that finite square free non degenerate involutive set theoretic solutions associated to a left brace are multipermutation solutions

regular subgroups of the affine group and asymmetric product of radical braces abstract in this paper we introduce the asymmetric product of radical braces a construction which extends the semidirect product of radical braces this new construction allows to obtain rather systematic constructions of regular subgroups of the affine group and in particular our approach allows to put in a more general context the regular subgroups constructed by heged s 2000 9

nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries we demonstrate the synthesis of sparse sampling and dimensionality reduction to characterize and model nonlinear dynamical systems over a range of bifurcation parameters first we construct modal libraries using the classical proper orthogonal decomposition in order to expose the dominant low rank coherent structures here libraries of the nonlinear terms are also constructed in order to take advantage of the discrete empirical interpolation method and projection that allows for the approximation of nonlinear terms from a sparse number of grid points the selected grid points are shown to be effective sensing and measurement locations for characterizing the underlying dynamics stability and bifurcations of nonlinear dynamical systems the use of empirical interpolation points and sparse representation facilitates a family of local reduced order models for each physical regime rather than a higher order global model which has the benefit of physical interpretability of energy transfer between coherent structures the method advocated also allows for orders of magnitude improvement in computational speed and memory requirements to illustrate the method the discrete interpolation points and nonlinear modal libraries are used for sparse representation in order to classify and reconstruct the dynamic bifurcation regimes in the complex ginzburg landau equation it is also shown that point measurements of the nonlinearity are more effective than linear measurements when sensor noise is present

a deterministic theory of low rank matrix completion the problem of completing a large low rank matrix using a subset of revealed entries has received much attention in the last ten years the main result of this paper gives a necessary and sufficient condition stated in the language of graph limit theory for a sequence of matrix completion problems with arbitrary missing patterns to be asymptotically solvable it is then shown that a small modification of the candes recht nuclear norm minimization algorithm provides the required asymptotic solution whenever the sequence of problems is asymptotically solvable the theory is fully deterministic with no assumption of randomness a number of open questions are listed

indefinite stochastic linear quadratic control and generalized differential riccati equation a stochastic linear quadratic lq control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite indefinite stochastic lq theory has been extensively developed and has found interesting applications in finance however there remains an outstanding open problem which is to identify an appropriate riccati type equation whose solvability is it equivalent to the solvability of the indefinite stochastic lq problem this paper solves this open problem for lq control in a finite time horizon a new type of differential riccati equation called the generalized differential riccati equation is introduced which involves algebraic equality inequality constraints and a matrix pseudoinverse it is then shown that the solvability of the generalized riccati equation is not only sufficient but also necessary for the well posedness of the indefinite lq problem and the existence of optimal feedback open loop controls moreover all of the optimal controls can be identified via the solution to the riccati equation an example is presented to illustrate the theory developed

on the limit of the largest eigenvalue of the large dimensional sample covariance matrix in this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity the above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products sp matrix exist

the optimal hard threshold for singular values is 4 sqrt 3 we consider recovery of low rank matrices from noisy data by hard thresholding of singular values in which empirical singular values below a threshold are set to 0 we study the asymptotic mean squared error amse in a framework where the matrix size is large compared with the rank of the matrix to be recovered and the signal to noise ratio of the low rank piece stays constant the amse optimal choice of hard threshold in the case of n by n matrix in white noise of level is simply 4 3 n 2 309 n when is known or simply 2 858 y med when is unknown where y med is the median empirical singular value for nonsquare m by n matrices with m n the thresholding coefficients 4 3 and 2 858 are replaced with different provided constants that depend on m n asymptotically this thresholding rule adapts to unknown rank and unknown noise level in an optimal manner it is always better than hard thresholding at any other value and is always better than ideal truncated singular value decomposition tsvd which truncates at the true rank of the low rank matrix we are trying to recover hard thresholding at the recommended value to recover an n by n matrix of rank r guarantees an amse at most 3 nr 2 in comparison the guarantees provided by tsvd optimally tuned singular value soft thresholding and the best guarantee achievable by any shrinkage of the data singular values are 5 nr 2 6 nr 2 and 2 nr 2 respectively the recommended value for hard threshold also offers among hard thresholds the best possible amse guarantees for recovering matrices with bounded nuclear norm empirical evidence suggests that performance improvement over tsvd and other popular shrinkage rules can be substantial for different noise distributions even in relatively small n

linear quadratic stochastic differential games open loop and closed loop saddle points in this paper we consider a linear quadratic stochastic two person zero sum differential game the controls for both players are allowed to appear in both drift and diffusion of the state equation the weighting matrices in the performance functional are not assumed to be definite non singular the existence of an open loop saddle point is characterized by the existence of an adapted solution to a linear forward backward stochastic differential equation with constraints together with a convexity concavity condition and the existence of a closed loop saddle point is characterized by the existence of a regular solution to a riccati differential equation it turns out that there is a significant difference between open loop and closed loop saddle points also it is found that there is an essential feature that prevents a linear quadratic optimal control problem from being a special case of linear quadratic two person zero sum differential games

risk sensitive mean field games in this paper we study a class of risk sensitive mean field stochastic differential games we show that under appropriate regularity conditions the mean field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a hamilton jacobi bellman hjb equation with an additional quadratic term we provide an explicit solution of the mean field best response when the instantaneous cost functions are log quadratic and the state dynamics are affine in the control an equivalent mean field risk neutral problem is formulated and the corresponding mean field equilibria are characterized in terms of backward forward macroscopic mckean vlasov equations fokker planck kolmogorov equations and hjb equations we provide numerical examples on the mean field behavior to illustrate both linear and mckean vlasov dynamics

counterexample to a conjecture about braces abstract we find an example of a finite solvable group in fact a finite p group in which is not possible to define a left brace structure or equivalently which is not an iyb group this answers a question posed by cedo jespers and del rio related to the yang baxter equation our argument is an improvement of an argument of rump using results about hopf galois extensions and lsa structures we explain explicitly the relation between these two areas of mathematics and brace theory hoping that it will be useful in the future

deterministic parallel analysis an improved method for selecting factors and principal components factor analysis and principal component analysis are used in many application areas the first step choosing the number of components remains a serious challenge our work proposes improved methods for this important problem one of the most popular state of the art methods is parallel analysis pa which compares the observed factor strengths with simulated strengths under a noise only model the paper proposes improvements to pa we first derandomize it proposing deterministic pa which is faster and more reproducible than pa both pa and deterministic pa are prone to a shadowing phenomenon in which a strong factor makes it difficult to detect smaller but more interesting factors we propose deflation to counter shadowing we also propose to raise the decision threshold to improve estimation accuracy we prove several consistency results for our methods and test them in simulations we also illustrate our methods on data from the human genome diversity project where they significantly improve the accuracy

stochastic maximum principle for mean field type optimal control under partial information this technical note is concerned with a partially observed optimal control problem whose novel feature is that the cost functional is of mean field type hence determining the optimal control is time inconsistent in the sense that bellman s dynamic programming principle does not hold a maximum principle is established using girsanov s theorem and convex variation some nonlinear filtering results for backward stochastic differential equations bsdes are developed by expressing the solutions of the bsdes as some ito s processes an illustrative example is demonstrated in terms of the maximum principle and the filtering

semigroups ofi type abstract assume thatsis a semigroup generated by x1 xn and let u be the multiplicative free commutative semigroup generated by u1 un we say thatsis ofi typeif there is a bijectivev u ssuch that for alla u v u1a v una x1v a xnv a this condition appeared naturally in the work on sklyanin algebras by john tate and the second author in this paper we show that the condition for a semigroup to be ofi type is related to various other mathematical notions found in the literature in particular we show that semigroups ofi type appear in the study of the set theoretic solutions of the yang baxter equation in the theory of bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author

solvability and asymptotic behavior of generalized riccati equations arising in indefinite stochastic lq controls the optimal control problem in a finite time horizon with an indefinite quadratic cost function for a linear system subject to multiplicative noise on both the state and control can be solved via a constrained matrix differential riccati equation in this paper we provide general necessary and sufficient conditions for the solvability of this generalized differential riccati equation furthermore its asymptotic behavior is investigated along with its connection to the generalized algebraic riccati equation associated with the linear quadratic control problem in finite time horizon examples are presented to illustrate the results established

large population cost coupled lqg problems with nonuniform agents individual mass behavior and decentralized varepsilon nash equilibria we consider linear quadratic gaussian lqg games in large population systems where the agents evolve according to nonuniform dynamics and are coupled via their individual costs a state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an epsiv nash equilibrium property a stability property of the mass behavior is established and the effect of inaccurate population statistics on an isolated agent is also analyzed by variational techniques

braces radical rings and the quantum yang baxter equation abstract non degenerate cycle sets are equivalent to non degenerate unitary set theoretical solutions of the quantum yang baxter equation we embed such cycle sets into generalized radical rings braces and study their interaction in this context we establish a galois theory between ideals of braces and quotient cycle sets our main result determines the relationship between two square free cycle sets operating transitively on each other