generalized blocks for symmetric groups we study via character theoretic methods an l analogue of the modular representation theory of the symmetric group for an arbitrary integer l 2 we find that many of the invariants of the usual block theory ie when l is prime generalize in a natural fashion to this new context

admm based fast algorithm for multi group multicast beamforming in large scale wireless systems multi group multicast beamforming in wireless systems with large antenna arrays and massive audience is investigated in this paper multicast beamforming design is a well known non convex quadratically constrained quadratic programming qcqp problem a conventional method to tackle this problem is to approximate it as a semi definite programming problem via semi definite relaxation whose performance however deteriorates considerably as the number of per group users goes large a recent attempt is to apply convex concave procedure ccp to find a stationary solution by treating it as a difference of convex programming problem whose complexity however increases dramatically as the problem size increases in this paper we propose a low complexity high performance algorithm for multi group multicast beamforming design in large scale wireless systems by leveraging the alternating direction method of multipliers admm together with ccp in specific the original non convex qcqp problem is first approximated as a sequence of convex subproblems via ccp each convex subproblem is then reformulated as a novel admm form our admm reformulation enables that each updating step is performed by solving multiple small size subproblems with closed form solutions in parallel numerical results show that our fast algorithm maintains the same favorable performance as state of the art algorithms but reduces the complexity by orders of magnitude

an error bound for a discrete reduced order model of a linear multivariable system the design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction in order for the design based on the order reduction to include a guarantee of stability it is sufficient to have a bound on the model error previous work has provided such a bound for continuous time systems for algorithms based on balancing in this note an linftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing

consensus admm for general quadratically constrained quadratic programming nonconvex quadratically constrained quadratic programming qcqp problems have numerous applications in signal processing machine learning and wireless communications albeit the general qcqp is np hard and several interesting special cases are np hard as well this paper proposes a new algorithm for general qcqp the problem is first reformulated in consensus optimization form to which the alternating direction method of multipliers can be applied the reformulation is done in such a way that each of the subproblems is a qcqp with only one constraint qcqp 1 which is efficiently solvable irrespective of non convexity the core components are carefully designed to make the overall algorithm more scalable including efficient methods for solving qcqp 1 memory efficient implementation parallel distributed implementation and smart initialization the proposed algorithm is then tested in two applications multicast beamforming and phase retrieval the results indicate superior performance over prior state of the art methods

parallel and distributed methods for constrained nonconvex optimization part i theory in this two part paper we propose a general algorithmic framework for the minimization of a nonconvex smooth function subject to nonconvex smooth constraints and also consider extensions to some structured nonsmooth problems the algorithm solves a sequence of separable strongly convex problems and maintains feasibility at each iteration convergence to a stationary solution of the original nonconvex optimization is established our framework is very general and flexible and unifies several existing successive convex approximation sca based algorithms more importantly and differently from current sca approaches it naturally leads to distributed and parallelizable implementations for a large class of nonconvex problems this part i is devoted to the description of the framework in its generality in part ii we customize our general methods to several multiagent optimization problems in communications networking and machine learning the result is a new class of centralized and distributed algorithms that compare favorably to existing ad hoc centralized schemes

fast approximation algorithms for a class of non convex qcqp problems using first order methods a number of important problems in engineering can be formulated as non convex quadratically constrained quadratic programming qcqp the general qcqp problem is np hard in this paper we consider a class of non convex qcqp problems that are expressible as the maximization of the point wise minimum of homogeneous convex quadratics over a simple convex set existing approximation strategies for such problems are generally incapable of achieving favorable performance complexity tradeoffs they are either characterized by good performance but high complexity and lack of scalability or low complexity but relatively inferior performance this paper focuses on bridging this gap by developing high performance low complexity successive non smooth convex approximation algorithms for problems in this class exploiting the structure inherent in each subproblem specialized first order methods are used to efficiently compute solutions multicast beamforming is considered as an application example to showcase the effectiveness of the proposed algorithms which achieve a very favorable performance complexity tradeoff relative to the existing state of the art

elementary divisors of the shapovalov form on the basic representation of kac moody algebras we calculate the invariant factors of the standard lattice vz inside the basic representation of untwisted kac moody algebras of ade type this in turn gives the invariant factors of cartan matrices for symmetric groups in prime characteristic as well as for the iwahori hecke algebra at roots of unity

balanced truncation of linear time varying systems in this paper balanced truncation of linear time varying systems is studied in discrete and continuous time based on relatively basic calculations with time varying lyapunov equations inequalities we are able to derive both upper and lower error bounds for the truncated models these results generalize well known time invariant formulas the case of time varying state dimension is considered input output stability of all truncated balanced realizations is also proven the method is finally successfully applied to a high order model

model reduction by nice selections for linear switched systems a moment matching method for model reduction of linear switched systems lsss is presented the method can be seen as a non trivial extension of the krylov subspace methods for linear time invariant lti systems the procedure is based on the so called nice selections which represent a choice of basis in the reachability or observability space of the lss the framework can also be used for exact matching of the input output behavior of an lss with a reduced order lss for a specific switching sequence conditions for applicability of the method for model reduction are stated and finally the results are illustrated on numerical examples

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

an efficient global algorithm for single group multicast beamforming consider the single group multicast beamforming problem where multiple users receive the same data stream simultaneously from a single transmitter the problem is np hard and all existing algorithms for the problem either find suboptimal approximate or local stationary solutions in this paper we propose an efficient branch and bound algorithm for the problem that is guaranteed to find its global solution to the best of our knowledge our proposed algorithm is the first tailored global algorithm for the single group multicast beamforming problem simulation results show that our proposed algorithm is computationally efficient albeit its theoretical worst case iteration complexity is exponential with respect to the number of receivers and it significantly outperforms a state of the art general purpose global optimization solver called baron our proposed algorithm provides an important benchmark for performance evaluation of existing algorithms for the same problem by using it as the benchmark we show that two state of the art algorithms semidefinite relaxation algorithm and successive linear approximation algorithm work well when the problem dimension i e the number of antennas at the transmitter and the number of receivers is small but their performance deteriorates quickly as the problem dimension increases

on graded cartan invariants of symmetric groups and hecke algebras we consider graded cartan matrices of the symmetric groups and the iwahori hecke algebras of type a at roots of unity these matrices are mathbb z v v 1 valued and may also be interpreted as gram matrices of the shapovalov form on sums of weight spaces of a basic representation of an affine quantum group we present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring mathbb z v v 1 this proves and generalizes a conjecture of ando suzuki yamada on the invariants of these matrices over mathbb q v v 1 and also generalizes the first author s recent proof of the kulshammer olsson robinson conjecture over mathbb z

multigroup multicast beamformer design for miso ofdm with antenna selection we study the problem of designing transmit beamformers for a multigroup multicasting by considering a multiple input single output orthogonal frequency division multiplexing framework the design objective involves either minimizing the total transmit power for certain guaranteed quality of service or maximizing the minimum achievable rate among the users for a given transmit power budget the problem of interest can be formulated as a nonconvex quadratically constrained quadratic programming qcqp for which the prevailing semidefinite relaxation sdr technique is inefficient for at least two reasons at first the relaxed problem cannot be reformulated as a semidefinite programming second even if the relaxed problem is solved the so called randomization procedure should be used to generate a feasible solution to the original qcqp which is difficult to derive for the considered problem to overcome these shortcomings we adopt successive convex approximation framework to find multicast beamformers directly the proposed method not only avoids the need of randomization search but also incurs less computational complexity compared to an sdr approach in addition we also extend multicasting beamformer design problem with an additional constraint on the number of active elements which is particularly relevant when the number of antennas is larger than that of radio frequency chains numerical results are used to demonstrate the superior performance of our proposed methods over the existing solutions

the one bit null space learning algorithm and its convergence this paper proposes a new algorithm for mimo cognitive radio secondary users su to learn the null space of the interference channel to the primary user pu without burdening the pu with any knowledge or explicit cooperation with the su the knowledge of this null space enables the su to transmit in the same band simultaneously with the pu by utilizing separate spatial dimensions than the pu specifically the su transmits in the null space of the interference channel to the pu we present a new algorithm called the one bit null space learning algorithm obnsla in which the su learns the pu s null space by observing a binary function that indicates whether the interference it inflicts on the pu has increased or decreased in comparison to the su s previous transmitted signal this function is obtained by listening to the pu transmitted signal or control channel and extracting information from it about whether the pu s signal to interference plus noise power ratio sinr has increased or decreased the obnsla is shown to have a linear convergence rate and an asymptotic quadratic convergence rate finally we derive bounds on the interference that the su inflicts on the pu as a function of a parameter determined by the su this lets the su control the maximum level of interference which enables it to protect the pu completely blindly with minimum complexity the asymptotic analysis and the derived bounds also apply to the recently proposed blind null space learning algorithm

optimal and successive approaches to signal design for multiple antenna physical layer multicasting in modern wireless communications systems that send a common information stream called multicasting systems are widely needed for distributing content such as television or radio in this paper the multiple antenna physical layer multicasting channel is considered in which a common message is simultaneously transmitted to multiple single antenna users to achieve the capacity of this multicasting set up the covariance matrix of the transmit signal vector needs to be determined to maximize the smallest maximum achievable rate among all the users in this paper we develop a simple successive algorithm to determine the covariance matrix of the signal vector under the assumption that channel state information csi is perfectly available at the transmitter we first characterize properties of the capacity achieving covariance matrix and derive a closed form expression for the matrix in the two user case we then derive a closed form expression for the rate maximizing beamformer in the case of two users and propose a successive beamforming algorithm that generates the beamforming vector with low computational complexity in the proposed precoding design scheme the covariance matrix is successively constructed by orthogonalizing the subspace spanned by each user s channel vector until the maximum number of recursions which is the same as the minimum of the number of transmit antennas and the number of users is reached the achievable rate of the proposed scheme is compared with the capacity of optimal transmission transmit beamforming antenna subset selection and open loop transmission which does not require csi at the transmitter as the number of users and or antennas grows large it is shown that the average achievable rate of the proposed scheme achieves the same average achievable rate scaling as the true capacity while reducing the computational complexity

robust multicast beamforming for spectrum sharing based cognitive radios we consider a robust downlink beamforming optimization problem for secondary multicast transmission in a multiple input multiple output mimo spectrum sharing cognitive radio cr network the minimization of transmit power is formulated subject to both quality of service qos constraints on the secondary receivers and interference temperature constraints on the primary users under the assumption of imperfect channel state information csi the problem is a nonconvex quadratically constrained quadratic program qcqp and in general it is hard to achieve the global optimality as a compromise we present two randomized approximation algorithms for the problem via convex optimization techniques apart from the general setting of the robust beamforming problem we identify one interesting special case the robust problem of which can be solved efficiently simulation results are presented to demonstrate the performance gains of the proposed algorithms over an existing robust design

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

balanced truncation for discrete time markov jump linear systems this technical note investigates the model reduction problem for mean square stable discrete time markov jump linear systems for this class of systems a balanced truncation algorithm is developed the reduced order model is suboptimal however the approximation error which is captured by means of the stochastic gain is bounded from above by twice the sum of singular numbers associated to the truncated states of each mode such a result allows rigorous simplification of the dynamics of each mode in an independent manner with respect to a metric which is relevant from a robust control point of view

dissipativity based asynchronous filtering for periodic markov jump systems abstract this paper is devoted to developing a periodic and asynchronous filter for periodic markov jump systems a periodic and asynchronous filter means that the filter parameters are s periodic for a given operating mode and its operating modes are unnecessary to be identical but form a periodic hidden markov model together with the modes of the plant a sufficient condition is proposed to guarantee that the filtering error system is exponentially mean square stable and strictly q s r dissipative to realize this filter a slack matrix is introduced to deal with the nonlinearity and finsler s lemma is applied such that the decision variables are reduced finally a numerical simulation provides verification for the effectiveness of our designed filter

generalized perfect isometries in some groups of lie rank one abstract in a recent paper b kulshammer j b olsson and g r robinson introduced notions of generalized blocks and generalized perfect isometries and studied them in the case of the symmetric group s n it is the purpose of this paper to investigate some families of groups of lie rank one exhibiting generalized perfect isometries where there are none in the sense given by m broue and thus proving a weaker analogue of one of broue s conjectures in these cases

h model reduction of markovian jump linear systems abstract in this paper the h model reduction problem for linear systems that possess randomly jumping parameters is studied the development includes both the continuous and discrete cases it is shown that the reduced order models exist if a set of matrix inequalities is feasible an effective iterative algorithm involving linear matrix inequalities is suggested to solve the matrix inequalities characterizing the model reduction solutions using the numerical solutions of the matrix inequalities the reduced order models can be obtained an example is given to illustrate the proposed model reduction method

parallel and distributed methods for constrained nonconvex optimization part ii applications in communications and machine learning in part i of this paper we proposed and analyzed a novel algorithmic framework for the minimization of a nonconvex objective function subject to nonconvex constraints based on inner convex approximations this part ii is devoted to the nontrivial application of the framework to the following relevant large scale problems ranging from communications to machine learning 1 generalizations of the rate profile maximization in mimo interference broadcast networks 2 the max min fair multicast multigroup beamforming problem in a multicell environment and 3 a general nonconvex constrained bi criteria formulation for k sparse variable selection in statistical learning the two criteria are a nonconvex loss objective function measuring the fitness of the model to data and the latter is a nonconvex sparsity inducing constraint in the general form of difference of convex dc functions which allows to accomodate in a unified fashion convex and nonconvex surrogates of the ell 0 function the proposed algorithms outperform current state of the art schemes for 1 3 both theoretically and numerically for instance they are the first distributed schemes for the class of problems 1 and 2 and they also lead to subproblems enjoying closed form solutions

high performance adaptive algorithms for single group multicast beamforming the single group multicast beamforming problem is np hard and the available approximations do not always achieve favorable performance complexity tradeoffs this paper introduces a new class of adaptive multicast beamforming algorithms that features guaranteed convergence and state of the art performance at low complexity each update takes a step in the direction of an inverse signal to noise ratio snr weighted linear combination of the snr gradient vectors of all the users convergence of this update to a karush kuhn tucker kkt point of proportionally fair beamforming is established simulations show that the proposed approach can enable better performance than the prior state of art in terms of multicast rate at considerably lower complexity this reveals an interesting link between max min fair and proportionally fair multicast beamforming formulations for cases where there is no initial channel state information at the transmitter an online algorithm is developed that simultaneously learns the user channel correlation matrices and adapts the beamforming vector to maximize the minimum long term average snr among the users using only periodic binary snr feedback from each receiver the online algorithm uses the analytic center cutting plane method to quickly learn the user correlation matrices with limited signaling overhead

generalised cartan invariants of symmetric groups kulshammer olsson and robinson developed an analogue of modular representation theory of symmetric groups where is not necessarily a prime they gave a conjectural combinatorial description for invariant factors of the cartan matrix in this context we confirm their conjecture by proving a more precise blockwise version of the conjecture due to bessenrodt and hill

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

an improved error estimate for reduced order models of discrete time systems the authors derive under weaker conditions a discrete time counterpart of k glover s 1984 error estimate for reduced order models following the same lines a simple proof of glover s result for continuous time systems can be given

balanced truncation for a class of stochastic jump linear systems and model reduction for hidden markov models this paper develops a generalization of the balanced truncation algorithm applicable to a class of discrete time stochastic jump linear systems the approximation error which is captured by means of the stochastic l 2 gain is bounded from above by twice the sum of singular numbers associated to the truncated states similar to the case of linear time invariant systems a two step model reduction algorithm for hidden markov models is also developed the first step relies on the aforementioned balanced truncation algorithm due to a topological equivalence established between hidden markov models and a subclass of stochastic jump linear systems in a second step the positivity constraints which reflect the hidden markov model structure are enforced by solving a low dimensional optimization problem

on the balanced truncation and coprime factors reduction of markovian jump linear systems abstract the paper deals with the balanced truncation and coprime factors reduction of markovian jump linear mjl systems which can have mode varying state input and output dimensions we develop machinery for balancing mean square stable mjl system realizations using generalized gramians and strict lyapunov inequalities and provide an improved a priori upper bound on the error induced in the balanced truncation process we also generalize the coprime factors reduction method and in doing so extend the applicability of the balanced truncation technique to the class of mean square stabilizable and detectable mjl systems we provide tools to establish mean square stabilizability and detectability of the considered mjl systems in addition a notion of right coprime factorization of mjl systems and methods to construct such factorizations are given the error measure in the coprime factors reduction approach while still norm based does not directly capture the mismatch between the nominal system and the reduced order model as is the case in the balanced truncation approach where mean square stable models are considered instead the error measure is given in terms of the distance between the coprime factors realizations and thus has an interpretation in terms of robust feedback stability the paper concludes with an illustrative example which demonstrates how to apply the coprime factors model reduction approach

reachability and observability reduction for linear switched systems with constrained switching we present an algorithm for reducing the number of continuous states of a discrete time linear switched system such that the reduced system has the same input output behavior as the original system for a subset of switching sequences the procedure can be interpreted as reachability and observability reduction for a linear switched system with constrained switching the proposed method is expected to be useful for abstraction based control synthesis methods for hybrid systems

cartan determinants and shapovalov forms we compute the determinant of the gram matrix of the shapovalov form on weight spaces of the basic representation of an affine kac moody algebra of ade type possibly twisted as a consequence we obtain explicit formulae for the determinants of the cartan matrices of p blocks of the symmetric group and its double cover and of the associated hecke algebras at roots of unity

on generalized blocks for alternating groups abstract in a recent paper kulshammer olsson robinson gave a d analogue for the nakayama conjecture for symmetric groups where d 2 is an arbitrary integer we prove that there is a natural d analogue of the nakayama conjecture for alternating groups whenever d is 2 or an arbitrary odd integer greater than 1 this generalizes an old result of kerber

isometries and extra special sylow groups of order p3 abstract a new type of isometry between the sets of irreducible characters of blocks is introduced and a conjecture is stated using the classifications of finite simple groups and saturated fusion systems over an extra special group p of order p 3 and exponent p we check that the conjecture is affirmatively answered for the principal blocks of finite groups with sylow p subgroup p

approximations of stochastic hybrid systems this paper develops a notion of approximation for a class of stochastic hybrid systems that includes as special cases both jump linear stochastic systems and linear stochastic hybrid automata our approximation framework is based on the recently developed notion of the so called stochastic simulation functions these lyapunov like functions can be used to rigorously quantify the distance or error between a system and its approximate abstraction for the class of jump linear stochastic systems and linear stochastic hybrid automata we show that the computation of stochastic simulation functions can be cast as a tractable linear matrix inequality problem this enables us to compute the modeling error incurred by abstracting some of the continuous dynamics or by neglecting the influence of stochastic noise or even the influence of stochastic discrete jumps

perfect isometries and murnaghan nakayama rules acknowledgement the authors wish to thank the referee for a very careful and precise reading of several earlier versions of this manuscript they are grateful for the number quality and helpfulness of comments and suggestions received

optimal filter approximations in conditionally gaussian pairwise markov switching models we consider a general triplet markov gaussian linear system bf x bf r bf y where bf x is an hidden continuous random sequence bf r is an hidden discrete markov chain bf y is an observed continuous random sequence when the triplet bf x bf r bf y is a classical conditionally gaussian linear state space model cglssm the mean square error optimal filter is not workable with a reasonable complexity and different approximate methods e g based on particle filters are used we propose two contributions the first one is to extend the cglssm to a new more general model called the conditionally gaussian pairwise markov switching model cgpmsm in which bf x is not necessarily markov given bf r the second contribution is to consider a particular case of cgpmsm in which bf r bf y is markov and in which an exact filter optimal in the sense of mean square error can be performed with linear time complexity some experiments show that the proposed method and the suited particle filter have comparable efficiency while the second one is much faster